From 95a289be7520200d8f09da38d9c83fe9f6cad436 Mon Sep 17 00:00:00 2001 From: Daniel Weschke Date: Sat, 21 Dec 2019 12:51:34 +0100 Subject: [PATCH] move geometry to geometry2d and add new geometry module (3d) --- docs/build/html/.buildinfo | 2 +- docs/build/html/_modules/data.html | 305 --- docs/build/html/_modules/date.html | 221 -- docs/build/html/_modules/function.html | 516 ----- docs/build/html/_modules/geometry_plot.html | 185 -- docs/build/html/_modules/index.html | 22 +- docs/build/html/_modules/mathematics.html | 1025 --------- docs/build/html/_modules/numerical/fit.html | 196 -- .../html/_modules/numerical/integration.html | 250 --- docs/build/html/_modules/numerical/ode.html | 540 ----- .../html/_modules/numerical/ode_model.html | 219 -- docs/build/html/_modules/pylib/data.html | 22 +- docs/build/html/_modules/pylib/date.html | 18 +- docs/build/html/_modules/pylib/drawblock.html | 25 +- docs/build/html/_modules/pylib/function.html | 18 +- docs/build/html/_modules/pylib/geometry.html | 686 ++---- .../{geometry.html => pylib/geometry2d.html} | 72 +- ...eometry_plot.html => geometry2d_plot.html} | 19 +- .../html/_modules/pylib/mathematics.html | 70 +- .../html/_modules/pylib/numerical/fit.html | 10 +- .../_modules/pylib/numerical/integration.html | 8 +- .../html/_modules/pylib/numerical/ode.html | 20 +- .../_modules/pylib/numerical/ode_model.html | 12 +- .../html/_modules/pylib/time_of_day.html | 26 +- docs/build/html/_modules/time_of_day.html | 253 --- docs/build/html/_sources/data.rst.txt | 7 - docs/build/html/_sources/date.rst.txt | 7 - docs/build/html/_sources/function.rst.txt | 7 - docs/build/html/_sources/geometry.rst.txt | 7 - docs/build/html/_sources/index.rst.txt | 2 + docs/build/html/_sources/mathematics.rst.txt | 7 - docs/build/html/_sources/numerical.rst.txt | 46 - docs/build/html/_sources/pylib.data.rst.txt | 7 + docs/build/html/_sources/pylib.date.rst.txt | 7 + .../html/_sources/pylib.drawblock.rst.txt | 7 + ...ry_plot.rst.txt => pylib.function.rst.txt} | 4 +- .../html/_sources/pylib.geometry.rst.txt} | 4 +- .../html/_sources/pylib.geometry2d.rst.txt | 7 + .../_sources/pylib.geometry2d_plot.rst.txt | 7 + .../html/_sources/pylib.mathematics.rst.txt | 7 + .../html/_sources/pylib.numerical.fit.rst.txt | 7 + .../pylib.numerical.integration.rst.txt | 7 + .../html/_sources/pylib.numerical.ode.rst.txt | 7 + .../pylib.numerical.ode_model.rst.txt | 7 + .../html/_sources/pylib.numerical.rst.txt | 49 +- docs/build/html/_sources/pylib.rst.txt | 86 +- .../html/_sources/pylib.time_of_day.rst.txt | 7 + docs/build/html/_sources/time_of_day.rst.txt | 7 - .../html/_static/documentation_options.js | 2 +- docs/build/html/genindex.html | 606 ++---- docs/build/html/index.html | 15 +- docs/build/html/modules.html | 40 +- docs/build/html/numerical.html | 589 ----- docs/build/html/objects.inv | Bin 1531 -> 1421 bytes docs/build/html/py-modindex.html | 127 +- .../build/html/{data.html => pylib.data.html} | 55 +- .../build/html/{date.html => pylib.date.html} | 47 +- docs/build/html/pylib.drawblock.html | 145 ++ .../{function.html => pylib.function.html} | 59 +- docs/build/html/pylib.geometry.html | 307 +++ .../{geometry.html => pylib.geometry2d.html} | 101 +- ...y_plot.html => pylib.geometry2d_plot.html} | 33 +- docs/build/html/pylib.html | 1909 +---------------- ...athematics.html => pylib.mathematics.html} | 151 +- docs/build/html/pylib.numerical.fit.html | 189 ++ docs/build/html/pylib.numerical.html | 507 +---- .../html/pylib.numerical.integration.html | 205 ++ docs/build/html/pylib.numerical.ode.html | 357 +++ .../build/html/pylib.numerical.ode_model.html | 194 ++ ...ime_of_day.html => pylib.time_of_day.html} | 61 +- docs/build/html/search.html | 6 +- docs/build/html/searchindex.js | 2 +- docs/source/conf.py | 2 +- docs/source/data.rst | 7 - docs/source/date.rst | 7 - docs/source/function.rst | 7 - docs/source/geometry.rst | 7 - docs/source/index.rst | 2 + docs/source/mathematics.rst | 7 - docs/source/numerical.rst | 46 - docs/source/pylib.data.rst | 7 + docs/source/pylib.date.rst | 7 + docs/source/pylib.drawblock.rst | 7 + docs/source/pylib.function.rst | 7 + docs/source/pylib.geometry.rst | 7 + docs/source/pylib.geometry2d.rst | 7 + docs/source/pylib.geometry2d_plot.rst | 7 + docs/source/pylib.mathematics.rst | 7 + docs/source/pylib.numerical.fit.rst | 7 + docs/source/pylib.numerical.integration.rst | 7 + docs/source/pylib.numerical.ode.rst | 7 + docs/source/pylib.numerical.ode_model.rst | 7 + docs/source/pylib.numerical.rst | 49 +- docs/source/pylib.rst | 86 +- docs/source/pylib.time_of_day.rst | 7 + docs/source/time_of_day.rst | 7 - pylib/drawblock.py | 17 +- pylib/geometry.py | 680 ++---- pylib/geometry2d.py | 511 +++++ .../{geometry_plot.py => geometry2d_plot.py} | 7 +- tests/test_geometry.py | 304 +-- tests/test_geometry2d.py | 259 +++ 102 files changed, 3771 insertions(+), 9082 deletions(-) delete mode 100644 docs/build/html/_modules/data.html delete mode 100644 docs/build/html/_modules/date.html delete mode 100644 docs/build/html/_modules/function.html delete mode 100644 docs/build/html/_modules/geometry_plot.html delete mode 100644 docs/build/html/_modules/mathematics.html delete mode 100644 docs/build/html/_modules/numerical/fit.html delete mode 100644 docs/build/html/_modules/numerical/integration.html delete mode 100644 docs/build/html/_modules/numerical/ode.html delete mode 100644 docs/build/html/_modules/numerical/ode_model.html rename docs/build/html/_modules/{geometry.html => pylib/geometry2d.html} (90%) rename docs/build/html/_modules/pylib/{geometry_plot.html => geometry2d_plot.html} (92%) delete mode 100644 docs/build/html/_modules/time_of_day.html delete mode 100644 docs/build/html/_sources/data.rst.txt delete mode 100644 docs/build/html/_sources/date.rst.txt delete mode 100644 docs/build/html/_sources/function.rst.txt delete mode 100644 docs/build/html/_sources/geometry.rst.txt delete mode 100644 docs/build/html/_sources/mathematics.rst.txt delete mode 100644 docs/build/html/_sources/numerical.rst.txt create mode 100644 docs/build/html/_sources/pylib.data.rst.txt create mode 100644 docs/build/html/_sources/pylib.date.rst.txt create mode 100644 docs/build/html/_sources/pylib.drawblock.rst.txt rename docs/build/html/_sources/{geometry_plot.rst.txt => pylib.function.rst.txt} (59%) rename docs/{source/geometry_plot.rst => build/html/_sources/pylib.geometry.rst.txt} (59%) create mode 100644 docs/build/html/_sources/pylib.geometry2d.rst.txt create mode 100644 docs/build/html/_sources/pylib.geometry2d_plot.rst.txt create mode 100644 docs/build/html/_sources/pylib.mathematics.rst.txt create mode 100644 docs/build/html/_sources/pylib.numerical.fit.rst.txt create mode 100644 docs/build/html/_sources/pylib.numerical.integration.rst.txt create mode 100644 docs/build/html/_sources/pylib.numerical.ode.rst.txt create mode 100644 docs/build/html/_sources/pylib.numerical.ode_model.rst.txt create mode 100644 docs/build/html/_sources/pylib.time_of_day.rst.txt delete mode 100644 docs/build/html/_sources/time_of_day.rst.txt delete mode 100644 docs/build/html/numerical.html rename docs/build/html/{data.html => pylib.data.html} (77%) rename docs/build/html/{date.html => pylib.date.html} (77%) create mode 100644 docs/build/html/pylib.drawblock.html rename docs/build/html/{function.html => pylib.function.html} (84%) create mode 100644 docs/build/html/pylib.geometry.html rename docs/build/html/{geometry.html => pylib.geometry2d.html} (76%) rename docs/build/html/{geometry_plot.html => pylib.geometry2d_plot.html} (63%) rename docs/build/html/{mathematics.html => pylib.mathematics.html} (75%) create mode 100644 docs/build/html/pylib.numerical.fit.html create mode 100644 docs/build/html/pylib.numerical.integration.html create mode 100644 docs/build/html/pylib.numerical.ode.html create mode 100644 docs/build/html/pylib.numerical.ode_model.html rename docs/build/html/{time_of_day.html => pylib.time_of_day.html} (70%) delete mode 100644 docs/source/data.rst delete mode 100644 docs/source/date.rst delete mode 100644 docs/source/function.rst delete mode 100644 docs/source/geometry.rst delete mode 100644 docs/source/mathematics.rst delete mode 100644 docs/source/numerical.rst create mode 100644 docs/source/pylib.data.rst create mode 100644 docs/source/pylib.date.rst create mode 100644 docs/source/pylib.drawblock.rst create mode 100644 docs/source/pylib.function.rst create mode 100644 docs/source/pylib.geometry.rst create mode 100644 docs/source/pylib.geometry2d.rst create mode 100644 docs/source/pylib.geometry2d_plot.rst create mode 100644 docs/source/pylib.mathematics.rst create mode 100644 docs/source/pylib.numerical.fit.rst create mode 100644 docs/source/pylib.numerical.integration.rst create mode 100644 docs/source/pylib.numerical.ode.rst create mode 100644 docs/source/pylib.numerical.ode_model.rst create mode 100644 docs/source/pylib.time_of_day.rst delete mode 100644 docs/source/time_of_day.rst create mode 100644 pylib/geometry2d.py rename pylib/{geometry_plot.py => geometry2d_plot.py} (96%) create mode 100644 tests/test_geometry2d.py diff --git a/docs/build/html/.buildinfo b/docs/build/html/.buildinfo index 2c89628..85ac018 100644 --- a/docs/build/html/.buildinfo +++ b/docs/build/html/.buildinfo @@ -1,4 +1,4 @@ # Sphinx build info version 1 # This file hashes the configuration used when building these files. When it is not found, a full rebuild will be done. -config: aaf67f6f94ce2e6ce3750a4b226f6461 +config: 2cbc17343901b023dfef01a2244f3120 tags: 645f666f9bcd5a90fca523b33c5a78b7 diff --git a/docs/build/html/_modules/data.html b/docs/build/html/_modules/data.html deleted file mode 100644 index c64a7a2..0000000 --- a/docs/build/html/_modules/data.html +++ /dev/null @@ -1,305 +0,0 @@ - - - - - - - data — pylib 2019.5.19 documentation - - - - - - - - - - - - - - - - - - - - -
-
-
- - -
- -

Source code for data

-#!/usr/bin/env python
-# -*- coding: utf-8 -*-
-"""Read and write data to or from file and manipulate data structures.
-
-:Date: 2019-10-11
-
-.. module:: data
-  :platform: *nix, Windows
-  :synopsis: Handle data files and structures.
-
-.. moduleauthor:: Daniel Weschke <daniel.weschke@directbox.de>
-"""
-import math
-import pickle
-
-
[docs]def read(file_name, x_column, y_column, default=None, verbose=False):
-  """Read ascii data file.
-
-  :param filename: file to read
-  :type filename: str
-  :param x_column: column index for the x data (first column is 0)
-  :type x_column: int
-  :param y_column: column index for the y data (first column is 0)
-  :type y_column: int
-  :param default: return object if data loading fails
-  :type default: object
-  :param verbose: verbose information (default = False)
-  :type verbose: bool
-
-  :returns: x and y
-  :rtype: tuple(list, list)
-  """
-  import re
-
-  x = default
-  y = default
-
-  if verbose:
-    print('check if data is available')
-  try:
-    file = open(file_name)
-    x = []
-    y = []
-    for row in file:
-      fields = re.split(r'\s+', row.strip())
-      x.append(float(fields[x_column]))
-      y.append(float(fields[y_column]))
-    file.close()
-  except IOError:
-    if verbose:
-      print('data file not found')
-  return x, y

-
-
[docs]def write(file_name, data):
-  """Write ascii file.
-
-  :param file_name: file to write
-  :type file_name: str
-  :param data: data to write
-  :type data: str
-  """
-  with open(file_name, 'w') as file:
-    file.write(data)

-
-
[docs]def load(file_name, default=None, verbose=False):
-  """Load stored program objects from binary file.
-
-  :param file_name: file to load
-  :type file_name: str
-  :param default: return object if data loading fails
-  :type default: object
-  :param verbose: verbose information (default = False)
-  :type verbose: bool
-
-  :returns: loaded data
-  :rtype: object
-  """
-  if verbose:
-    print('check if data is available')
-  try:
-    with open(file_name, 'rb') as input:
-      # one load for every dump is needed to load all the data
-      object_data = pickle.load(input)
-    if verbose:
-      print('found:')
-      print(object_data)
-  except IOError:
-    object_data = default
-    if verbose:
-      print('no saved datas found')
-  return object_data

-
-
[docs]def store(file_name, object_data):
-  """Store program objects to binary file.
-
-  :param file_name: file to store
-  :type file_name: str
-  :param object_data: data to store
-  :type object_data: object
-  """
-  with open(file_name, 'wb') as output:
-    # every dump needs a load
-    pickle.dump(object_data, output, pickle.HIGHEST_PROTOCOL)

-
-
[docs]def fold_list(lst, n):
-  """Convert one-dimensional kx1 array (list) to two-dimensional mxn
-  array. m = k / n
-
-  :param lst: list to convert
-  :type lst: list
-  :param n: length of the second dimenson
-  :type n: int
-
-  :returns: two-dimensional array (list of lists)
-  :rtype: list
-  """
-  k = len(lst)
-  if k % n == 0:
-    length = int(k/n)
-    return [lst[i*n:i*n+n] for i in range(length)]

-
-
[docs]def seq(start, stop=None, step=1):
-  r"""Create an arithmetic bounded sequence.
-
-  The sequence is one of the following;
-
-  - empty :math:`\{\}=\emptyset`, if start and stop are the same
-  - degenerate :math:`\{a\}`, if the sequence has only one element.
-  - left-close and right-open :math:`[a, b)`
-
-  :param start: start of the sequence, the lower bound. If only start
-    is given than it is interpreted as stop and start will be 0.
-  :type start: int or float
-  :param stop: stop of sequence, the upper bound.
-  :type stop: int or float
-  :param step: step size, the common difference (constant difference
-    between consecutive terms).
-  :type step: int or float
-  :returns: arithmetic bounded sequence
-  :rtype: list
-  """
-  # example of seq(4, 0, -0.4)
-  #   without round:
-  #     [4.0, 3.6, 3.2, 2.8, 2.4, 2.0, 1.5999999999999996,
-  #      1.1999999999999997, 0.7999999999999998, 0.3999999999999999]
-  #   with round:
-  #     [4.0, 3.6, 3.2, 2.8, 2.4, 2.0, 1.6, 1.2, 0.8, 0.4]
-  # example of seq(4, 0, -0.41)
-  #   without round:
-  #     [4.0, 3.59, 3.18, 2.77, 2.3600000000000003,
-  #      1.9500000000000002, 1.54, 1.1300000000000003,
-  #      0.7200000000000002, 0.31000000000000005]
-  #   with round:
-  #     [4.0, 3.59, 3.18, 2.77, 2.36, 1.95, 1.54, 1.13, 0.72, 0.31]
-  if stop is None:
-    return seq(0, start, step)
-
-  start_str = str(start)
-  start_exp = 0
-  if '.' in start_str:
-    start_exp = len(start_str.split('.')[1])
-
-  step_str = str(step)
-  step_exp = 0
-  if '.' in step_str:
-    step_exp = len(step_str.split('.')[1])
-
-  exponent = max(start_exp, step_exp)  # no stop because it is an open bound
-
-  n = int(math.ceil((stop - start)/float(step)))
-  lst = []
-  if n > 0:
-    lst = [round(start + step*i, exponent) for i in range(n)]
-  return lst

-
-
[docs]def unique_ending(ids, n=1):
-  """From id list get list with unique ending.
-
-  :param ids: ids
-  :type ids: list
-  :param n: minumum chars or ints
-  :type n: int
-
-  :returns: unique ending of ids
-  :rtype: list
-  """
-  if ids is not None:
-    x = [idi[-n:] for idi in ids]
-    if len(x) > len(set(x)):
-      return unique_ending(ids, n+1)
-    else:
-      return x

-
-
[docs]def get_id(ids, uide):
-  """Get full id from unique id ending.
-
-  :param ids: ids
-  :type ids: list
-  :param uide: unique id ending
-  :type uide: str
-
-  :returns: full id
-  :rtype: str or int
-  """
-  # take first element, because we know it is a unique ending
-  return [idi for idi in ids if idi.endswith(uide)][0]

-
- -
- -
-
- -
-
-
- ©2019, Daniel Weschke. - - | - Powered by Sphinx 2.2.1 - & Alabaster 0.7.12 - -
- - - - - - \ No newline at end of file diff --git a/docs/build/html/_modules/date.html b/docs/build/html/_modules/date.html deleted file mode 100644 index 9a842aa..0000000 --- a/docs/build/html/_modules/date.html +++ /dev/null @@ -1,221 +0,0 @@ - - - - - - - date — pylib 2019.5.19 documentation - - - - - - - - - - - - - - - - - - - - -
-
-
- - -
- -

Source code for date

-#!/usr/bin/env python
-# -*- coding: utf-8 -*-
-"""Calculate spacial dates.
-
-:Date: 2018-01-15
-
-.. module:: date
-  :platform: *nix, Windows
-  :synopsis: Special dates.
-   
-.. moduleauthor:: Daniel Weschke <daniel.weschke@directbox.de>
-"""
-
-
[docs]def gaußsche_osterformel(year):
-  """Gaußsche Osterformel.
-
-  :param year: the year to calculate the Easter Sunday
-  :type year: int
-  
-  :returns: the day of Easter Sunday as a day in march.
-  :rtype: int
-  
-  :ivar X: Das Jahr / year
-  :vartype X: int
-  :ivar K(X): Die Säkularzahl
-  :vartype K(X): int
-  :ivar M(X): Die säkulare Mondschaltung
-  :vartype M(X): int
-  :ivar S(K): Die säkulare Sonnenschaltung
-  :vartype S(K): int
-  :ivar A(X): Den Mondparameter
-  :vartype A(X): int
-  :ivar D(A,M): Den Keim für den ersten Vollmond im Frühling
-  :vartype D(A,M): int
-  :ivar R(D,A): Die kalendarische Korrekturgröße
-  :vartype R(D,A): int
-  :ivar OG(D,R): Die Ostergrenze
-  :vartype OG(D,R): int
-  :ivar SZ(X,S): Den ersten Sonntag im März
-  :vartype SZ(X,S): int
-  :ivar OE(OG,SZ): Die Entfernung des Ostersonntags von der Ostergrenze (Osterentfernung in Tagen)
-  :vartype OE(OG,SZ): int
-  :ivar OS(OG,OE): Das Datum des Ostersonntags als Märzdatum (32. März = 1. April usw.)
-  :vartype OS(OG,OE): int
-  
-  Algorithmus gilt für den Gregorianischen Kalender.
-
-  source: https://de.wikipedia.org/wiki/Gau%C3%9Fsche_Osterformel
-  """
-  x = year
-  k = x // 100
-  m = 15 + (3*k + 3) // 4 - (8*k + 13) // 25
-  s = 2 - (3*k + 3) // 4
-  a = x % 19
-  d = (19*a + m) % 30
-  r = (d + a // 11) // 29
-  og = 21 + d - r
-  sz = 7 - (x + x // 4 + s) % 7
-  oe = 7 - (og - sz) % 7
-  os = og + oe
-  return os

-
-
[docs]def easter_sunday(year):
-  """Easter Sunday.
-
-  :param year: the year to calculate the Easter Sunday
-  :type year: int
-  
-  :returns: the day of Easter Sunday
-  :rtype: datetime.date"""
-  import datetime
-  march = datetime.date(year, 3, 1)
-  day = march + datetime.timedelta(days=gaußsche_osterformel(year))
-  return day

-
-
[docs]def easter_friday(year):
-  """Easter Friday.
-
-  :param year: the year to calculate the Easter Friday
-  :type year: int
-  
-  :returns: the day of Easter Friday
-  :rtype: datetime.date"""
-  import datetime
-  day = easter_sunday(year) + datetime.timedelta(days=-2)
-  return day

-
-
[docs]def easter_monday(year):
-  """Easter Monday.
-
-  :param year: the year to calculate the Easter Monday
-  :type year: int
-  
-  :returns: the day of Easter Monday
-  :rtype: datetime.date"""
-  import datetime
-  day = easter_sunday(year) + datetime.timedelta(days=+1)
-  return day

-
-
[docs]def ascension_of_jesus(year):
-  """Ascension of Jesus.
-
-  :param year: the year to calculate the ascension of Jesus
-  :type year: int
-  
-  :returns: the day of ascension of Jesus
-  :rtype: datetime.date"""
-  import datetime
-  day = easter_sunday(year) + datetime.timedelta(days=+39)
-  return day

-
-
[docs]def pentecost(year):
-  """Pentecost.
-
-  :param year: the year to calculate the Pentecost
-  :type year: int
-  
-  :returns: the day of Pentecost
-  :rtype: datetime.date"""
-  import datetime
-  day = easter_sunday(year) + datetime.timedelta(days=+49)
-  return day

-
- -
- -
-
- -
-
-
- ©2019, Daniel Weschke. - - | - Powered by Sphinx 2.2.1 - & Alabaster 0.7.12 - -
- - - - - - \ No newline at end of file diff --git a/docs/build/html/_modules/function.html b/docs/build/html/_modules/function.html deleted file mode 100644 index d3cf54c..0000000 --- a/docs/build/html/_modules/function.html +++ /dev/null @@ -1,516 +0,0 @@ - - - - - - - function — pylib 2019.5.19 documentation - - - - - - - - - - - - - - - - - - - - -
-
-
- - -
- -

Source code for function

-#!/usr/bin/env python
-# -*- coding: utf-8 -*-
-"""Mathematical equations.
-
-:Date: 2019-11-15
-
-.. module:: function
-  :platform: *nix, Windows
-  :synopsis: Mathematical equations.
-
-.. moduleauthor:: Daniel Weschke <daniel.weschke@directbox.de>
-"""
-import math
-from pylib.data import seq
-from pylib.mathematics import lcm
-
-
[docs]def transformation(f, scale_vertical=1, scale_horizontal=1,
-                   shift_horizontal=0, shift_vertical=0):
-  r"""Transform functions.
-
-  :param f: function or list of functions
-  :type f: function or list
-  :param scale_vertical: "a" scale factor in vertical direction (default
-    = 1)
-  :type height: float
-  :param scale_horizontal: "b" scale factor in horizontal direction
-    (default = 1)
-  :type height: float
-  :param shift_horizontal: "c" shift factor in horizontal direction
-    (default = 0)
-  :type height: float
-  :param shift_vertical: "d" shift factor in vertical direction (default
-    = 0)
-  :type height: float
-
-  :returns: transformed function or list of transformed functions
-  :rtype: function or list
-
-  .. math::
-    y = a \, f(b\,(x-c)) + d
-  """
-  # shorter variables
-  a = scale_vertical
-  b = scale_horizontal
-  c = shift_horizontal
-  d = shift_vertical
-
-  # check if f is a function than put it in a list and return only
-  # the function, not the one element list
-  if callable(f):
-    return transformation(
-      [f], scale_vertical=a, scale_horizontal=b, shift_horizontal=c, shift_vertical=d)[0]
-
-  # otherwise assume list of functions
-  if not f:  # if f is empty. End of the recursive fucntion
-    return []
-  return [lambda x, t: a*f[0](b*(x-c), t)+d] +\
-    transformation(
-      f[1:], scale_vertical=a, scale_horizontal=b, shift_horizontal=c, shift_vertical=d)

-
-
[docs]def sine_wave(A=1, k=1, f=1, phi=0, D=0, degree=False):
-  r"""A sine wave or sinusoid is a mathematical curve that describes a
-  smooth periodic oscillation.
-
-  :param A: amplitude
-  :type A: float or int
-  :param k: (angular) wave number
-  :type k: float or int
-  :param f: ordinary frequency
-  :type f: float or int
-  :param phi: phase
-  :type phi: float or int
-  :param D: non-zero center amplitude
-  :type D: float or int
-  :param degree: boolean to switch between radians and degree. If
-    False phi is interpreted in radians and if True then phi is
-    interpreted in degrees.
-  :type degree: bool
-
-  :results: sine wave function of spatial variable x and optional
-    time t
-
-  In general, the function is:
-
-  .. math::
-    y(x,t) = A\sin(kx + 2\pi f t + \varphi) + D \\
-    y(x,t) = A\sin(kx + \omega t + \varphi) + D
-
-  where:
-
-    * A, amplitude, the peak deviation of the function from zero.
-    * f, ordinary frequency, the number of oscillations (cycles) that
-      occur each second of time.
-    * ω = 2πf, angular frequency, the rate of change of the function
-      argument in units of radians per second. If ω < 0 the wave is
-      moving to the right, if ω > 0 the wave is moving to the left.
-    * φ, phase, specifies (in radians) where in its cycle the
-      oscillation is at t = 0.
-    * x, spatial variable that represents the position on the
-      dimension on which the wave propagates.
-    * k, characteristic parameter called wave number (or angular wave
-      number), which represents the proportionality between the
-      angular frequency ω and the linear speed (speed of propagation)
-      ν.
-    * D, non-zero center amplitude.
-
-  The wavenumber is related to the angular frequency by:
-
-  .. math::
-    k={\omega \over v}={2\pi f \over v}={2\pi \over \lambda }
-
-  where λ (lambda) is the wavelength, f is the frequency, and v is the
-  linear speed.
-
-  .. seealso::
-    :meth:`cosine_wave`
-  """
-  if degree:
-    phi = math.radians(phi)
-  return lambda x, t=0: A*math.sin(k*x + 2*math.pi*f*t + phi) + D

-
-
[docs]def cosine_wave(A=1, k=1, f=1, phi=0, D=0, degree=False):
-  r"""A cosine wave is said to be sinusoidal, because,
-  :math:`\cos(x) = \sin(x + \pi/2)`, which is also a sine wave with a
-  phase-shift of π/2 radians. Because of this head start, it is often
-  said that the cosine function leads the sine function or the sine
-  lags the cosine.
-
-  :param A: amplitude
-  :type A: float or int
-  :param k: (angular) wave number
-  :type k: float or int
-  :param f: ordinary frequency
-  :type f: float or int
-  :param phi: phase
-  :type phi: float or int
-  :param D: non-zero center amplitude
-  :type D: float or int
-  :param degree: boolean to switch between radians and degree. If
-    False phi is interpreted in radians and if True then phi is
-    interpreted in degrees.
-  :type degree: bool
-
-  :results: sine wave function of spatial variable x and optional
-    time t
-
-  .. seealso::
-    :meth:`sine_wave`
-  """
-  if degree:
-    phi = phi + 90
-  else:
-    phi = phi + math.pi/2
-  return sine_wave(A=A, k=k, f=f, phi=phi, D=D, degree=degree)

-
-
-#
-# Parametric equations
-# roulette
-#
-
-
[docs]def hypotrochoid(R, r, d):
-  r"""Hypotrochoid
-
-  A point is attached with a distance d from the center of a circle
-  of radius r. The circle is rolling around the inside of a fixed
-  circle of radius R.
-
-  :param R: radius of the fixed exterior circle
-  :type R: float
-  :param r: radius of the rolling circle inside of the fixed circle
-  :typre r: float
-  :param d: distance from the center of the interior circle
-  :type d: float
-
-  :results: functions for x of theta and y of theta
-  :rtype: tuple
-
-  .. math::
-    x(\theta) = (R - r)\cos\theta + d\cos\left(\frac{R-r}{r}\theta\right) \\
-    y(\theta) = (R - r)\sin\theta - d\sin\left(\frac{R-r}{r}\theta\right) \\
-    \theta = \left[0, 2\pi\frac{\mathrm{lcm}(r, R)}{R}\right]
-
-  ::
-
-           *  *  *
-        * R         *
-      *               *
-     *           *  *  *
-    *         * r      **
-    *        *     .... *
-    *        *      d   *
-    *         *        **
-     *           *  *  *
-      *               *
-        *          *
-           *  *  *
-
-  >>> x, y = hyotrochoid(20, 6, 6)[:1]
-  >>> x, y, theta_end = hyotrochoid(20, 6, 6)
-
-  .. seealso::
-    :meth:`mathematics.lcm`
-  """
-  x = lambda theta: (R-r)*math.cos(theta) + d*math.cos((R-r)/r * theta)
-  y = lambda theta: (R-r)*math.sin(theta) - d*math.sin((R-r)/r * theta)
-  theta_end = 2*math.pi*lcm(r, R)/R
-  return x, y, theta_end

-
-
[docs]def epitrochoid(R, r, d):
-  r"""Epitrochoid
-
-  A point is attached with a distance d from the center of a circle
-  of radius r. The circle is rolling around the outside of a fixed
-  circle of radius R.
-
-  :param R: radius of the fixed interior circle
-  :type R: float
-  :param r: radius of the rolling circle outside of the fixed circle
-  :typre r: float
-  :param d: distance from the center of the exterior circle
-  :type d: float
-
-  :results: functions for x of theta and y of theta
-  :rtype: tuple
-
-  .. math::
-    x(\theta) = (R + r)\cos\theta - d\cos\left(\frac{R+r}{r}\theta\right) \\
-    y(\theta) = (R + r)\sin\theta - d\sin\left(\frac{R+r}{r}\theta\right) \\
-    \theta = \left[0, 2\pi\right]
-
-  ::
-
-           *  *  *
-        * R         *
-      *               *
-     *                 *     *  *
-    *                   * * r      *
-    *                   ** ....     *
-    *                   **   d      *
-    *                   * *        *
-     *                 *     *  *
-      *               *
-        *          *
-           *  *  *
-
-  >>> x, y = epitrochoid(3, 1, 0.5)[:1]
-  >>> x, y, theta_end = epitrochoid(3, 1, 0.5)
-  """
-  x = lambda theta: (R+r)*math.cos(theta) - d*math.cos((R+r)/r * theta)
-  y = lambda theta: (R+r)*math.sin(theta) - d*math.sin((R+r)/r * theta)
-  theta_end = 2*math.pi
-  return x, y, theta_end

-
-
[docs]def to_str(f, x=None, x_0=0, x_1=1, t=None, h=10, w=80, density=1, char_set="line"):
-  """Represent functions as string frame with a specific character set.
-  which are normed to the range of [0, 1] to
-
-  :param f: function or list of functions normed to the range of [0, 1]
-  :type f: function or list
-  :param h: number of chars in vertical direction
-  :type h: int
-  :param w: number of chars in horizontal direction
-  :type w: int
-
-  :param char_set: either "braille" or "block". "braille" uses Unicode
-    Characters in the Braille Patterns Block (fisrt index U+2800, last
-    index U+28FF [CUDB]_) and the "block" uses part of the Unicode
-    Characters in the Block Elements Block (fisrt index U+2580, last
-    index U+259F [CUDB]_). Alias for braille is line and alias for
-    block is histogram
-  :type char_set: str
-
-  Usage:
-    * case dependent arguments
-
-      * 1 quasi line plot (braille characters)
-        f = lambda function (lambda x, t=0: ...)
-        x_1 = max x value
-        x_0 = min x value
-        t = time (animation)
-      * 2 histogram (block characters)
-        f = lambda function (lambda x, t=0: ...)
-        x_1 = max x value
-        x_0 = min x value
-        t = time (animation)
-        chart="histogram"
-
-    * general arguments
-      w = window width in number of chars
-      density = number of data point per pixel (for line chart higher density makes thicker lines)
-      chart = either "line" or "histogram"
-
-
-  .. rubric:: Braille Patterns Block
-
-  * Dots or pixels per character in vertical direction: 6
-  * Dots or pixels per character in horizontal direction: 2
-
-  First dot (bottom left) is the zero (0) position of the
-  function. So, a function value of zero does not mean to have zero
-  dots but one.
-  Example of 3 columns and 3 rows (upside down view of 'normal' braille/drawille position)
-
-  ::
-
-       |            ^ y axis
-       |            |
-       |   ,_____,,_____,,_____,
-     7 +   | *  *|| *  *|| *  *|
-     6 +   | *  *|| *  *|| *  *|
-     5 +   | *  *|| *  *|| *  *|
-     4 +   | *  *|| *  *|| *  *|
-       |   ;=====;;=====;;=====;
-     3 +   | *  *|| *  *|| *  *|
-     2 +   | *  *|| *  *|| *  *|
-     1 +   | *  *|| *  *|| *  *|
-     0 + - | *  *|| *  *|| *  *| ---> x axis
-       |   ;=====;;=====;;=====;
-    -1 +   | *  *|| *  *|| *  *|
-    -2 +   | *  *|| *  *|| *  *|
-    -3 +   | *  *|| *  *|| *  *|
-    -4 +   | *  *|| *  *|| *  *|
-       |   `````````````````````
-       |            |
-       `-----+--+---+--+---+--+-------------
-            -2 -1   0  1   2  3
-
-
-  .. rubric:: Block Elements Block
-
-  * Dots or pixels per character in vertical direction: 8
-  * Dots or pixels per character in horizontal direction: 1
-
-  Example of 3 columns and 3 rows
-
-  ::
-
-       |             ^ y axis
-       |             |
-       |   ,_____,,_____,,_____,
-    15 +   | --- || --- || --- |
-    14 +   | --- || --- || --- |
-    13 +   | --- || --- || --- |
-    12 +   | --- || --- || --- |
-    11 +   | --- || --- || --- |
-    10 +   | --- || --- || --- |
-     9 +   | --- || --- || --- |
-     8 +   | --- || --- || --- |
-       |   ;=====;;=====;;=====;
-     7 +   | --- || --- || --- |
-     6 +   | --- || --- || --- |
-     5 +   | --- || --- || --- |
-     4 +   | --- || --- || --- |
-     3 +   | --- || --- || --- |
-     2 +   | --- || --- || --- |
-     1 +   | --- || --- || --- |
-     0 + - | --- || --- || --- | ---> x axis
-       |   ;=====;;=====;;=====;
-    -1 +   | --- || --- || --- |
-    -2 +   | --- || --- || --- |
-    -3 +   | --- || --- || --- |
-    -4 +   | --- || --- || --- |
-    -5 +   | --- || --- || --- |
-    -6 +   | --- || --- || --- |
-    -7 +   | --- || --- || --- |
-    -8 +   | --- || --- || --- |
-       |   `````````````````````
-       |             |
-       `------+------+------+---------------
-             -1      0      1
-
-
-  .. rubric:: References
-
-  .. [CUDB] `Unicode Database - Blocks <ftp://ftp.unicode.org/Public/UNIDATA/Blocks.txt>`_
-
-
-  .. seealso::
-    :meth:`pylib.function.transformation`
-  """
-  # scale function to used chars and dots/pixel in y direction (4 for braille characters): from [0, 1] to [0, chars*4-1]
-  # negate the function because the y axis is pointing downwards: from [0, 1] to [-1, 0] or from [0, chars*4-1] to [-(chars*4-1), 0]
-  # and becasue of the negation shift the function by one dot/pixel, otherwise the function would need another char, because the 0 is on an another char on the "positive side".
-  # devide the time by the number of dots/pixel in x direction (2 for braille characters)
-  window_factor = w/(x_1-x_0)
-  frame = ""
-  if char_set in ["line", "braille"]:
-    import drawille
-
-    pixels_horizontal = 2
-    pixels_vertical = 4
-    a = -(h*pixels_vertical-1)
-    b = 1/pixels_horizontal
-    f = transformation(f, a, b, 0, -1)
-
-    canvas = drawille.Canvas()
-    # divide step width of the sequence by 2 (double density, 2 dots/pixel per char)
-    # multiplicate x by 2 (2 dots/pixel per char)
-    for x_i in seq(x_0*window_factor*pixels_horizontal, x_1*window_factor*pixels_horizontal, 1/density):
-      canvas.set(x_i, f(x_i, t))
-    #frame = canvas.frame(min_x=a*pixel_per_char, min_y=1, max_x=b*pixel_per_char, max_y=21)
-    frame = canvas.frame()
-  elif char_set in ["histogram", "block"]:
-    import drawblock
-
-    pixels_horizontal = 1
-    pixels_vertical = 8
-    a = h*pixels_vertical-1
-    f = transformation(f, a, 1, 0, 0)
-
-    density = min(density, 1)  # density max 1!
-    x = seq(x_0*window_factor, x_1*window_factor, 1/pixels_horizontal/density)
-    f = [f(xi, t) for xi in x]
-    frame = drawblock.histogram(f, x)
-
-  return frame

-
- -
- -
-
- -
-
-
- ©2019, Daniel Weschke. - - | - Powered by Sphinx 2.2.1 - & Alabaster 0.7.12 - -
- - - - - - \ No newline at end of file diff --git a/docs/build/html/_modules/geometry_plot.html b/docs/build/html/_modules/geometry_plot.html deleted file mode 100644 index 3d3ae3d..0000000 --- a/docs/build/html/_modules/geometry_plot.html +++ /dev/null @@ -1,185 +0,0 @@ - - - - - - - geometry_plot — pylib 2019.5.19 documentation - - - - - - - - - - - - - - - - - - - - -
-
-
- - -
- -

Source code for geometry_plot

-#!/usr/bin/env python
-# -*- coding: utf-8 -*-
-"""2D geometry plotter using  matplotlib (pylab).
-
-:Date: 2019-08-20
-
-.. module:: geometry_plot
-  :platform: *nix, Windows
-  :synopsis: Geometry plotting.
-
-.. moduleauthor:: Daniel Weschke <daniel.weschke@directbox.de>
-"""
-import math
-import pylab
-from geometry import (
-  distance, angle, line, interpolate_hermite, rotate_xy, translate_xy
-)
-
-pylab.style.use('dark_background')
-pylab.rcParams['grid.color'] = 'gray'
-
-# The kwargs are Line2D properties:
-# https://matplotlib.org/3.1.1/_modules/matplotlib/lines.html#Line2D
-# https://matplotlib.org/devdocs/_modules/matplotlib/lines.html#Line2D
-pylab_Line2D_properties = [
-  'linewidth', 'linestyle', 'color',
-  'marker', 'markersize', 'markeredgewidth', 'markeredgecolor',
-  'markerfacecolor', 'markerfacecoloralt',
-  'fillstyle', 'antialiased',
-  'dash_capstyle', 'solid_capstyle',
-  'dash_joinstyle', 'solid_joinstyle',
-  'pickradius', 'drawstyle', 'markevery',
-]
-
-
-
[docs]def plot_lines(lns, **kwargs):
-  if 'color' not in kwargs:
-    # Colors from "tab10" colormap
-    cmap = pylab.get_cmap("tab10")
-    kwargs['color'] = cmap(0)
-
-    # Colors from color cycle
-    cycle = pylab.rcParams['axes.prop_cycle'].by_key()['color']
-    kwargs['color'] = cycle[0]
-
-    # Colors from CN notation (same as the first 10 colors from the color cycle)
-    kwargs['color'] = "C0"
-
-  for ln in lns:
-    x, y = line(*ln)
-    pylab.plot(x, y, **kwargs)

-
-
-
[docs]def plot_cubic_lines(lns, **kwargs):
-  if 'color' not in kwargs:
-    # Colors from "tab10" colormap
-    cmap = pylab.get_cmap("tab10")
-    kwargs['color'] = cmap(0)
-
-    # Colors from color cycle
-    cycle = pylab.rcParams['axes.prop_cycle'].by_key()['color']
-    kwargs['color'] = cycle[0]
-
-    # Colors from CN notation (same as the first 10 colors from the color cycle)
-    kwargs['color'] = "C0"
-
-  factor = 1
-  if 'factor' in kwargs:
-    factor = kwargs['factor']
-
-  samples = 10
-  if 'samples' in kwargs:
-    samples = kwargs['samples']
-
-  kwargs = {k: v for k, v in kwargs.items() if k in pylab_Line2D_properties}
-
-  for ln in lns:
-    l, r, d = ln
-    L = distance(l, r)
-    ang = angle(l, r)
-    x, y = interpolate_hermite(d[1], d[2], d[4], d[5], d[0], d[3],
-                               scale_x=L, scale_y=factor, samples=samples)
-    x, y = rotate_xy((0, 0), ang, x, y)
-    x, y = translate_xy(l, x, y)
-
-    pylab.plot(x, y, **kwargs)

-
- -
- -
-
- -
-
-
- ©2019, Daniel Weschke. - - | - Powered by Sphinx 2.2.1 - & Alabaster 0.7.12 - -
- - - - - - \ No newline at end of file diff --git a/docs/build/html/_modules/index.html b/docs/build/html/_modules/index.html index 3509ca8..34e9952 100644 --- a/docs/build/html/_modules/index.html +++ b/docs/build/html/_modules/index.html @@ -4,7 +4,7 @@ - Overview: module code — pylib 2019.5.19 documentation + Overview: module code — pylib 2019.12.21 documentation @@ -33,29 +33,19 @@

All modules for which code is available

-
@@ -74,6 +64,10 @@

Navigation

+

Contents:

+

Related Topics

diff --git a/docs/build/html/_modules/mathematics.html b/docs/build/html/_modules/mathematics.html deleted file mode 100644 index 8c007d8..0000000 --- a/docs/build/html/_modules/mathematics.html +++ /dev/null @@ -1,1025 +0,0 @@ - - - - - - - mathematics — pylib 2019.5.19 documentation - - - - - - - - - - - - - - - - - - - - -
-
-
- - -
- -

Source code for mathematics

-#!/usr/bin/env python
-# -*- coding: utf-8 -*-
-"""Mathematical functions and objects.
-
-:Date: 2019-12-12
-
-.. module:: mathematics
-  :platform: *nix, Windows
-  :synopsis: Mathematical functions and objects.
-
-.. moduleauthor:: Daniel Weschke <daniel.weschke@directbox.de>
-"""
-import math
-from math import gcd
-
-
[docs]def lcm(a, b):
-  """Compute the lowest common multiple of a and b"""
-  return a/gcd(a, b)*b

-
-
-
[docs]class vector(list):
-  """Use/create vector like lists
-
-  * size  -> len(a)
-  * abs   -> abs(a)
-  * dot   -> a * b
-  * outer -> a @ b
-
-  use super constructor
-
-  use super __iter__
-
-  use super __setitem__
-
-  >>> v = vector([1,2,3,4,5])
-  >>> v[3:5] = [1,2]
-  >>> print(v)
-  [1, 2, 3, 1, 2]
-  >>> isinstance(v, vector)
-  True
-
-  use super __lt__(a, b)
-
-  use super __le__(a, b)
-
-  use super __eq__(a, b)
-
-  >>> v = vector([1,2,3,1,2])
-  >>> v2 = vector([1,2,3,1,2])
-  >>> v == v2
-  True
-
-  use super __ne__(a, b)
-
-  use super __ge__(a, b)
-
-  use super __gt__(a, b)
-
-  use super __contains__
-
-  >>> 2 in vector([1,2,3])
-  True
-
-  __isub__
-
-  >>> v = vector([1,2,3])
-  >>> v -= vector([3,3,3])
-  >>> print(v)
-  [-2, -1, 0]
-
-  __imul__
-
-  >>> v = vector([1,2,3])
-  >>> v *= vector([3,3,3])
-  >>> print(v)
-  18
-
-  __imatmul__
-
-  >>> m = vector([1,2,3])
-  >>> m *= vector([3,3,3])
-  >>> print(v)
-  [[3, 3, 3], [6, 6, 6], [9, 9, 9]]
-  """
-
-  def __getitem__(self, index):
-    """
-    For index return value, for range return new vector object.
-
-    :Example:
-
-    >>> v = vector([1, 2, 3, 4, 5])
-    >>> v[1:3]
-    [2, 3]
-    >>> v = vector([1, 2, 3, 4, 5])
-    >>> v[3]
-    4
-    """
-    # use the list.__getslice__ method and convert result to vector
-    item = super().__getitem__(index)
-    return vector(item) if isinstance(item, list) else item
-
-  def __pos__(self):
-    """+ a (new object)
-    """
-    return vector([*self])
-
-  def __neg__(self):
-    """- a (new object)
-    """
-    return vector([-i for i in self])
-
-  def __add__(self, other):
-    """a + b (new object)
-
-    :Example:
-
-    >>> v = vector([1,2,3]) + vector([1,2,3])
-    >>> print(v)
-    [2, 4, 6]
-    """
-    return vector([i+j for i, j in zip(self, other)])
-
-  def __iadd__(self, other):
-    """a += b (new object)
-
-    :Example:
-
-    >>> v = vector([1,2,3])
-    >>> v += vector([3,3,3])
-    >>> print(v)
-    [4, 5, 6]
-    """
-    return vector([i+j for i, j in zip(self, other)])
-
-  def __sub__(self, other):
-    """a - b (new object)
-
-    :Example:
-
-    >>> v = vector([1,2,3]) - vector([1,2,3])
-    >>> print(v)
-    [0, 0, 0]
-    """
-    return vector([i-j for i, j in zip(self, other)])
-
-  def __mul__(self, other):
-    r"""Scalar multiplication, dot product (inner product) or
-    vector-matrix multiplication.
-
-    a * b (new object)
-
-    :type other: scalar, vector (or 1d list), matrix (or 2d list)
-
-    .. math::
-      \mathbf{c} &= \mathbf{a} \cdot b \\
-      \mathbf{c} &= \mathbf{a} \cdot \mathbf{b} \\
-      \mathbf{c} &= \mathbf{a} \cdot \mathbf{B}
-
-    .. note::
-      No size checking will be conducted, therefore no exceptions for
-      wrong usage (result will be nonsense).
-
-    :Example:
-
-    >>> v = vector([1,2,3,4,5])*3
-    >>> print(v)
-    [3, 6, 9, 12, 15]
-    >>> v = vector([1,2,3,4,5])*3.
-    >>> print(v)
-    [3.0, 6.0, 9.0, 12.0, 15.0]
-    >>> s = vector([1,2,3])*vector([1,2,3])
-    >>> print(s)
-    14
-
-    .. seealso::
-      :meth:`__rmul__`
-    """
-    try:       # vector * vector
-      return sum([i*j for i, j in zip(self, other)])
-    except:
-      try:     # vector * matrix
-        return vector([sum(c*d for c, d in zip(self, b_col)) for b_col in zip(*other)])
-      except:  # vector * scalar
-        return vector([i*other for i in self])
-
-  def __rmul__(self, other):
-    r"""Scalar multiplication, dot product (inner product) or
-    matrix-vector multiplication.
-
-    a * b (new object)
-
-    :type other: scalar (or 1d list and 2d list)
-
-    .. math::
-      \mathbf{c} &= a \cdot \mathbf{b} \\
-      \mathbf{c} &= \mathbf{a} \cdot \mathbf{b} \\
-      \mathbf{c} &= \mathbf{A} \cdot \mathbf{b}
-
-    .. note::
-      No size checking will be conducted, therefore no exceptions for
-      wrong usage (result will be nonsense).
-
-    :Example:
-
-    >>> v = 3*vector([1,2,3,4,5])
-    >>> print(v)
-    [3, 6, 9, 12, 15]
-    >>> v = 3.*vector([1,2,3,4,5])
-    >>> print(v)
-    [3.0, 6.0, 9.0, 12.0, 15.0]
-
-    .. seealso::
-      :meth:`__mul__`
-      :meth:`matrix.__mul__` for matrix * vector
-    """
-    try:     # 2d list * vector (matrix * vector see matrix.__mul__)
-      return vector([sum(c*d for c, d in zip(a_row, self)) for a_row in other])
-    except:  # scalar * vector
-      return self*other
-
-  def __matmul__(self, other):
-    r"""Outer product a @ b (new object)
-
-    .. math::
-      \vec{a} \otimes \vec{b}
-      =
-      \begin{pmatrix}
-        a_{1}\\
-        a_{2}\\
-        a_{3}
-      \end{pmatrix}
-      \otimes
-      \begin{pmatrix}
-        b_{1}\\
-        b_{2}\\
-        b_{3}
-      \end{pmatrix}
-      =
-      \begin{pmatrix}
-        a_{1}b_{1}&a_{1}b_{2}&a_{1}b_{3}\\
-        a_{2}b_{1}&a_{2}b_{2}&a_{2}b_{3}\\
-        a_{3}b_{1}&a_{3}b_{2}&a_{3}b_{3}
-      \end{pmatrix}
-
-
-    :Example:
-
-    >>> m = vector([1,2,3]) @ vector([1,2,4])
-    >>> print(m)
-    [[1, 2, 4], [2, 4, 8], [3, 6, 12]]
-    """
-    try:     # vector * vector
-      return matrix([[i*j for j in other] for i in self])
-    except:  # vector * number
-      return self*other
-
-  def __abs__(self):
-    r"""Magnitude / norm of a vector
-
-    .. math::
-      b = \sqrt{\sum a_i^2}
-
-    :Example:
-
-    >>> norm([3, 4])
-    5
-    >>> norm(vector([3, 4]))
-    5
-    """
-    return math.sqrt(self * self)
-
-  def __str__(self):
-    return str([*self])
-
-  def __repr__(self):
-    return "vector(" + str(self) + ")"
-
-
[docs]  @staticmethod
-  def full(length, fill_value):
-    """Returns a vector of length m or matrix of size m rows, n
-    columns filled with v.
-
-    :param length: length of the vector, e. g. 3
-    :type length: int
-    :param fill_value: Fill value
-    :Type fill_value: scalar
-
-    :Example:
-
-    >>> v = vector.full(3, 7)
-    >>> print(v)
-    [7, 7, 7]
-    """
-    return vector([fill_value for i in range(length)])

-
-
[docs]  @staticmethod
-  def zeros(length):
-    """Returns a zero vector of length m or matrix of size rows, n
-    columns filled with zeros.
-
-    :param length: length of the vector, e. g. 3
-    :type length: int
-
-    :Example:
-
-    >>> v = zeros(3)
-    >>> print(v)
-    [0.0, 0.0, 0.0]
-    """
-    return vector.full(length, 0.)

-
-
[docs]  @staticmethod
-  def ones(length):
-    """Returns a vector of length m or matrix of size rows, n
-    columns filled with ones.
-
-    :param length: lhape of the vector, e. g. 3
-    :type length: int
-
-    :Example:
-
-    >>> v = ones(3)
-    >>> print(v)
-    [1.0, 1.0, 1.0]
-    """
-    return vector.full(length, 1.)

-
-
[docs]  @staticmethod
-  def random(shape, lmin=0.0, lmax=1.0):
-    """Returns a random vector of length n or matrix of size m rows, n
-    columns filled with random numbers.
-
-    :Example:
-
-    >>> v = random(3)
-    >>> print(v)
-    [0.9172905912930438, 0.8908124278322492, 0.5256002790725927]
-    >>> v = random(3, 1, 2)
-    >>> print(v)
-    [1.2563665665080803, 1.9270454509964547, 1.2381672401270487]
-    """
-    import random
-    dl = lmax-lmin
-    return vector([dl*random.random()+lmin for i in range(shape)])

-
-
[docs]  @staticmethod
-  def normalize(a):
-    r"""Normalize a vector (i. e. the vector has a length of 1)
-
-    :type a: vector
-
-    .. math::
-      \vec{e}_a = \frac{\vec{a}}{|\vec{a}|}
-
-    .. seealso::
-      :meth:`norm` for a norm (magnitude) of a vector
-    """
-    a_mag = abs(a)
-    return vector([i/a_mag for i in a])

-
-
[docs]  @staticmethod
-  def ang(a, b):
-    return math.acos(a * b / (abs(a) * abs(b)))

-
-
[docs]  @staticmethod
-  def conjugate(a):
-    """
-    New vector object
-
-    :type a: list
-    """
-    return vector([i.conjugate() for i in a])

-
-
[docs]  @staticmethod
-  def re(a):
-    """Return the real parts of a complex vector
-
-    :type a: list
-    """
-    return vector([i.real for i in a])

-
-
[docs]  @staticmethod
-  def im(a):
-    """Return the imaginary parts of a complex vector
-
-    :type a: list
-    """
-    return vector([i.imag for i in a])

-
-
[docs]  @staticmethod
-  def abs(a):
-    """Return modulus parts of a complex vector
-
-    :type a: list
-    """
-    return vector([abs(i) for i in a])

-
-
[docs]  @staticmethod
-  def arg(a):
-    """Return phase parts of a complex vector
-
-    :type a: list
-    """
-    return vector([math.atan2(i.imag, i.real) for i in a])

-
-
[docs]  @staticmethod
-  def cross(a, b):
-    r"""Cross product
-
-    :type a: list
-    :type b: list
-
-    c is orthogonal to both a and b.
-    The direction of c can be found with the right-hand rule.
-
-    .. math::
-      \vec{c} = \vec{a} \times \vec{b}
-    """
-    return [a[1]*b[2] - a[2]*b[1],
-            a[2]*b[0] - a[0]*b[2],
-            a[0]*b[1] - a[1]*b[0]]

-
-
[docs]  def xyz(self):
-    return self[:3]

-
-
[docs]  def rotate_x(self, theta):
-    r"""Rotation about the x dirction.
-
-    .. math::
-      \begin{bmatrix}x' \\ y' \\ z' \\ h'\end{bmatrix} =
-      \begin{bmatrix}
-        1 & 0 & 0 & 0 \\
-        0 & \cos \theta & -\sin \theta & 0 \\
-        0 & \sin \theta & \cos \theta & 0 \\
-        0 & 0 & 0 & 1
-      \end{bmatrix}
-      \begin{bmatrix}x \\ y \\ z \\ h\end{bmatrix}
-    """
-    self[:] = matrix.rx(theta) * self
-    return self

-
-
[docs]  def rotate_y(self, theta):
-    r"""Rotation about the y dirction.
-
-    .. math::
-      \begin{bmatrix}x' \\ y' \\ z' \\ h'\end{bmatrix} =
-      \begin{bmatrix}
-        \cos \theta & 0 & \sin \theta & 0 \\
-        0 & 1 & 0 & 0 \\
-        -\sin \theta & 0 & \cos \theta & 0 \\
-        0 & 0 & 0 & 1
-      \end{bmatrix}
-      \begin{bmatrix}x \\ y \\ z \\ h\end{bmatrix}
-    """
-    self[:] = matrix.ry(theta) * self
-    return self

-
-
[docs]  def rotate_z(self, theta):
-    r"""Rotation about the z dirction.
-
-    .. math::
-      \begin{bmatrix}x' \\ y' \\ z' \\ h'\end{bmatrix} =
-      \begin{bmatrix}
-        \cos \theta & -\sin \theta & 0 & 0 \\
-        \sin \theta & \cos \theta & 0 & 0 \\
-        0 & 0 & 1 & 0 \\
-        0 & 0 & 0 & 1
-      \end{bmatrix}
-      \begin{bmatrix}x \\ y \\ z \\ h\end{bmatrix}
-    """
-    self[:] = matrix.rz(theta) * self
-    return self

-
-
[docs]  def translate(self, tx, ty, tz):
-    r"""Translation
-
-    .. math::
-      \begin{bmatrix}x' \\ y' \\ z' \\ h'\end{bmatrix} =
-      \begin{bmatrix}
-        1 & 0 & 0 & t_x \\
-        0 & 1 & 0 & t_y \\
-        0 & 0 & 1 & t_z \\
-        0 & 0 & 0 & 1
-      \end{bmatrix}
-      \begin{bmatrix}x \\ y \\ z \\ h\end{bmatrix}
-    """
-    self[:] = matrix.t(tx, ty, tz) * self
-    return self

-
-
[docs]  def scale(self, sx, sy=None, sz=None):
-    r"""Scaling
-
-    uniform scaling if sx=sy=sz=s.
-    Note that scaling happens around the origin, so objects not
-    centered at the origin will have their centers move. To avoid this,
-    either scale the object when it's located at the origin, or
-    perform a translation afterwards to move the object back to where
-    it should be.
-
-    .. math::
-      \begin{bmatrix}x' \\ y' \\ z' \\ h'\end{bmatrix} =
-      \begin{bmatrix}
-        s_x & 0 & 0 & 0 \\
-        0 & s_y & 0 & 0 \\
-        0 & 0 & s_z & 0 \\
-        0 & 0 & 0 & 1
-      \end{bmatrix}
-      \begin{bmatrix}x \\ y \\ z \\ h\end{bmatrix}
-    """
-    if not sy:
-      sy = sx
-      sz = sx
-    self[:] = matrix.s(sx, sy, sz) * self
-    return self

-
-
[docs]  def ch_cs(self, cs):
-    r"""Transform this vector from its defined coordinate system to a
-    new coordinate system, defined by the given coordinate system (u,
-    v and w direction vectors).
-
-    .. math::
-      \begin{bmatrix}x' \\ y' \\ z' \\ h'\end{bmatrix} =
-      \begin{bmatrix}
-        u_x & u_y & u_z & 0 \\
-        v_x & v_y & v_z & 0 \\
-        w_x & w_y & w_z & 0 \\
-        0 & 0 & 0 & 1
-      \end{bmatrix}
-      \begin{bmatrix}x \\ y \\ z \\ h\end{bmatrix}
-    """
-    self[:] = cs * self
-    return self

-
-
[docs]class matrix(list):
-  """Use/create matrix like list of lists
-  """
-
-  def __getitem__(self, index):
-    """
-    For index return value, for range return new vector object.
-
-    :Example:
-
-    >>> m = matrix([[1, 2, 3, 0], [4, 5, 6, 0], [7, 8, 9, 0], [0, 0, 0, 0]])
-    >>> print(m[2])
-    [7, 8, 9, 0]
-    >>> print(m[:,1:3])
-    [[2, 3], [5, 6], [8, 9], [0, 0]]
-    >>> print(m[0:2,1:3])
-    [[2, 3], [5, 6]]
-    >>> print(m[::2,::2])
-    [[1, 3], [7, 9]]
-    """
-    # index: slice(stop), slice(start, stop[, step])
-    # for e. g. m[(1,3),:] -> index = ((1, 3), slice(None, None, None))
-    # use the list.__getslice__ method and convert result to vector
-    try:    # 2d slicing (tuple of sclices)
-      item = [row.__getitem__(index[1]) for row in super().__getitem__(index[0])]
-    except: # 1d slicing
-      item = super().__getitem__(index)
-    return matrix(item) if isinstance(item, list) else item
-
-
[docs]  @staticmethod
-  def rx(theta):
-    r"""Rotation matrix about the x direction.
-
-    Rotates the coordinate system of vectors
-
-    .. math::
-      R_{x}(\theta) =
-      \begin{bmatrix}
-        1 & 0 & 0 & 0 \\
-        0 & \cos \theta & -\sin \theta & 0 \\
-        0 & \sin \theta & \cos \theta & 0 \\
-        0 & 0 & 0 & 1
-      \end{bmatrix}
-    """
-    s = math.sin(theta)
-    c = math.cos(theta)
-    T = matrix([[ 1, 0, 0, 0],
-                [ 0, c,-s, 0],
-                [ 0, s, c, 0],
-                [ 0, 0, 0, 1]])
-    return T

-
-
[docs]  @staticmethod
-  def ry(theta):
-    r"""Rotation matrix about the y direction.
-
-    Rotates the coordinate system of vectors
-
-    .. math::
-      R_{y}(\theta) =
-      \begin{bmatrix}
-        \cos \theta & 0 & \sin \theta & 0 \\
-        0 & 1 & 0 & 0 \\
-        -\sin \theta & 0 & \cos \theta & 0 \\
-        0 & 0 & 0 & 1
-      \end{bmatrix}
-    """
-    s = math.sin(theta)
-    c = math.cos(theta)
-    T = matrix([[ c, 0, s, 0],
-                [ 0, 1, 0, 0],
-                [-s, 0, c, 0],
-                [ 0, 0, 0, 1]])
-    return T

-
-
[docs]  @staticmethod
-  def rz(theta):
-    r"""Rotation matrix about the z direction.
-
-    Rotates the coordinate system of vectors
-
-    .. math::
-      R_{z}(\theta) =
-      \begin{bmatrix}
-        \cos \theta & -\sin \theta & 0 & 0 \\
-        \sin \theta & \cos \theta & 0 & 0 \\
-        0 & 0 & 1 & 0 \\
-        0 & 0 & 0 & 1
-      \end{bmatrix}
-
-    :Example:
-
-    .. math::
-      R_{z}(\theta) \begin{bmatrix}1 \\ 0 \\ 0 \\ 1\end{bmatrix} =
-      \begin{bmatrix}
-        \cos 90° & -\sin 90° & 0 & 0 \\
-        \sin 90° & \cos 90° & 0 & 0 \\
-        0 & 0 & 1 & 0 \\
-        0 & 0 & 0 & 1
-      \end{bmatrix}
-      \begin{bmatrix}1 \\ 0 \\ 0 \\ 1\end{bmatrix} =
-      \begin{bmatrix}
-        0 & -1 & 0 & 0 \\
-        1 & 0 & 0 & 0 \\
-        0 & 0 & 1 & 0 \\
-        0 & 0 & 0 & 1
-      \end{bmatrix}
-      \begin{bmatrix}1 \\ 0 \\ 0 \\ 1\end{bmatrix} =
-      \begin{bmatrix}0 \\ 1 \\ 0 \\ 1\end{bmatrix}
-    """
-    s = math.sin(theta)
-    c = math.cos(theta)
-    T = matrix([[ c,-s, 0, 0],
-                [ s, c, 0, 0],
-                [ 0, 0, 1, 0],
-                [ 0, 0, 0, 1]])
-    return T

-
-
[docs]  @staticmethod
-  def t(tx, ty, tz):
-    r"""Translation matrix
-
-    .. math::
-      T =
-      \begin{bmatrix}
-        1 & 0 & 0 & t_x \\
-        0 & 1 & 0 & t_y \\
-        0 & 0 & 1 & t_z \\
-        0 & 0 & 0 & 1
-      \end{bmatrix}
-    """
-    T = matrix([[ 1, 0, 0,tx],
-                [ 0, 1, 0,ty],
-                [ 0, 0, 1,tz],
-                [ 0, 0, 0, 1]])
-    return T

-
-
[docs]  @staticmethod
-  def s(sx, sy=None, sz=None):
-    r"""Scaling matrix
-
-    uniform scaling if sx=sy=sz=s.
-    Note that scaling happens around the origin, so objects not
-    centered at the origin will have their centers move. To avoid this,
-    either scale the object when it's located at the origin, or
-    perform a translation afterwards to move the object back to where
-    it should be.
-
-    .. math::
-      S =
-      \begin{bmatrix}
-        s_x & 0 & 0 & 0 \\
-        0 & s_y & 0 & 0 \\
-        0 & 0 & s_z & 0 \\
-        0 & 0 & 0 & 1
-      \end{bmatrix}
-    """
-    if not sy:
-      sy = sx
-      sz = sx
-    T = matrix([[sx, 0, 0, 0],
-                [ 0,sy, 0, 0],
-                [ 0, 0,sz, 0],
-                [ 0, 0, 0, 1]])
-    return T

-
-  def __mul__(self, other):
-    r"""Scalar multiplication, dot product (inner product) or
-    matrix-vector multiplication.
-
-    a * b (new object)
-
-    :type other: scalar, vector (or 1d list), matrix (or 2d list)
-
-    .. math::
-      \mathbf{C} &= \mathbf{A} \cdot b \\
-      \mathbf{c} &= \mathbf{A} \cdot \mathbf{b} \\
-      \mathbf{C} &= \mathbf{A} \cdot \mathbf{B}
-
-    .. note::
-      No size checking will be conducted, therefore no exceptions for
-      wrong usage (result will be nonsense).
-
-    :Example:
-
-    >>> m = matrix([[1, 2, 3, 4], [5, 6, 7, 8], [9, 10, 11, 12], [0, 0, 0, 1]]) * 5
-    >>> print(m)
-    [[5, 10, 15, 20], [25, 30, 35, 40], [45, 50, 55, 60], [0, 0, 0, 5]]
-    >>> m = matrix([[1, 2, 3, 4], [5, 6, 7, 8], [9, 10, 11, 12], [0, 0, 0, 1]]) * 5.
-    >>> print(m)
-    [[5.0, 10.0, 15.0, 20.0], [25.0, 30.0, 35.0, 40.0], [45.0, 50.0, 55.0, 60.0], [0.0, 0.0, 0.0, 5.0]]
-    >>> v = matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]]) * vector([12, 12, 13])
-    >>> print(v)
-    [75, 186, 297]
-    >>> m = matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]]) * matrix([[12, 12, 13], [14, 15, 16], [17, 18, 19]])
-    >>> print(m)
-    [[91, 96, 102], [220, 231, 246], [349, 366, 390]]
-
-    .. seealso::
-      :meth:`__rmul__`
-    """
-    try:       # matrix * matrix
-      return matrix([[sum(c*d for c, d in zip(a_row, b_col)) for b_col in zip(*other)] for a_row in self])
-    except:
-      try:     # matrix * vector
-        return vector([sum(c*d for c, d in zip(a_row, other)) for a_row in self])
-      except:  # matrix * scalar
-        return matrix([[a*other for a in a_row] for a_row in self])
-
-  def __rmul__(self, other):
-    r"""Scalar multiplication, dot product (inner product) or
-    vector-matrix multiplication.
-
-    a * b (new object)
-
-    :type other: scalar (or 1d list and 2d list)
-
-    .. math::
-      \mathbf{C} &= a \cdot \mathbf{B} \\
-      \mathbf{c} &= \mathbf{a} \cdot \mathbf{B} \\
-      \mathbf{C} &= \mathbf{A} \cdot \mathbf{B}
-
-    .. note::
-      No size checking will be conducted, therefore no exceptions for
-      wrong usage (result will be nonsense).
-
-    :Example:
-
-    >>> m = 5 * matrix([[1, 2, 3, 4], [5, 6, 7, 8], [9, 10, 11, 12], [0, 0, 0, 1]]) * 5
-    >>> print(m)
-    [[5, 10, 15, 20], [25, 30, 35, 40], [45, 50, 55, 60], [0, 0, 0, 5]]
-    >>> m = 5. * matrix([[1, 2, 3, 4], [5, 6, 7, 8], [9, 10, 11, 12], [0, 0, 0, 1]])
-    >>> print(m)
-    [[5.0, 10.0, 15.0, 20.0], [25.0, 30.0, 35.0, 40.0], [45.0, 50.0, 55.0, 60.0], [0.0, 0.0, 0.0, 5.0]]
-
-    .. seealso::
-      :meth:`__mul__`
-      :meth:`vector.__mul__` for vector * matrix
-    """
-    try:       # 2d list * matrix (matrix * matrix see matrix.__mul__)
-      return matrix([[sum(c*d for c, d in zip(a_row, b_col)) for b_col in zip(*self)] for a_row in other])
-    except:
-      try:     # 1d list * matrix (vector * matrix see vector.__mul__)
-        return vector([sum(c*d for c, d in zip(other, b_col)) for b_col in zip(*self)])
-      except:  # scalar * vector
-        return self*other
-
-  def __str__(self):
-    return str([*self])
-
-  def __repr__(self):
-    return "matrix(" + str(self) + ")"
-
-
[docs]  def rotate_x(self, theta):
-    r"""Rotation about the x dirction.
-
-    .. math::
-      \begin{bmatrix}
-        xx & xy & xz & t_x \\
-        yx' & yy' & yz' & t_y' \\
-        zx' & zy' & zz' & t_z' \\
-        0 & 0 & 0 & h
-      \end{bmatrix} =
-      \begin{bmatrix}
-        1 & 0 & 0 & 0 \\
-        0 & \cos \theta & -\sin \theta & 0 \\
-        0 & \sin \theta & \cos \theta & 0 \\
-        0 & 0 & 0 & h
-      \end{bmatrix}
-      \begin{bmatrix}
-        xx & xy & xz & t_x \\
-        yx & yy & yz & t_y \\
-        zx & zy & zz & t_z \\
-        0 & 0 & 0 & h
-      \end{bmatrix}
-    """
-    self[:] = matrix.rx(theta) * self
-    return self

-
-
[docs]  def rotate_y(self, theta):
-    r"""Rotation about the y dirction.
-
-    .. math::
-      \begin{bmatrix}
-        xx' & xy' & xz' & t_x' \\
-        yx & yy & yz & t_y \\
-        zx' & zy' & zz' & t_z' \\
-        0 & 0 & 0 & h
-      \end{bmatrix} =
-      \begin{bmatrix}
-        \cos \theta & 0 & \sin \theta & 0 \\
-        0 & 1 & 0 & 0 \\
-        -\sin \theta & 0 & \cos \theta & 0 \\
-        0 & 0 & 0 & h
-      \end{bmatrix}
-      \begin{bmatrix}
-        xx & xy & xz & t_x \\
-        yx & yy & yz & t_y \\
-        zx & zy & zz & t_z \\
-        0 & 0 & 0 & h
-      \end{bmatrix}
-    """
-    self[:] = matrix.ry(theta) * self
-    return self

-
-
[docs]  def rotate_z(self, theta):
-    r"""Rotation about the z dirction.
-
-    .. math::
-      \begin{bmatrix}
-        xx' & xy' & xz' & t_x' \\
-        yx' & yy' & yz' & t_y' \\
-        zx & zy & zz & t_z \\
-        0 & 0 & 0 & h
-      \end{bmatrix} =
-      \begin{bmatrix}
-        \cos \theta & -\sin \theta & 0 & 0 \\
-        \sin \theta & \cos \theta & 0 & 0 \\
-        0 & 0 & 1 & 0 \\
-        0 & 0 & 0 & 1
-      \end{bmatrix}
-      \begin{bmatrix}
-        xx & xy & xz & t_x \\
-        yx & yy & yz & t_y \\
-        zx & zy & zz & t_z \\
-        0 & 0 & 0 & h
-      \end{bmatrix}
-    """
-    self[:] = matrix.rz(theta) * self
-    return self

-
-
[docs]  def translate(self, tx, ty, tz):
-    r"""Translation
-
-    .. math::
-      \begin{bmatrix}
-        xx & xy & xz & t_x' \\
-        yx & yy & yz & t_y' \\
-        zx & zy & zz & t_z' \\
-        0 & 0 & 0 & h
-      \end{bmatrix} =
-      \begin{bmatrix}
-        1 & 0 & 0 & t_x \\
-        0 & 1 & 0 & t_y \\
-        0 & 0 & 1 & t_z \\
-        0 & 0 & 0 & 1
-      \end{bmatrix}
-      \begin{bmatrix}
-        xx & xy & xz & t_x \\
-        yx & yy & yz & t_y \\
-        zx & zy & zz & t_z \\
-        0 & 0 & 0 & h
-      \end{bmatrix}
-    """
-    self[:] = matrix.t(tx, ty, tz) * self
-    return self

-
-
[docs]  def scale(self, sx, sy=None, sz=None):
-    r"""Scaling
-
-    uniform scaling if sx=sy=sz=s.
-    Note that scaling happens around the origin, so objects not
-    centered at the origin will have their centers move. To avoid this,
-    either scale the object when it's located at the origin, or
-    perform a translation afterwards to move the object back to where
-    it should be.
-
-    .. math::
-      \begin{bmatrix}
-        xx' & xy' & xz' & t_x' \\
-        yx' & yy' & yz' & t_y' \\
-        zx' & zy' & zz' & t_z' \\
-        0 & 0 & 0 & h
-      \end{bmatrix} =
-      \begin{bmatrix}
-        s_x & 0 & 0 & 0 \\
-        0 & s_y & 0 & 0 \\
-        0 & 0 & s_z & 0 \\
-        0 & 0 & 0 & 1
-      \end{bmatrix}
-      \begin{bmatrix}
-        xx & xy & xz & t_x \\
-        yx & yy & yz & t_y \\
-        zx & zy & zz & t_z \\
-        0 & 0 & 0 & h
-      \end{bmatrix}
-    """
-    if not sy:
-      sy = sx
-      sz = sx
-    self[:] = matrix.s(sx, sy, sz) * self
-    return self

-
- -
- -
-
- -
-
-
- ©2019, Daniel Weschke. - - | - Powered by Sphinx 2.2.1 - & Alabaster 0.7.12 - -
- - - - - - \ No newline at end of file diff --git a/docs/build/html/_modules/numerical/fit.html b/docs/build/html/_modules/numerical/fit.html deleted file mode 100644 index b2dbf92..0000000 --- a/docs/build/html/_modules/numerical/fit.html +++ /dev/null @@ -1,196 +0,0 @@ - - - - - - - numerical.fit — pylib 2019.5.19 documentation - - - - - - - - - - - - - - - - - - - - -
-
-
- - -
- -

Source code for numerical.fit

-#!/usr/bin/env python
-# -*- coding: utf-8 -*-
-"""Function and approximation.
-
-:Date: 2019-10-15
-
-.. module:: fit
-  :platform: *nix, Windows
-  :synopsis: Function and approximation.
-
-.. moduleauthor:: Daniel Weschke <daniel.weschke@directbox.de>
-"""
-from __future__ import print_function
-from pylab import array, argmax, gradient, exp, sqrt, log, linspace
-from scipy.optimize import curve_fit
-
-
[docs]def gauss(x, *p):
-  """Gauss distribution function.
-
-  .. math::
-    f(x)=ae^{-(x-b)^{2}/(2c^{2})}
-
-  :param x: positions where the gauss function will be calculated
-  :type x: int or float or list or numpy.ndarray
-  :param p: gauss parameters [a, b, c, d]:
-
-    * a -- amplitude (:math:`\\int y \\,\\mathrm{d}x=1 \\Leftrightarrow a=1/(c\\sqrt{2\\pi})` )
-    * b -- expected value :math:`\\mu` (position of maximum, default = 0)
-    * c -- standard deviation :math:`\\sigma` (variance :math:`\\sigma^2=c^2`)
-    * d -- vertical offset (default = 0)
-  :type p: list
-
-  :returns: gauss values at given positions x
-  :rtype: numpy.ndarray
-  """
-  x = array(x)  # cast e. g. list to numpy array
-  a, b, c, d = p
-  return a*exp(-(x - b)**2./(2. * c**2.)) + d

-
-
[docs]def gauss_fit(x, y, e=None, x_fit=None, verbose=False):
-  """Fit Gauss distribution function to data.
-
-  :param x: positions
-  :type x: int or float or list or numpy.ndarray
-  :param y: values
-  :type y: int or float or list or numpy.ndarray
-  :param e: error values (default = None)
-  :type e: int or float or list or numpy.ndarray
-  :param x_fit: positions of fitted function (default = None, if None then x
-    is used)
-  :type x_fit: int or float or list or numpy.ndarray
-  :param verbose: verbose information (default = False)
-  :type verbose: bool
-
-  :returns:
-    * numpy.ndarray -- fitted values (y_fit)
-    * numpy.ndarray -- parameters of gauss distribution function (popt:
-      amplitude a, expected value :math:`\\mu`, standard deviation
-      :math:`\\sigma`, vertical offset d)
-    * numpy.float64 -- full width at half maximum (FWHM)
-  :rtype: tuple
-
-  .. seealso::
-    :meth:`gauss`
-  """
-  x = array(x)  # cast e. g. list to numpy array
-  y = array(y)  # cast e. g. list to numpy array
-  y_max = max(y)
-  y_max_pos = argmax(y)
-  x_y_max = x[y_max_pos]
-
-  # starting parameter
-  p0 = [y_max, x_y_max, .1, y[0]]
-  if verbose:
-    print('p0:', end=' ')
-    print(p0)
-  if e is not None:
-    popt, pcov = curve_fit(gauss, x, y, p0=p0, sigma=e)
-  else:
-    popt, pcov = curve_fit(gauss, x, y, p0=p0)
-  if verbose:
-    print('popt:', end=' ')
-    print(popt)
-    #print(pcov)
-
-  FWHM = 2*sqrt(2*log(2))*popt[2]
-  if verbose:
-    print('FWHM', FWHM)
-
-  if x_fit is None:
-    x_fit = x
-  y_fit = gauss(x_fit, *popt)
-
-  return y_fit, popt, FWHM

-
-if __name__ == "__main__":
-  pass
-
- -
- -
-
- -
-
-
- ©2019, Daniel Weschke. - - | - Powered by Sphinx 2.2.1 - & Alabaster 0.7.12 - -
- - - - - - \ No newline at end of file diff --git a/docs/build/html/_modules/numerical/integration.html b/docs/build/html/_modules/numerical/integration.html deleted file mode 100644 index b39524c..0000000 --- a/docs/build/html/_modules/numerical/integration.html +++ /dev/null @@ -1,250 +0,0 @@ - - - - - - - numerical.integration — pylib 2019.5.19 documentation - - - - - - - - - - - - - - - - - - - - -
-
-
- - -
- -

Source code for numerical.integration

-# -*- coding: utf-8 -*-
-"""Numerical integration, numerical quadrature.
-
-de: numerische Integration, numerische Quadratur.
-
-:Date: 2015-10-15
-
-.. module:: integration
-  :platform: *nix, Windows
-  :synopsis: Numerical integration.
-
-.. moduleauthor:: Daniel Weschke <daniel.weschke@directbox.de>
-"""
-
-from __future__ import division
-from numpy import linspace, trapz, zeros
-
-
[docs]def trapez(f, a=0, b=1, N=10, x=None, verbose=False,
-           save_values=False):
-  r"""
-  Integration of :math:`f(x)` using the trapezoidal rule
-  (Simpson's rule, Kepler's rule).
-
-  de: Trapezregel, Simpsonregel (Thomas Simpson), Keplersche
-  Fassregel (Johannes Kepler)
-
-  :param f: function to integrate.
-  :type f: function or list
-  :param a: lower limit of integration (default = 0).
-  :type a: float
-  :param b: upper limit of integration (default = 1).
-  :type b: float
-  :param N: specify the number of subintervals.
-  :type N: int
-  :param x: variable of integration, necessary if f is a list
-    (default = None).
-  :type x: list
-  :param verbose: print information (default = False)
-  :type verbose: bool
-
-  :returns: the definite integral as approximated by trapezoidal
-    rule.
-  :rtype: float
-
-  The trapezoidal rule approximates the integral by the area of a
-  trapezoid with base h=b-a and sides equal to the values of the
-  integrand at the two end points.
-
-  .. math::
-    f_n(x) = f(a)+\frac{f(b)-f(a)}{b-a}(x-a)
-
-  .. math::
-    I &= \int\limits_a^b f(x) \,\mathrm{d}x \\
-    I &\approx \int\limits_a^b f_n(x) \,\mathrm{d}x \\
-      &= \int\limits_a^b
-         \left( f(a)+\frac{f(b)-f(a)}{b-a}(x-a) \right)
-         \mathrm{d}x \\
-      &= \left.\left( f(a)-a\frac{f(b)-f(a)}{b-a} \right)
-         x \right\vert_a^b +
-         \left. \frac{f(b)-f(a)}{b-a} \frac{x^2}{2}
-         \right\vert_a^b \\
-      &= \frac{b-a}{2}\left[f(a)+f(b)\right]
-
-  The composite trapezium rule. If the interval is divided into n
-  segments (not necessarily equal)
-
-  .. math::
-    a = x_0 \leq x_1 \leq x_2 \leq \ldots \leq x_n = b
-
-  .. math::
-    I &\approx \sum\limits_{i=0}^{n-1} \frac{1}{2} (x_{i+1}-x_i)
-    \left[f(x_{i+1})+f(x_i)\right] \\
-
-  Special Case (Equaliy spaced base points)
-
-  .. math::
-    x_{i+1}-x_i = h \quad \forall i
-
-  .. math::
-    I \approx  h \left\{ \frac{1}{2} \left[f(x_0)+f(x_n)\right] +
-      \sum\limits_{i=1}^{n-1} f(x_i) \right\}
-
-  .. rubric:: Example
-
-  .. math::
-    I &= \int\limits_a^b f(x) \,\mathrm{d}x \\
-    f(x) &= x^2 \\
-    a &= 0 \\
-    b &= 1
-
-  analytical solution
-
-  .. math::
-    I = \int\limits_{0}^{1} x^2 \,\mathrm{d}x
-      = \left. \frac{1}{3} x^3 \right\vert_0^1
-      = \frac{1}{3}
-
-  numerical solution
-
-  >>> f = lambda(x): x**2
-  >>> trapez(f, 0, 1, 1)
-  0.5
-  >>> trapez(f, 0, 1, 10)
-  0.3350000000000001
-  >>> trapez(f, 0, 1, 100)
-  0.33335000000000004
-  """
-  N = int(N)
-  # f is function or list
-  if callable(f):
-    # h width of each subinterval
-    h = (b-a)/N
-
-    # x variable of integration
-    x = linspace(a, b, N+1)
-    if save_values:
-      # ff contribution from the points
-      ff = zeros((N+1))
-      for n in linspace(0, N, N+1):
-        ff[n] = f(x[n])
-      T = (ff[0]/2.+sum(ff[1:N])+ff[N]/2.)*h
-    else:
-      TL = f(x[0])
-      TR = f(x[N])
-      TI = 0
-      for n in range(1, N):
-        TI = TI + f(x[n])
-      T = (TL/2.+TI+TR/2.)*h
-  else:
-    N = len(f)-1
-    T = 0
-    for n in range(N):
-      T = T + (x[n+1]-x[n])/2*(f[n+1]+f[n])
-
-  if verbose:
-    print(T)
-
-  return T

-
-
-if __name__ == '__main__':
-  func = lambda x: x**2
-  trapez(func, 0, 1, 1e6, verbose=True)
-  #print(trapz(func, linspace(0,1,10)))
-
-  trapez([0, 1, 4, 9], x=[0, 1, 2, 3], verbose=True)
-  #print(trapz([0,1,4,9]))
-
-  trapez([2, 2], x=[-1, 1], verbose=True)
-
-  trapez([-1, 1, 1, -1], x=[-1, -1, 1, 1], verbose=True)
-
- -
- -
-
- -
-
-
- ©2019, Daniel Weschke. - - | - Powered by Sphinx 2.2.1 - & Alabaster 0.7.12 - -
- - - - - - \ No newline at end of file diff --git a/docs/build/html/_modules/numerical/ode.html b/docs/build/html/_modules/numerical/ode.html deleted file mode 100644 index 7a0edb1..0000000 --- a/docs/build/html/_modules/numerical/ode.html +++ /dev/null @@ -1,540 +0,0 @@ - - - - - - - numerical.ode — pylib 2019.5.19 documentation - - - - - - - - - - - - - - - - - - - - -
-
-
- - -
- -

Source code for numerical.ode

-#!/usr/bin/env python
-# -*- coding: utf-8 -*-
-"""Numerical solver of ordinary differential equations.
-
-Solves the initial value problem for systems of first order
-ordinary differential equations.
-
-:Date: 2015-09-21
-
-.. module:: ode
-  :platform: *nix, Windows
-  :synopsis: Numerical solver.
-
-.. moduleauthor:: Daniel Weschke <daniel.weschke@directbox.de>
-"""
-
-from __future__ import division, print_function
-from numpy import array, isnan, sum, zeros, dot
-from numpy.linalg import norm, inv
-
-
[docs]def e1(f, x0, t, *p, verbose=False):
-  r"""Explicit first-order method /
-  (standard, or forward) Euler method /
-  Runge-Kutta 1st order method.
-
-  de:
-  Euler'sche Polygonzugverfahren / explizite Euler-Verfahren /
-  Euler-Cauchy-Verfahren / Euler-vorwärts-Verfahren
-
-  :param f: the function to solve
-  :type f: function
-  :param x0: initial condition
-  :type x0: list
-  :param t: time
-  :type t: list
-  :param `*p`: parameters of the function (thickness, diameter,
-    ...)
-  :param verbose: print information (default = False)
-  :type verbose: bool
-
-  Approximate the solution of the initial value problem
-
-  .. math ::
-    \dot{x} &= f(t,x) \\
-    x(t_0) &= x_0
-
-  Choose a value h for the size of every step and set
-
-  .. math ::
-    t_i = t_0 + i h ~,\quad i=1,2,\ldots,n
-
-  The derivative of the solution is approximated as the forward
-  difference equation
-  
-  .. math ::
-    \dot{x}_i = f(t_i, x_i) = \frac{x_{i+1} - x_i}{t_{i+1}-t_i}
-
-  Therefore one step :math:`h` of the Euler method from
-  :math:`t_i` to :math:`t_{i+1}` is
-
-  .. math ::
-    x_{i+1} &= x_i + (t_{i+1}-t_i) f(t_i, x_i) \\
-    x_{i+1} &= x_i + h f(t_i, x_i) \\
-
-  Example 1:
-
-  .. math ::
-    m\ddot{u} + d\dot{u} + ku = f(t) \\
-    \ddot{u} = m^{-1}(f(t) - d\dot{u} - ku) \\
-
-  with
-
-  .. math ::
-    x_1 &= u        &\quad \dot{x}_1 = \dot{u} = x_2 \\
-    x_2 &= \dot{u}  &\quad \dot{x}_2 = \ddot{u} \\
-
-  becomes
-
-  .. math ::
-    \dot{x}_1 &= x_2 \\
-    \dot{x}_2 &= m^{-1}(f(t) - d x_2 - k x_1) \\
-
-  or
-
-  .. math ::
-    \dot{x} &= f(t,x) \\
-    \begin{bmatrix} \dot{x}_1 \\ \dot{x}_2 \end{bmatrix} &=
-    \begin{bmatrix} x_2 \\ m^{-1}(f(t) - d x_2 - k x_1)
-    \end{bmatrix} \\
-    &=
-    \begin{bmatrix} 0 \\ m^{-1} f(t) \end{bmatrix} +
-    \begin{bmatrix} 0 & 1 \\ -m^{-1} k & -m^{-1} d \end{bmatrix}
-    \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}
-
-  Example 2:
-
-  .. math ::
-    m(u)\ddot{u} + d(u,\dot{u})\dot{u} + k(u)u = f(t) \\
-    \ddot{u} = m^{-1}(u)(f(t) - d(u,\dot{u})\dot{u} - k(u)u) \\
-
-  with
-
-  .. math ::
-    x_1 &= u        &\quad \dot{x}_1 = \dot{u} = x_2 \\
-    x_2 &= \dot{u}  &\quad \dot{x}_2 = \ddot{u} \\
-
-  becomes
-
-  .. math ::
-    \dot{x}_1 &= x_2 \\
-    \dot{x}_2 &=
-      m^{-1}(x_1)(f(t) - d(x_1,x_2) x_2 - k(x_1) x_1) \\
-
-  or
-
-  .. math ::
-    \dot{x} &= f(t,x) \\
-    \begin{bmatrix} \dot{x}_1 \\ \dot{x}_2 \end{bmatrix} &=
-    \begin{bmatrix}
-      x_2 \\ m^{-1}(x_1)(f(t) - d(x_1,x_2) x_2 - k(x_1) x_1)
-      \end{bmatrix} \\
-    &=
-    \begin{bmatrix} 0 \\ m^{-1}(x_1) f(t) \end{bmatrix} +
-    \begin{bmatrix}
-      0 & 1 \\ -m^{-1}(x_1) k(x_1) & -m^{-1} d(x_1,x_2)
-      \end{bmatrix}
-    \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}
-
-  The Euler method is a first-order method, which means that the
-  local error (error per step) is proportional to the square of
-  the step size, and the global error (error at a given time) is
-  proportional to the step size.
-  """
-  x = zeros((len(t), len(x0)))  # Preallocate array
-  x[0,:] = x0  # Initial condition gives solution at first t
-  for i in range(len(t)-1):     # Calculation loop
-    Dt = t[i+1]-t[i]
-    dxdt = array(f(x[i,:], t[i], *p))
-    # Approximate solution at next value of x
-    x[i+1,:] = x[i,:] + dxdt*Dt
-  if verbose:
-    print('Numerical integration of ODE using explicit ' +
-      'first-order method (Euler / Runge-Kutta) was successful.')
-  return x

-
-
[docs]def e2(f, x0, t, *p, verbose=False):
-  r"""Explicit second-order method / Runge-Kutta 2nd order method.
-
-  :param f: the function to solve
-  :type f: function
-  :param x0: initial condition
-  :type x0: list
-  :param t: time
-  :type t: list
-  :param `*p`: parameters of the function (thickness, diameter,
-    ...)
-  :param verbose: print information (default = False)
-  :type verbose: bool
-  """
-  x = zeros((len(t), len(x0)))  # Preallocate array
-  x[0,:] = x0  # Initial condition gives solution at first t
-  for i in range(len(t)-1):     # Calculation loop
-    Dt = t[i+1]-t[i]
-    k_1 = array(f(x[i,:], t[i], *p))
-    k_2 = array(f(x[i,:]+0.5*Dt*k_1, t[i]+0.5*Dt, *p))
-    # Approximate solution at next value of x
-    x[i+1,:] = x[i,:] + k_2*Dt
-  if verbose:
-    print('Numerical integration of ODE using explicit ' +
-      '2th-order method (Runge-Kutta) was successful.')
-  return x

-
-
[docs]def e4(f, x0, t, *p, verbose=False):
-  r"""Explicit fourth-order method / Runge-Kutta 4th order method.
-
-  :param f: the function to solve
-  :type f: function
-  :param x0: initial condition
-  :type x0: list
-  :param t: time
-  :type t: list
-  :param `*p`: parameters of the function (thickness, diameter,
-    ...)
-  :param verbose: print information (default = False)
-  :type verbose: bool
-  """
-  x = zeros((len(t), len(x0)))        # Preallocate array
-  x[0,:] = x0                          # Initial condition
-  for i in range(len(t)-1):           # Calculation loop
-    Dt = t[i+1]-t[i]
-    k_1 = array(f(x[i,:], t[i], *p))
-    k_2 = array(f(x[i,:]+0.5*Dt*k_1, t[i]+0.5*Dt, *p))
-    k_3 = array(f(x[i,:]+0.5*Dt*k_2, t[i]+0.5*Dt, *p))
-    k_4 = array(f(x[i,:]+k_3*Dt, t[i]+Dt, *p))
-    # Approximate solution at next value of x
-    x[i+1,:] = x[i,:] + 1./6*(k_1+2*k_2+2*k_3+k_4)*Dt
-  if verbose:
-    print('Numerical integration of ODE using explicit ' +
-      '4th-order method (Runge-Kutta) was successful.')
-  return x

-
-
[docs]def fpi(f, xi, ti, ti1, *p, max_iterations=1000, tol=1e-9,
-  verbose=False):
-  r"""Fixed-point iteration.
-  
-  :param f: the function to iterate :math:`f = \dot{x}(x,t)`
-  :type f: function
-  :param xi: initial condition :math:`x_i`
-  :type xi: list
-  :param ti: time :math:`t_i`
-  :type ti: float
-  :param ti1: time :math:`t_{i+1}`
-  :type ti1: float
-  :param `*p`: parameters of the function (thickness, diameter,
-    ...)
-  :param max_iterations: maximum number of iterations
-  :type max_iterations: int
-  :param tol: tolerance against residuum :math:`\varepsilon`
-    (default = 1e-9)
-  :type tol: float
-  :param verbose: print information (default = False)
-  :type verbose: bool
-  
-  :returns: :math:`x_{i}`
-  
-  .. math ::
-    x_{i,j=0} = x_{i}
-  
-  .. math ::
-    x_{i,j+1} = x_i + \dot{x}(x_{i,j}, t_{i+1})\cdot(t_{i+1}-t_i)
-  
-  .. math ::
-    \text{residuum} = \frac{\lVert x_{i,j+1}-x_{i,j}\rVert}
-      {\lVert x_{i,j+1} \rVert} < \varepsilon
-  
-  .. math ::
-    x_{i} = x_{i,j=\text{end}}
-  """
-  xij = xi
-  for j in range(max_iterations):
-    dxdt = array(f(xij, ti1, *p))
-    # Approximate solution at next value of x
-    xij1 = xi + dxdt * (ti1-ti)
-    residuum = norm(xij1-xij)/norm(xij1)
-    xij = xij1
-    if residuum < tol:
-      break
-  iterations = j+1  # number beginning with 1 therefore + 1
-  return xij, iterations

-
-
[docs]def i1(f, x0, t, *p, max_iterations=1000, tol=1e-9,
-  verbose=False):
-  r"""Implicite first-order method / backward Euler method.
-
-  :param f: the function to solve
-  :type f: function
-  :param x0: initial condition
-  :type x0: list
-  :param t: time
-  :type t: list
-  :param `*p`: parameters of the function (thickness, diameter,
-    ...)
-  :param max_iterations: maximum number of iterations
-  :type max_iterations: int
-  :param tol: tolerance against residuum (default = 1e-9)
-  :type tol: float
-  :param verbose: print information (default = False)
-  :type verbose: bool
-  
-  The backward Euler method has order one and is A-stable.
-  """
-  iterations = zeros((len(t), 1))
-  x = zeros((len(t), len(x0)))     # Preallocate array
-  x[0,:] = x0       # Initial condition gives solution at first t
-  # x(i+1) = x(i) + f(x(i+1), t(i+1)), exact value of
-  # f(x(i+1), t(i+1)) is not available therefore using
-  # Newton-Raphson method
-  for i in range(len(t)-1):
-    Dt = t[i+1]-t[i]
-    xi = x[i,:]
-    xi, iteration = fpi(f, xi, t[i], t[i+1], *p, max_iterations,
-      tol, verbose)
-    x[i+1,:] = xi
-    iterations[i] = iteration
-  if verbose:
-    print('Numerical integration of ODE using implicite ' +
-      'first-order method (Euler) was successful.')
-  return x, iterations

-
-
[docs]def newmark_newtonraphson(f, x0, xp0, xpp0, t, *p, gamma=.5,
-  beta=.25, max_iterations=1000, tol=1e-9, verbose=False):
-  r"""Newmark method.
-
-  :param f: the function to solve
-  :type f: function
-  :param x0: initial condition
-  :type x0: list
-  :param xp0: initial condition
-  :type xp0: list
-  :param xpp0: initial condition
-  :type xpp0: list
-  :param t: time
-  :type t: list
-  :param `*p`: parameters of the function (thickness, diameter,
-    ...)
-  :param gamma: newmark parameter for velocity (default = 0.5)
-  :type gamma: float
-  :param beta: newmark parameter for displacement (default = 0.25)
-  :type beta: float
-  :param max_iterations: maximum number of iterations
-  :type max_iterations: int
-  :param tol: tolerance against residuum (default = 1e-9)
-  :type tol: float
-  :param verbose: print information (default = False)
-  :type verbose: bool
-  """
-  iterations = zeros((len(t), 1))
-  x = zeros((len(t), len(x0)))     # Preallocate array
-  xp = zeros((len(t), len(xp0)))   # Preallocate array
-  xpp = zeros((len(t), len(xpp0))) # Preallocate array
-  x[0,:] = x0       # Initial condition gives solution at first t
-  xp[0,:] = xp0     # Initial condition gives solution at first t
-  xpp[0,:] = xpp0   # Initial condition gives solution at first t
-  for i in range(len(t)-1):
-    Dt = t[i+1]-t[i]
-
-    xi = x[i,:].reshape(3,1)
-    xpi = xp[i,:].reshape(3,1)
-    xppi = xpp[i,:].reshape(3,1)
-    x1 = xi
-    xp1 = xpi
-    xpp1 = xppi
-    j = 0
-    for j in range(max_iterations):  # Fixed-point iteration
-      #dxdt = array(f(t[i+1], x1, p))
-      # Approximate solution at next value of x
-      #x11 = x[i,:] + dxdt*Dt
-
-      N, dN, dNp, dNpp = f(x1.reshape(-1,).tolist(),
-        xp1.reshape(-1,).tolist(), xpp1.reshape(-1,).tolist(),
-        t[i], *p)
-      if isnan(sum(dN)) or isnan(sum(dNp)) or isnan(sum(dNpp)):
-        print('divergiert')
-        break
-
-      xpp11 = xpp1 - dot(inv(dNpp), (N + dot(dN, (x1-xi)) + \
-                                          dot(dNp, (xp1-xpi))))
-      xp1 = xpi + Dt*( (1-gamma)*xppi + gamma*xpp11 )
-      x1 = xi + Dt*xpi + Dt**2*( (.5-beta)*xppi + beta*xpp11 )
-            
-      residuum = norm(xpp11-xpp1)/norm(xpp11)
-      xpp1 = xpp11
-      if residuum < tol:
-        break
-    iterations[i] = j+1
-
-    xpp[i+1,:] = xpp1.reshape(-1,).tolist()
-    xp[i+1,:] = xp1.reshape(-1,).tolist()
-    x[i+1,:] = x1.reshape(-1,).tolist()
-  if verbose:
-    print('Numerical integration of ODE using explicite ' +
-      'newmark method was successful.')
-  return x, xp, xpp, iterations

-  # x = concatenate((x, xp, xpp), axis=1)
-
-
[docs]def newmark_newtonraphson_rdk(fnm, x0, xp0, xpp0, t, *p, gamma=.5,
-  beta=.25, max_iterations=1000, tol=1e-9, verbose=False):
-  r"""Newmark method.
-
-  :param f: the function to solve
-  :type f: function
-  :param x0: initial condition
-  :type x0: list
-  :param xp0: initial condition
-  :type xp0: list
-  :param xpp0: initial condition
-  :type xpp0: list
-  :param t: time
-  :type t: list
-  :param `*p`: parameters of the function (thickness, diameter,
-    ...)
-  :param gamma: newmark parameter for velocity (default = 0.5)
-  :type gamma: float
-  :param beta: newmark parameter for displacement (default = 0.25)
-  :type beta: float
-  :param max_iterations: maximum number of iterations
-  :type max_iterations: int
-  :param tol: tolerance against residuum (default = 1e-9)
-  :type tol: float
-  :param verbose: print information (default = False)
-  :type verbose: bool
-  """
-  iterations = zeros((len(t), 1))
-  x = zeros((len(t), len(x0)))    # Preallocate array
-  xp = zeros((len(t), len(xp0)))  # Preallocate array
-  xpp = zeros((len(t), len(xpp0)))  # Preallocate array
-  x[0,:] = x0       # Initial condition gives solution at first t
-  xp[0,:] = xp0     # Initial condition gives solution at first t
-  xpp[0,:] = xpp0   # Initial condition gives solution at first t
-  for i in range(len(t)-1):
-    Dt = t[i+1]-t[i]
-
-    rm, rmx, rmxpp, rd, rdx, rdxp, rk, rkx, f = fnm(x[i,:],
-      xp[i,:], xpp[i,:], t[i], *p)
-        
-    xi = x[i,:].reshape(3,1)
-    xpi = xp[i,:].reshape(3,1)
-    xppi = xpp[i,:].reshape(3,1)
-    x1 = xi
-    xp1 = xpi
-    xpp1 = xppi
-    j = 0
-    for j in range(max_iterations): # Fixed-point iteration
-      #dxdt = array(f(t[i+1], x1, p))
-      # Approximate solution at next value of x
-      #x11 = x[i,:] + dxdt*Dt
-
-      r = (rmx+rdx+rkx)*Dt**2./4 + rdxp*Dt/2 + rmxpp 
-      rp = f - (rm + dot(rmx, (Dt*xpi+Dt**2./4*xppi)) - \
-                     dot(rmxpp, xppi) + \
-                rd + dot(rdx, (Dt*xpi+Dt**2./4*xppi)) + \
-                     dot(rdxp, Dt/2*xppi) + \
-                rk + dot(rkx, (Dt*xpi+Dt**2./4*xppi)) )
-      xpp11 = dot(inv(r), rp)
-      xp1 = xpi + Dt*( (1-gamma)*xppi + gamma*xpp11 )
-      x1 = xi + Dt*xpi + Dt**2*( (.5-beta)*xppi + beta*xpp11 )
-
-      residuum = norm(xpp11-xpp1)/norm(xpp11)
-      xpp1 = xpp11
-      if residuum < tol:
-        break
-    iterations[i] = j+1
-
-    xpp[i+1,:] = xpp1.reshape(-1,).tolist()
-    xp[i+1,:] = xp1.reshape(-1,).tolist()
-    x[i+1,:] = x1.reshape(-1,).tolist()
-  if verbose:
-    print('Numerical integration of ODE using explicite ' +
-      'newmark method was successful.')
-  return x, xp, xpp, iterations

-  # x = concatenate((x, xp, xpp), axis=1)
-
- -
- -
-
- -
-
-
- ©2019, Daniel Weschke. - - | - Powered by Sphinx 2.2.1 - & Alabaster 0.7.12 - -
- - - - - - \ No newline at end of file diff --git a/docs/build/html/_modules/numerical/ode_model.html b/docs/build/html/_modules/numerical/ode_model.html deleted file mode 100644 index 454d466..0000000 --- a/docs/build/html/_modules/numerical/ode_model.html +++ /dev/null @@ -1,219 +0,0 @@ - - - - - - - numerical.ode_model — pylib 2019.5.19 documentation - - - - - - - - - - - - - - - - - - - - -
-
-
- - -
- -

Source code for numerical.ode_model

-#!/usr/bin/env python
-# -*- coding: utf-8 -*-
-"""Mathmatical models governed by ordinary differential equations.
-
-Describes initial value problems as systems of first order ordinary differential
-equations.
-
-:Date: 2019-05-25
-
-.. module:: ode_model
-  :platform: *nix, Windows
-  :synopsis: Models of ordinary differential equations.
-
-.. moduleauthor:: Daniel Weschke <daniel.weschke@directbox.de>
-"""
-from __future__ import division, print_function
-from numpy import array, cos, sin, dot, square
-from numpy.linalg import inv
-
-
[docs]def disk(x, t, *p):
-  """Rotation of an eccentric disk.
-
-  :param x: values of the function
-  :type x: list
-  :param t: time
-  :type t: list
-  :param `*p`: parameters of the function
-
-    * diameter
-    * eccentricity
-    * torque
-  """
-  qp1 = x[3]
-  qp2 = x[4]
-  qp3 = x[5]
-  M = array([[1, 0, cos(x[2])], [0, 1, -sin(x[2])], [0, 0, 1]])
-  y = array([[-2*p[0]*x[3]+sin(x[2])*x[5]**2-2*p[0]*cos(x[2])*x[5]-x[0]], \
-             [-2*p[0]*x[4]+cos(x[2])*x[5]**2-2*p[0]*sin(x[2])*x[5]-x[1]], \
-             [p[2]-p[1]*x[1]*sin(x[2])+p[1]*x[0]*cos(x[2])]])
-  qp46 = dot(inv(M), y)
-  qp4, qp5, qp6 = qp46.reshape(-1,).tolist()  # 2d array to 1d array to list
-  return qp1, qp2, qp3, qp4, qp5, qp6

-
-
[docs]def disk_nm(xn, xpn, xppn, t, *p):
-  """Rotation of an eccentric disk.
-
-  :param xn: values of the function
-  :type xn: list
-  :param xpn: first derivative values of the function
-  :type xpn: list
-  :param xppn: second derivative values of the function
-  :type xppn: list
-  :param t: time
-  :type t: list
-  :param `*p`: parameters of the function
-
-    * diameter
-    * eccentricity
-    * torque
-  """
-  N = array([[xppn[0]+cos(xn[2])*xppn[2]+2*p[0]*xpn[0]+2*p[0]*cos(xn[2])*xpn[2]-sin(xn[2])*square(xpn[2])+xn[0]],
-             [xppn[1]-sin(xn[2])*xppn[2]+2*p[0]*xpn[1]-2*p[0]*sin(xn[2])*xpn[2]-cos(xn[2])*square(xpn[2])+xn[1]],
-             [xppn[2]+p[1]*(-cos(xn[2])*xn[0]+sin(xn[2])*xn[1])-p[2]]])
-  dN = array([[1,                0,               -sin(xn[2]*xppn[2])-2*p[0]*sin(xn[2])*xpn[2]-cos(xn[2])*square(xpn[2])],
-              [0,                1,               -cos(xn[2]*xppn[2])-2*p[0]*cos(xn[2])*xpn[2]+sin(xn[2])*square(xpn[2])],
-              [-p[1]*cos(xn[2]), p[1]*cos(xn[2]),  p[1]*(sin(xn[2])*xn[0]+cos(xn[2])*xn[1])]])
-  dNp = array([[2*p[0], 0,       2*p[0]*cos(xn[2])-2*sin(xn[2])*xpn[2]],
-               [0,      2*p[0], -2*p[0]*sin(xn[2])-2*cos(xn[2])*xpn[2]],
-               [0,      0,       0]])
-  dNpp = array([[1, 0,  cos(xn[2])],
-                [0, 1, -sin(xn[2])],
-                [0, 0,  1]])
-  return N, dN, dNp, dNpp

-
-
[docs]def disk_nmmdk(xn, xpn, xppn, t, *p):
-  """Rotation of an eccentric disk.
-
-  :param xn: values of the function
-  :type xn: list
-  :param xpn: derivative values of the function
-  :type xpn: list
-  :param xppn: second derivative values of the function
-  :type xppn: list
-  :param t: time
-  :type t: list
-  :param `*p`: parameters of the function
-
-    * diameter
-    * eccentricity
-    * torque
-  """
-  rm = array([[xppn[0]+cos(xn[2])*xppn[2]],
-              [xppn[1]-sin(xn[2])*xppn[2]],
-              [xppn[2]]])
-  rmx = array([[0, 0, -sin(xn[2]*xppn[2])],
-               [0, 0, -cos(xn[2]*xppn[2])],
-               [0, 0,  0]])
-  rmxpp = array([[1, 0,  cos(xn[2])],
-                 [0, 1, -sin(xn[2])],
-                 [0, 0,  1]])
-  rd = array([[2*p[0]*xpn[0]+2*p[0]*cos(xn[2])*xpn[2]-sin(xn[2])*square(xpn[2])],
-              [2*p[0]*xpn[1]-2*p[0]*sin(xn[2])*xpn[2]-cos(xn[2])*square(xpn[2])],
-              [0]])
-  rdx = array([[0, 0, -2*p[0]*sin(xn[2])*xpn[2]-cos(xn[2])*square(xpn[2])],
-               [0, 0, -2*p[0]*cos(xn[2])*xpn[2]+sin(xn[2])*square(xpn[2])],
-               [0, 0,  0]])
-  rdxp = array([[2*p[0], 0,       2*p[0]*cos(xn[2])-2*sin(xn[2])*xpn[2]],
-                [0,      2*p[0], -2*p[0]*sin(xn[2])-2*cos(xn[2])*xpn[2]],
-                [0,      0,       0]])
-  rk = array([[xn[0]],
-              [xn[1]],
-              [p[1]*(-cos(xn[2])*xn[0]+sin(xn[2])*xn[1])]])
-  rkx = array([[ 1,               0,               0],
-               [ 0,               1,               0],
-               [-p[1]*cos(xn[2]), p[1]*cos(xn[2]), p[1]*(sin(xn[2])*xn[0]+cos(xn[2])*xn[1])]])
-  f = array([[0], [0], [p[2]]])
-  return rm, rmx, rmxpp, rd, rdx, rdxp, rk, rkx, f

-
-if __name__ == '__main__':
-  True
-
- -
- -
-
- -
-
-
- ©2019, Daniel Weschke. - - | - Powered by Sphinx 2.2.1 - & Alabaster 0.7.12 - -
- - - - - - \ No newline at end of file diff --git a/docs/build/html/_modules/pylib/data.html b/docs/build/html/_modules/pylib/data.html index 6c1bef2..4c0b454 100644 --- a/docs/build/html/_modules/pylib/data.html +++ b/docs/build/html/_modules/pylib/data.html @@ -4,7 +4,7 @@ - pylib.data — pylib 2019.5.19 documentation + pylib.data — pylib 2019.12.21 documentation @@ -48,7 +48,7 @@ import math import pickle -
[docs]def read(file_name, x_column, y_column, default=None, verbose=False): +
[docs]def read(file_name, x_column, y_column, default=None, verbose=False): """Read ascii data file. :param filename: file to read @@ -86,7 +86,7 @@ print('data file not found') return x, y
-
[docs]def write(file_name, data): +
[docs]def write(file_name, data): """Write ascii file. :param file_name: file to write @@ -97,7 +97,7 @@ with open(file_name, 'w') as file: file.write(data)
-
[docs]def load(file_name, default=None, verbose=False): +
[docs]def load(file_name, default=None, verbose=False): """Load stored program objects from binary file. :param file_name: file to load @@ -125,7 +125,7 @@ print('no saved datas found') return object_data
-
[docs]def store(file_name, object_data): +
[docs]def store(file_name, object_data): """Store program objects to binary file. :param file_name: file to store @@ -137,7 +137,7 @@ # every dump needs a load pickle.dump(object_data, output, pickle.HIGHEST_PROTOCOL)
-
[docs]def fold_list(lst, n): +
[docs]def fold_list(lst, n): """Convert one-dimensional kx1 array (list) to two-dimensional mxn array. m = k / n @@ -154,7 +154,7 @@ length = int(k/n) return [lst[i*n:i*n+n] for i in range(length)]
-
[docs]def seq(start, stop=None, step=1): +
[docs]def seq(start, stop=None, step=1): r"""Create an arithmetic bounded sequence. The sequence is one of the following; @@ -208,7 +208,7 @@ lst = [round(start + step*i, exponent) for i in range(n)] return lst
-
[docs]def unique_ending(ids, n=1): +
[docs]def unique_ending(ids, n=1): """From id list get list with unique ending. :param ids: ids @@ -226,7 +226,7 @@ else: return x
-
[docs]def get_id(ids, uide): +
[docs]def get_id(ids, uide): """Get full id from unique id ending. :param ids: ids @@ -257,6 +257,10 @@

Navigation

+

Contents:

+

Related Topics

diff --git a/docs/build/html/_modules/pylib/date.html b/docs/build/html/_modules/pylib/date.html index 9c6de39..39ced54 100644 --- a/docs/build/html/_modules/pylib/date.html +++ b/docs/build/html/_modules/pylib/date.html @@ -4,7 +4,7 @@ - pylib.date — pylib 2019.5.19 documentation + pylib.date — pylib 2019.12.21 documentation @@ -46,7 +46,7 @@ .. moduleauthor:: Daniel Weschke <daniel.weschke@directbox.de> """ -
[docs]def gaußsche_osterformel(year): +
[docs]def gaußsche_osterformel(year): """Gaußsche Osterformel. :param year: the year to calculate the Easter Sunday @@ -95,7 +95,7 @@ os = og + oe return os
-
[docs]def easter_sunday(year): +
[docs]def easter_sunday(year): """Easter Sunday. :param year: the year to calculate the Easter Sunday @@ -108,7 +108,7 @@ day = march + datetime.timedelta(days=gaußsche_osterformel(year)) return day
-
[docs]def easter_friday(year): +
[docs]def easter_friday(year): """Easter Friday. :param year: the year to calculate the Easter Friday @@ -120,7 +120,7 @@ day = easter_sunday(year) + datetime.timedelta(days=-2) return day
-
[docs]def easter_monday(year): +
[docs]def easter_monday(year): """Easter Monday. :param year: the year to calculate the Easter Monday @@ -132,7 +132,7 @@ day = easter_sunday(year) + datetime.timedelta(days=+1) return day
-
[docs]def ascension_of_jesus(year): +
[docs]def ascension_of_jesus(year): """Ascension of Jesus. :param year: the year to calculate the ascension of Jesus @@ -144,7 +144,7 @@ day = easter_sunday(year) + datetime.timedelta(days=+39) return day
-
[docs]def pentecost(year): +
[docs]def pentecost(year): """Pentecost. :param year: the year to calculate the Pentecost @@ -173,6 +173,10 @@

Navigation

+

Contents:

+

Related Topics

diff --git a/docs/build/html/_modules/pylib/drawblock.html b/docs/build/html/_modules/pylib/drawblock.html index d163117..e1dbd2a 100644 --- a/docs/build/html/_modules/pylib/drawblock.html +++ b/docs/build/html/_modules/pylib/drawblock.html @@ -4,7 +4,7 @@ - pylib.drawblock — pylib 2019.5.19 documentation + pylib.drawblock — pylib 2019.12.21 documentation @@ -35,12 +35,23 @@

Source code for pylib.drawblock

 #!/usr/bin/env python
 # -*- coding: utf-8 -*-
+"""Draw with block characters.
 
+:Date: 2019-11-15
 
-
[docs]def histogram(f, x=None):
+.. module:: drawblock
+  :platform: *nix, Windows
+  :synopsis: Draw with block characters.
+
+.. moduleauthor:: Daniel Weschke <daniel.weschke@directbox.de>
+"""
+
+
[docs]def histogram(f, x=None):
   """Histogram chart with block symbols.
 
-  dots:
+  ::
+
+    dots
      ,_,
      |8|
      |7|
@@ -52,8 +63,8 @@
      |1|
      ```
 
-  ▁▂▃▄▅▆▇█
-  12345678
+    ▁▂▃▄▅▆▇█
+    12345678
   """
   from collections import defaultdict
   chars = defaultdict(lambda: defaultdict(int))
@@ -176,6 +187,10 @@
 
 
 
Navigation

+
Contents:

+
 
 

 
Related Topics

diff --git a/docs/build/html/_modules/pylib/function.html b/docs/build/html/_modules/pylib/function.html
index fd5df8d..028da16 100644
--- a/docs/build/html/_modules/pylib/function.html
+++ b/docs/build/html/_modules/pylib/function.html
@@ -4,7 +4,7 @@
 
   
     
-    pylib.function — pylib 2019.5.19 documentation
+    pylib.function — pylib 2019.12.21 documentation
     
     
     
@@ -49,7 +49,7 @@
 from pylib.data import seq
 from pylib.mathematics import lcm
 
-
[docs]def transformation(f, scale_vertical=1, scale_horizontal=1,
+
[docs]def transformation(f, scale_vertical=1, scale_horizontal=1,
                    shift_horizontal=0, shift_vertical=0):
   r"""Transform functions.
 
@@ -93,7 +93,7 @@
     transformation(
       f[1:], scale_vertical=a, scale_horizontal=b, shift_horizontal=c, shift_vertical=d)

 
-
[docs]def sine_wave(A=1, k=1, f=1, phi=0, D=0, degree=False):
+
[docs]def sine_wave(A=1, k=1, f=1, phi=0, D=0, degree=False):
   r"""A sine wave or sinusoid is a mathematical curve that describes a
   smooth periodic oscillation.
 
@@ -154,7 +154,7 @@
     phi = math.radians(phi)
   return lambda x, t=0: A*math.sin(k*x + 2*math.pi*f*t + phi) + D

 
-
[docs]def cosine_wave(A=1, k=1, f=1, phi=0, D=0, degree=False):
+
[docs]def cosine_wave(A=1, k=1, f=1, phi=0, D=0, degree=False):
   r"""A cosine wave is said to be sinusoidal, because,
   :math:`\cos(x) = \sin(x + \pi/2)`, which is also a sine wave with a
   phase-shift of π/2 radians. Because of this head start, it is often
@@ -194,7 +194,7 @@
 # roulette
 #
 
-
[docs]def hypotrochoid(R, r, d):
+
[docs]def hypotrochoid(R, r, d):
   r"""Hypotrochoid
 
   A point is attached with a distance d from the center of a circle
@@ -242,7 +242,7 @@
   theta_end = 2*math.pi*lcm(r, R)/R
   return x, y, theta_end

 
-
[docs]def epitrochoid(R, r, d):
+
[docs]def epitrochoid(R, r, d):
   r"""Epitrochoid
 
   A point is attached with a distance d from the center of a circle
@@ -287,7 +287,7 @@
   theta_end = 2*math.pi
   return x, y, theta_end

 
-
[docs]def to_str(f, x=None, x_0=0, x_1=1, t=None, h=10, w=80, density=1, char_set="line"):
+
[docs]def to_str(f, x=None, x_0=0, x_1=1, t=None, h=10, w=80, density=1, char_set="line"):
   """Represent functions as string frame with a specific character set.
   which are normed to the range of [0, 1] to
 
@@ -468,6 +468,10 @@
 
 
 
Navigation

+
Contents:

+
 
 

 
Related Topics

diff --git a/docs/build/html/_modules/pylib/geometry.html b/docs/build/html/_modules/pylib/geometry.html
index 3787977..c32d4c7 100644
--- a/docs/build/html/_modules/pylib/geometry.html
+++ b/docs/build/html/_modules/pylib/geometry.html
@@ -4,7 +4,7 @@
 
   
     
-    pylib.geometry — pylib 2019.5.19 documentation
+    pylib.geometry — pylib 2019.12.21 documentation
     
     
     
@@ -37,7 +37,7 @@
 # -*- coding: utf-8 -*-
 """2D geometry objects.
 
-:Date: 2019-08-28
+:Date: 2019-12-21
 
 .. module:: geometry
   :platform: *nix, Windows
@@ -46,504 +46,218 @@
 .. moduleauthor:: Daniel Weschke <daniel.weschke@directbox.de>
 """
 import math
-import numpy as np
+import copy
+from mathematics import vector, matrix
 
+
[docs]class Direction(vector):
+  """Direction in local coordinate system"""
+  def __init__(self, x=1, y=0, z=0):
+    super().__init__([x, y, z, 0])
 
-
[docs]def distance(point1, point2):
-  """Distance between two points (or length of a straight line).
+  def __str__(self):
+    return str(self.xyz())

 
-  :param point1: first point (first end point of straight line)
-  :type point1: tuple
-  :param point2: second point (second end point of straight line)
-  :type point2: tuple
+
[docs]class Point(vector):
+  """Point in local coordinate system"""
+  def __init__(self, x=0, y=0, z=0):
+    super().__init__([x, y, z, 1])
 
-  :returns: distance between the two points
-  :rtype: float
+  def __str__(self):
+    return str(self.xyz())

+
+
[docs]class CS(matrix):
+  """Coordinate system
   """
-  return math.sqrt((point2[0]-point1[0])**2 + (point2[1]-point1[1])**2)

+  def __init__(self, x=[1, 0, 0], y=[0, 1, 0], z=[0, 0, 1]):
+    super().__init__([[*x, 0], [*y, 0], [*z, 0], [0, 0, 0, 1]])
 
+  def __str__(self):
+    return '[' + ', '.join([str(i) for i in self.get_coordinates()]) + ']'
 
-
[docs]def angle(point1, point2=None):
-  """Angle of point or between two points.
+
[docs]  @staticmethod
+  def x90():
+    return CS((1, 0, 0), (0, 0, -1), (0, 1, 0))

 
-  :param point1: (first) point
-  :type point1: tuple
-  :param point2: second point (default = None)
-  :type point2: tuple
+
[docs]  @staticmethod
+  def xm90():
+    return CS((1, 0, 0), (0, 0, 1), (0, -1, 0))

 
-  :returns: angle of point or between two points
-  :rtype: float
+
[docs]  @staticmethod
+  def y90():
+    return CS((0, 0, 1), (0, 1, 0), (-1, 0, 0))

+
+
[docs]  @staticmethod
+  def ym90():
+    return CS((0, 0, -1), (0, 1, 0), (1, 0, 0))

+
+
[docs]  def get_coordinates(self):
+    """Get coordinates in 3d space"""
+    return self[:3,:3]

+
+
[docs]class World():
+  """World-space with world-space coordinates
   """
-  if point2 is None:
-    return math.atan2(point1[1], point1[0])
-  return math.atan2(point2[1]-point1[1], point2[0]-point1[0])

+  def __init__(self):
+    self._cs = CS()      # Camera
+    self._objects = []
+    self._store_init()
 
+  def __iter__(self):
+    """Returns the Iterator object"""
+    return iter(self.objects())
 
-
[docs]def translate(vec, *pts):
-  """Translate a point or polygon by a given vector.
+  def _store_init(self):
+    """Initialize or reset calculated values, because a new object was added.
+    """
+    self._bb = None
+    self._sd = None
 
-  :param vec: translation vector
-  :type vec: tuple
-  :param `*pts`: points to translate
+
[docs]  def cs(self, cs=None):
+    if cs:
+      self._cs = cs
+    return self._cs

 
-  :returns: (point_x, point_y) or (point1, point2, ...)
-  :rtype: tuple
+
[docs]  def ch_cs(self, cs):
+    self._cs = self._cs * cs
+    return self

 
-  .. seealso::
-    :meth:`translate_xy`
+
[docs]  def rotate_x(self, theta):
+    self._cs.rotate_x(theta)
+    return self

+
+
[docs]  def rotate_y(self, theta):
+    self._cs.rotate_y(theta)
+    return self

+
+
[docs]  def rotate_z(self, theta):
+    self._cs.rotate_z(theta)
+    return self

+
+
[docs]  def translate(self, tx, ty, tz):
+    self._cs.translate(tx, ty, tz)
+    return self

+
+
[docs]  def scale(self, sx, sy=None, sz=None):
+    self._cs.scale(sx, sy, sz)
+    return self

+
+
[docs]  def objects(self):
+    return [copy.deepcopy(i).ch_cs(self._cs) for i in self._objects]

+
+
[docs]  def add(self, *objects):
+    self._store_init()  # calculated values are not correct anymore
+    [self._objects.append(i) for i in objects]
+    return self

+
+
[docs]  def bounding_box(self):
+    if self._bb is not None:
+      return self._bb
+
+    xmin = math.inf
+    ymin = math.inf
+    zmin = math.inf
+    xmax = -math.inf
+    ymax = -math.inf
+    zmax = -math.inf
+    for i in self._objects:
+      xi, yi, zi = map(min, i.xyz())
+      xs, ys, zs = map(max, i.xyz())
+      #xmax = x if x > xmax: xmax = x
+      xmin = xi if xi < xmin else xmin
+      ymin = yi if yi < ymin else ymin
+      zmin = zi if zi < zmin else zmin
+      xmax = xs if xs > xmax else xmax
+      ymax = ys if ys > ymax else ymax
+      zmax = zs if zs > zmax else zmax
+
+    self._bb = xmin, xmax, ymin, ymax, zmin, zmax
+    return self._bb

+
+
[docs]  def space_diagonal(self):
+    if self._sd is not None:
+      return self._sd
+
+    bb = self.bounding_box()
+    a, b, c = bb[1]-bb[0], bb[3]-bb[2], bb[5]-bb[4]
+    return math.sqrt(a**2+b**2+c**2)

+
+
[docs]  def center(self):
+    bb = self.bounding_box()
+    self.ch_cs([[1,0,0,-(bb[1]-bb[0])/2],[0,1,0,-(bb[3]-bb[2])/2],[0,0,1,-(bb[5]-bb[4])/2],[0,0,0,1]])
+    return self

+
+# TODO: Wireframe(list) or Wireframe(matrix) ?
+# list of Points
+
[docs]class Wireframe():
+  """Open and closed wireframe object in local coordinate system
+
+  This class create its own points (copy).
   """
-  vx, vy = vec
-  return tuple([(x+vx, y+vy) for (x, y) in pts])

-
-
-
[docs]def translate_xy(vec, x, y):
-  """Translate a point or polygon by a given vector.
-
-  :param vec: translation vector
-  :type vec: tuple
-  :param x: points to translate
-  :type x: int or float or list
-  :param y: points to translate
-  :type y: int or float or list
-
-  :returns: (x', y')
-  :rtype: tuple
-
-  .. seealso::
-    :meth:`translate`
-  """
-  vx, vy = vec
-
-  if not hasattr(x, "__len__"):
-    x = [x]
-
-  if not hasattr(y, "__len__"):
-    y = [y]
-
-  xp = [xi+vx for xi in x]
-  yp = [yi+vy for yi in y]
-
-  # no list if it is only one value
-  if len(xp) == 1:
-    return xp[0], yp[0]
-
-  return xp, yp

-
-
-
[docs]def rotate(origin, angle, *pts, **kwargs):
-  """Rotate a point or polygon counterclockwise by a given angle
-  around a given origin. The angle should be given in radians.
-
-  :param origin: the center of rotation
-  :type origin: tuple
-  :param angle: the rotation angle
-  :type angle: int or float
-  :param `*pts`: points to rotate
-  :param `**kwargs`: options
-
-  :returns: (point_x, point_y) or (point1, point2, ...)
-  :rtype: tuple
-
-  .. seealso::
-    :meth:`rotate_xy`
-  """
-  ox, oy = origin
-
-  # add first point to the end
-  if "closed" in kwargs and kwargs["closed"] is True:
-    pts += (pts[0],)
-
-  result = tuple([(ox + math.cos(angle) * (px - ox) - math.sin(angle) * (py - oy),
-                   oy + math.sin(angle) * (px - ox) + math.cos(angle) * (py - oy))
-                  for (px, py) in pts])
-
-  # no tuple in tuple if it is only one point
-  if len(pts) == 1:
-    return result[0][0], result[0][1]
-
-  return result

-
-
-
[docs]def rotate_deg(origin, angle, *pts, **kwargs):
-  """Rotate a point or polygon counterclockwise by a given angle
-  around a given origin. The angle should be given in degrees.
-
-  :param origin: the center of rotation
-  :type origin: tuple
-  :param angle: the rotation angle
-  :type angle: int or float
-  :param `*pts`: points to rotate
-  :param `**kwargs`: options
-
-  :returns: (point_x, point_y) or (point1, point2, ...)
-  :rtype: tuple
-
-  .. seealso::
-    :meth:`rotate`
-  """
-  return rotate(origin, angle*math.pi/180, *pts, **kwargs)

-
-
-
[docs]def rotate_xy(origin, angle, x, y, **kwargs):
-  """Rotate x and y coordinates counterclockwise by a given angle
-  around a given origin. The angle should be given in radians.
-
-  :param origin: the center of rotation
-  :type origin: tuple
-  :param angle: the rotation angle
-  :type angle: int or float
-  :param x: x coordinates
-  :type x: int or float or list
-  :param y: y coordinates
-  :type y: int or float or list
-  :param `**kwargs`: options
-
-  .. seealso::
-    :meth:`rotate`
-  """
-  ox, oy = origin
-
-  if not hasattr(x, "__len__"):
-    x = [x]
-
-  if not hasattr(y, "__len__"):
-    y = [y]
-
-  # add first point to the end
-  if "closed" in kwargs and kwargs["closed"] is True:
-    x.append(x[0])
-    y.append(y[0])
-
-  x_result = [ox + math.cos(angle) * (xi - ox) - math.sin(angle) * (yi - oy)
-              for xi, yi in zip(x, y)]
-
-  y_result = [oy + math.sin(angle) * (xi - ox) + math.cos(angle) * (yi - oy)
-              for xi, yi in zip(x, y)]
-
-  # no list if it is only one value
-  if len(x_result) == 1:
-    return x_result[0], y_result[0]
-
-  return x_result, y_result

-
-
-
[docs]def rectangle(width, height):
-  """\
-  :param width: the width of the rectangle
-  :type width: int or float
-  :param height: the height of the rectangle
-  :type height: int or float
-
-  :returns: (point1, point2, point3, point4)
-  :rtype: tuple
-  """
-  pt1 = (-width/2, -height/2)
-  pt2 = (width/2, -height/2)
-  pt3 = (width/2, height/2)
-  pt4 = (-width/2, height/2)
-  return pt1, pt2, pt3, pt4, pt1

-
-
-
[docs]def square(width):
-  """\
-  :param width: the edge size of the square
-  :type width: int or float
-
-  :returns: (point1, point2, point3, point4)
-  :rtype: tuple
-
-  .. seealso::
-    :meth:`rectangle`
-  """
-  return rectangle(width, width)

-
-
-
[docs]def lines(pts, **kwargs):
-  """Lines defined by a list of end points.
-
-  :param pts: list of points in absolute global coordinate system. If
-    keyword inc is given than the inc decides what the left and the
-    right end point of the line is, otherwise it is assumed that the
-    points build a solid line, that is lines between the given points
-    in given order.
-  :type pts: list
-
-  :param `**kwargs`: options:
-
-    * deformation -- list of points. Additional deformation
-      (translation) at point.
-    * factor -- factor of the deformation (default = 1).
-    * inc -- the incidence table, a list of 2 element lists. The inc
-      decides what the left and the right end point of the line is.
-    * index_offset -- starting index of lists (default = 0).
-
-  :returns: list of endpoints for each line;
-    [((point1_x, point1_y), (point2_x, point2_y)),
-    (p1, p2),
-    ...]
-  :rtype: list
-
-  .. seealso::
-    :meth:`~geometry_plot.plot_lines` of the :mod:`geometry_plot`
-    module to plot the lines
-  """
-  if 'index_offset' not in kwargs:
-    kwargs['index_offset'] = 0
-  if 'deformation' in kwargs:
-    if 'factor' not in kwargs:
-      kwargs['factor'] = 1
-    pts = [(p[0]+d[0]*kwargs['factor'], p[1]+d[1]*kwargs['factor']) for
-           p, d in zip(pts, kwargs['deformation'])]
-  if 'inc' in kwargs:
-    return [(pts[l-kwargs['index_offset']],
-             pts[r-kwargs['index_offset']]) for l, r in kwargs['inc']]
-  return list(zip(pts[:-1], pts[1:]))

-
-
-
[docs]def cubics(pts, **kwargs):
-  """Cubic lines defined by a list of two end points. The deformation
-  as displacement and rotation (radians) is defined element wise as
-  keyword argument deformation or global node wise as
-  global_deformation. The global coordinate system is xy. x in the
-  right direction and y in the top direction.
-
-  :param pts: list of points in absolute global coordinate system. If
-    keyword inc is given than the inc decides what the left and the
-    right end point of the line is, otherwise it is assumed that the
-    points build a solid line, that is lines between the given points
-    in given order.
-  :type pts_rot: list
-
-  :param `**kwargs`: options:
-
-    * deformation -- list of deformation element wise. Additional
-      deformation (translation and rotation in radians) at element
-      left and right node.
-    * rotation_plane -- rotation plane of the element wise
-      deformation defined by a string; either 'xy' or 'xz' (default
-      = 'xy'). x in the right direction and y in the top direction
-      or z in the bottom direction.
-    * global_deformation -- list of deformation global node wise.
-      Additional deformation (horizontal translation, vertical
-      translation and rotation in radians) at node.
-    * factor -- factor of the derformation (default = 1).
-    * inc -- the incidence table, a list of 2 element lists. The inc
-      decides what the left and the right end point of the line is.
-    * index_offset -- starting index of lists (default = 0).
-
-  :returns: list of endpoints for each line;
-    [(((point1_x, point1_y) angle1), ((point2_x, point2_y), angle2),
-    (p1, angle1, p2, angle2),
-    ...]
-  :rtype: list
-  """
-  if 'index_offset' not in kwargs:
-    kwargs['index_offset'] = 0
-  if 'deformation' in kwargs or 'global_deformation' in kwargs:
-    if 'factor' not in kwargs:
-      kwargs['factor'] = 1
-    if 'inc' in kwargs:
-      if 'global_deformation' in kwargs:
-        lr = [(pts[l-kwargs['index_offset']],
-               pts[r-kwargs['index_offset']]) for l, r in kwargs['inc']]
-        ang = [angle(l, r) for l, r in lr]
-        # system deformation
-        U = kwargs['global_deformation']
-        if 'rotation_plane' in kwargs and 'xz' == kwargs['rotation_plane']:
-          # system deformation left X Z RY right X Z RY element wise
-          # Z downwards -> convert to X Y RZ
-          Ue = [[ U[(l-kwargs['index_offset'])][0],
-                 -U[(l-kwargs['index_offset'])][1],
-                  U[(l-kwargs['index_offset'])][2],
-                  U[(r-kwargs['index_offset'])][0],
-                 -U[(r-kwargs['index_offset'])][1],
-                  U[(r-kwargs['index_offset'])][2]] for l, r in kwargs['inc']]
-        else:
-          # system deformation left X Y RZ right X Y RZ element wise
-          Ue = [[U[(l-kwargs['index_offset'])][0],
-                 U[(l-kwargs['index_offset'])][1],
-                 U[(l-kwargs['index_offset'])][2],
-                 U[(r-kwargs['index_offset'])][0],
-                 U[(r-kwargs['index_offset'])][1],
-                 U[(r-kwargs['index_offset'])][2]] for l, r in kwargs['inc']]
-        # element deformation, X Y RZ to x r rz
-        # back transformation T^T = [[c, s, 0], [-s, c, 0], [0, 0, 1]]
-        u = [[( math.cos(angi)*Uei[0]+math.sin(angi)*Uei[1])*kwargs['factor'],
-              (-math.sin(angi)*Uei[0]+math.cos(angi)*Uei[1])*kwargs['factor'],
-                               Uei[2]                       *kwargs['factor'],
-              ( math.cos(angi)*Uei[3]+math.sin(angi)*Uei[4])*kwargs['factor'],
-              (-math.sin(angi)*Uei[3]+math.cos(angi)*Uei[4])*kwargs['factor'],
-                               Uei[5]                       *kwargs['factor']]
-        for Uei, angi in zip(Ue, ang)]
-      else:  # deformation
-        # the deformation is in element coordinate system, therefore the angle is needed
-        if 'rotation_plane' in kwargs and 'xz' == kwargs['rotation_plane']:
-          u = [[ ue[0]*kwargs['factor'],
-                -ue[1]*kwargs['factor'],
-                 ue[2]*kwargs['factor'],
-                 ue[3]*kwargs['factor'],
-                -ue[4]*kwargs['factor'],
-                 ue[5]*kwargs['factor']] for ue in kwargs['deformation']]
-        else:
-          u = [[ui*kwargs['factor'] for ui in ue] for ue in kwargs['deformation']]
-      return [(pts[l-kwargs['index_offset']],
-               pts[r-kwargs['index_offset']],
-               d) for (l, r), d in zip(kwargs['inc'], u)]
-  return list(zip(pts[:-1], pts[1:]))

-
-
-
[docs]def interpolate_hermite(lvd, lr, rvd, rr, lhd=0, rhd=0, scale_x=1, scale_y=1, samples=10):
-  r"""Interpolate cubic line with hermite boundary conditions.
-
-  :param lvd: left vertcal deflection
-  :type lvd: int or float
-  :param lr: left rotation
-  :type lr: int or float
-  :param rvd: right vertical deflection
-  :type rvd: int or float
-  :param rr: right rotation
-  :type rr: int or float
-  :param lhd: left horizontal deformation (default = 0)
-  :type lhd: int or float
-  :param rhd: right horizontal deformation (default = 0)
-  :type rhd: int or float
-  :param scale_x: length of element (default = 1)
-  :type scale_x: int or float
-  :param scale_y: factor of the deformation (default = 1).
-    This does not change the length.
-  :type scale_y: int or float
-  :param samples: number of sampling points (default = 10)
-  :type samples: int
-
-  .. math::
-    s = \frac{x - x_1}{L} \\
-    x = s\,L + x_1
-
-  """
-  L = scale_x
-  Lp = L + rhd - lhd
-
-  # x=[0,1] in non-dimensional coordinates
-  x = np.linspace(0, 1, num=samples)
-  N1 =  1     - 3*x**2 + 2*x**3
-  N2 = (    x - 2*x**2 +   x**3)*Lp
-  N3 =          3*x**2 - 2*x**3
-  N4 = (      -   x**2 +   x**3)*Lp
-  x = x*Lp
-
-
-  # x=[0,L] in global coordinates
-  x = np.linspace(0, Lp, num=samples)
-  N1 = 1     - 3*x**2/Lp**2 + 2*x**3/Lp**3
-  N2 =     x - 2*x**2/Lp    +   x**3/Lp**2
-  N3 =         3*x**2/Lp**2 - 2*x**3/Lp**3
-  N4 =       -   x**2/Lp    +   x**3/Lp**2
-
-
-  v = N1*lvd + N2*lr + N3*rvd + N4*rr
-  x = x + lhd
-  y = v*scale_y
-
-  return x, y

-
-
-#
-# matplotlib format, return lists for x and y
-#
-
-
[docs]def line(point1, point2, samples=2):
-  """Line defined by two end points.
-
-  .. math::
-    y = \\frac{y_2-y_1}{x_2-x_1}(x-x_1) + y_1
-
-  :param point1: one end point
-  :type point1: tuple
-  :param point2: other end point
-  :type point2: tuple
-  :param samples: number of sampling points (default = 2)
-  :type samples: int
-
-  :returns: ((point1_x, point2_x), (points1_y, point2_y)) or
-    ([sample_point1_x, sample_point2_x, ...],
-    [sample_points1_y, sample_point2_y, ...])
-  :rtype: tuple
-
-  :Example:
-
-  >>> x, y = line((0, 0), (1, 0))
-  >>> print(x, y)
-  ((0, 1), (0, 0))
-  """
-  p1x, p1y = point1
-  p2x, p2y = point2
-
-  denominator = (p1x - p2x)
-
-  if samples > 2 and denominator > 0:
-    x = np.linspace(p1x, p2x, samples)
-    a = (p1y - p2y) / denominator
-    b = (p1x*p2y - p2x*p1y) / denominator
-    y = a*x + b
-    return x, y
-  return (p1x, p2x), (p1y, p2y)  # matplotlib format

-
-
-
[docs]def cubic(point1, angle1, point2, angle2, samples=10):
-  """Cubic line defined by two end points and the rotation in radians
-  at the points.
-
-  :param point1: one end point
-  :type point1: tuple
-  :param angle1: the slope at the one end point
-  :type angle1: int or float
-  :param point2: other end point
-  :type point2: tuple
-  :param angle2: the slope at the other end point
-  :type angle2: int or float
-  :param samples: number of sampling points (default = 10)
-  :type samples: int
-
-  :returns: ([sample_point1_x, sample_point2_x, ...],
-    [sample_points1_y, sample_point2_y, ...])
-  :rtype: tuple
-  """
-  p1x, p1y = point1
-  p2x, p2y = point2
-
-  x = np.linspace(p1x, p2x, num=samples)
-
-  p1ys = math.tan(angle1)
-  p2ys = math.tan(angle2)
-  a = (p1x*p1ys + p1x*p2ys - p2x*p1ys - p2x*p2ys - 2*p1y + 2*p2y)/(p1x**3 - 3*p1x**2*p2x + 3*p1x*p2x**2 - p2x**3)
-  b = (- p1x**2*p1ys - 2*p1x**2*p2ys - p1x*p2x*p1ys + p1x*p2x*p2ys + 3*p1x*p1y - 3*p1x*p2y + 2*p2x**2*p1ys + p2x**2*p2ys + 3*p2x*p1y - 3*p2x*p2y)/(p1x**3 - 3*p1x**2*p2x + 3*p1x*p2x**2 - p2x**3)
-  c = (p1x**3*p2ys + 2*p1x**2*p2x*p1ys + p1x**2*p2x*p2ys - p1x*p2x**2*p1ys - 2*p1x*p2x**2*p2ys - 6*p1x*p2x*p1y + 6*p1x*p2x*p2y - p2x**3*p1ys)/(p1x**3 - 3*p1x**2*p2x + 3*p1x*p2x**2 - p2x**3)
-  d = (- p1x**3*p2x*p2ys + p1x**3*p2y - p1x**2*p2x**2*p1ys + p1x**2*p2x**2*p2ys - 3*p1x**2*p2x*p2y + p1x*p2x**3*p1ys + 3*p1x*p2x**2*p1y - p2x**3*p1y)/(p1x**3 - 3*p1x**2*p2x + 3*p1x*p2x**2 - p2x**3)
-  y = a*x**3 + b*x**2 + c*x + d
-  return x, y

-
-
-
[docs]def cubic_deg(point1, angle1, point2, angle2):
-  """Cubic line defined by two end points and the roation in degree
-  at the points.
-
-  :param point1: one end point
-  :type point1: tuple
-  :param angle1: the slope at the one end point
-  :type angle1: int or float
-  :param point2: other end point
-  :type point2: tuple
-  :param angle2: the slope at the other end point
-  :type angle2: int or float
-
-  :returns: ([sample_point1_x, sample_point2_x, ...],
-    [sample_points1_y, sample_point2_y, ...])
-  :rtype: tuple
-
-  .. seealso::
-    :meth:`cubic`
-  """
-  return cubic(point1, angle1 * math.pi/180, point2, angle2 * math.pi/180)

+  def __init__(self, *points, closed=False):
+    self._points = [copy.copy(i) for i in points]
+    self.closed = closed
+
+  def __str__(self):
+    return '[' + ', '.join([str(point) for point in self._points]) + ']'
+
+  def __iter__(self):
+    """Returns the Iterator object"""
+    return iter(self.points())
+
+
[docs]  def points(self):
+    """Get coordinates in 3d space"""
+    result = [i for i in self._points]
+    return result if not self.closed else result + [result[0]]

+
+
[docs]  def xyz(self):
+    return zip(*[i.xyz() for i in self.points()])

+
+
[docs]  def rotate_x(self, theta):
+    self._points = [point.rotate_x(theta) for point in self._points]
+    return self

+
+
[docs]  def rotate_y(self, theta):
+    self._points = [point.rotate_y(theta) for point in self._points]
+    return self

+
+
[docs]  def rotate_z(self, theta):
+    self._points = [point.rotate_z(theta) for point in self._points]
+    return self

+
+
[docs]  def translate(self, tx, ty, tz):
+    self._points = [point.translate(tx, ty, tz) for point in self._points]
+    return self

+
+
[docs]  def scale(self, sx, sy=None, sz=None):
+    if not sy:
+      sy = sx
+      sz = sx
+    self._points = [point.scale(sx, sy, sz) for point in self._points]
+    return self

+
+
[docs]  def ch_cs(self, cs):
+    self._points = [point.ch_cs(cs) for point in self._points]
+    return self

+
+
[docs]class Line(Wireframe):
+  """Line a open wireframe object in local coordinate system"""
+  def __init__(self, point1=Point(-1, 0, 0), point2=Point(1, 0, 0)):
+    super().__init__(point1, point2)

+
+
[docs]class Polygon(Wireframe):
+  """Polygon as closed wireframe object in local coordinate system"""
+  def __init__(self, *points):
+    super().__init__(*points, closed=True)

+
+
[docs]class Circle(Polygon):
+  """Circle a closed wireframe object in local coordinate system"""
+  def __init__(self, radius=1, n=10):
+    points = []
+    for i in range(n):
+      x = radius * math.sin(i*2*math.pi/n)
+      y = radius * math.cos(i*2*math.pi/n)
+      points.append(Point(x, y, 0))
+    super().__init__(*points)
@@ -562,6 +276,10 @@

Navigation

+

Contents:

+

Related Topics

diff --git a/docs/build/html/_modules/geometry.html b/docs/build/html/_modules/pylib/geometry2d.html similarity index 90% rename from docs/build/html/_modules/geometry.html rename to docs/build/html/_modules/pylib/geometry2d.html index fcc5e19..7324af0 100644 --- a/docs/build/html/_modules/geometry.html +++ b/docs/build/html/_modules/pylib/geometry2d.html @@ -4,20 +4,20 @@ - geometry — pylib 2019.5.19 documentation - - - - - - - - + pylib.geometry2d — pylib 2019.12.21 documentation + + + + + + + + - - + + - + @@ -32,16 +32,16 @@
-

Source code for geometry

+  
Source code for pylib.geometry2d
 #!/usr/bin/env python
 # -*- coding: utf-8 -*-
 """2D geometry objects.
 
 :Date: 2019-08-28
 
-.. module:: geometry
+.. module:: geometry2d
   :platform: *nix, Windows
-  :synopsis: Geometry objects.
+  :synopsis: 2D geometry objects.
 
 .. moduleauthor:: Daniel Weschke <daniel.weschke@directbox.de>
 """
@@ -49,7 +49,7 @@
 import numpy as np
 
 
-
[docs]def distance(point1, point2):
+
[docs]def distance(point1, point2):
   """Distance between two points (or length of a straight line).
 
   :param point1: first point (first end point of straight line)
@@ -63,7 +63,7 @@
   return math.sqrt((point2[0]-point1[0])**2 + (point2[1]-point1[1])**2)

 
 
-
[docs]def angle(point1, point2=None):
+
[docs]def angle(point1, point2=None):
   """Angle of point or between two points.
 
   :param point1: (first) point
@@ -79,7 +79,7 @@
   return math.atan2(point2[1]-point1[1], point2[0]-point1[0])

 
 
-
[docs]def translate(vec, *pts):
+
[docs]def translate(vec, *pts):
   """Translate a point or polygon by a given vector.
 
   :param vec: translation vector
@@ -96,7 +96,7 @@
   return tuple([(x+vx, y+vy) for (x, y) in pts])

 
 
-
[docs]def translate_xy(vec, x, y):
+
[docs]def translate_xy(vec, x, y):
   """Translate a point or polygon by a given vector.
 
   :param vec: translation vector
@@ -130,7 +130,7 @@
   return xp, yp

 
 
-
[docs]def rotate(origin, angle, *pts, **kwargs):
+
[docs]def rotate(origin, angle, *pts, **kwargs):
   """Rotate a point or polygon counterclockwise by a given angle
   around a given origin. The angle should be given in radians.
 
@@ -164,7 +164,7 @@
   return result

 
 
-
[docs]def rotate_deg(origin, angle, *pts, **kwargs):
+
[docs]def rotate_deg(origin, angle, *pts, **kwargs):
   """Rotate a point or polygon counterclockwise by a given angle
   around a given origin. The angle should be given in degrees.
 
@@ -184,7 +184,7 @@
   return rotate(origin, angle*math.pi/180, *pts, **kwargs)

 
 
-
[docs]def rotate_xy(origin, angle, x, y, **kwargs):
+
[docs]def rotate_xy(origin, angle, x, y, **kwargs):
   """Rotate x and y coordinates counterclockwise by a given angle
   around a given origin. The angle should be given in radians.
 
@@ -227,7 +227,7 @@
   return x_result, y_result

 
 
-
[docs]def rectangle(width, height):
+
[docs]def rectangle(width, height):
   """\
   :param width: the width of the rectangle
   :type width: int or float
@@ -244,7 +244,7 @@
   return pt1, pt2, pt3, pt4, pt1

 
 
-
[docs]def square(width):
+
[docs]def square(width):
   """\
   :param width: the edge size of the square
   :type width: int or float
@@ -258,7 +258,7 @@
   return rectangle(width, width)

 
 
-
[docs]def lines(pts, **kwargs):
+
[docs]def lines(pts, **kwargs):
   """Lines defined by a list of end points.
 
   :param pts: list of points in absolute global coordinate system. If
@@ -300,7 +300,7 @@
   return list(zip(pts[:-1], pts[1:]))

 
 
-
[docs]def cubics(pts, **kwargs):
+
[docs]def cubics(pts, **kwargs):
   """Cubic lines defined by a list of two end points. The deformation
   as displacement and rotation (radians) is defined element wise as
   keyword argument deformation or global node wise as
@@ -392,7 +392,7 @@
   return list(zip(pts[:-1], pts[1:]))

 
 
-
[docs]def interpolate_hermite(lvd, lr, rvd, rr, lhd=0, rhd=0, scale_x=1, scale_y=1, samples=10):
+
[docs]def interpolate_hermite(lvd, lr, rvd, rr, lhd=0, rhd=0, scale_x=1, scale_y=1, samples=10):
   r"""Interpolate cubic line with hermite boundary conditions.
 
   :param lvd: left vertcal deflection
@@ -451,7 +451,7 @@
 # matplotlib format, return lists for x and y
 #
 
-
[docs]def line(point1, point2, samples=2):
+
[docs]def line(point1, point2, samples=2):
   """Line defined by two end points.
 
   .. math::
@@ -489,7 +489,7 @@
   return (p1x, p2x), (p1y, p2y)  # matplotlib format

 
 
-
[docs]def cubic(point1, angle1, point2, angle2, samples=10):
+
[docs]def cubic(point1, angle1, point2, angle2, samples=10):
   """Cubic line defined by two end points and the rotation in radians
   at the points.
 
@@ -523,7 +523,7 @@
   return x, y

 
 
-
[docs]def cubic_deg(point1, angle1, point2, angle2):
+
[docs]def cubic_deg(point1, angle1, point2, angle2):
   """Cubic line defined by two end points and the roation in degree
   at the points.
 
@@ -552,7 +552,7 @@
-
-static abs(a)[source]
+
+static abs(a)[source]

Return modulus parts of a complex vector

-
-static ang(a, b)[source]
+
+static ang(a, b)[source]
-
-static arg(a)[source]
+
+static arg(a)[source]

Return phase parts of a complex vector

-
-ch_cs(cs)[source]
+
+ch_cs(cs)[source]

Transform this vector from its defined coordinate system to a new coordinate system, defined by the given coordinate system (u, v and w direction vectors).

@@ -383,16 +385,16 @@ v and w direction vectors).

-
-static conjugate(a)[source]
+
+static conjugate(a)[source]

New vector object

-
-static cross(a, b)[source]
+
+static cross(a, b)[source]

Cross product

@@ -403,8 +405,8 @@ The direction of c can be found with the right-hand rule.

-
-static full(length, fill_value)[source]
+
+static full(length, fill_value)[source]

Returns a vector of length m or matrix of size m rows, n columns filled with v.

@@ -428,16 +430,16 @@ columns filled with v.

-
-static im(a)[source]
+
+static im(a)[source]

Return the imaginary parts of a complex vector

-
-static normalize(a)[source]
+
+static normalize(a)[source]

Normalize a vector (i. e. the vector has a length of 1)

@@ -450,8 +452,8 @@ columns filled with v.

-
-static ones(length)[source]
+
+static ones(length)[source]

Returns a vector of length m or matrix of size rows, n columns filled with ones.

@@ -469,8 +471,8 @@ columns filled with ones.

-
-static random(shape, lmin=0.0, lmax=1.0)[source]
+
+static random(shape, lmin=0.0, lmax=1.0)[source]

Returns a random vector of length n or matrix of size m rows, n columns filled with random numbers.

@@ -488,16 +490,16 @@ columns filled with random numbers.

-
-static re(a)[source]
+
+static re(a)[source]

Return the real parts of a complex vector

-
-rotate_x(theta)[source]
+
+rotate_x(theta)[source]

Rotation about the x dirction.

\[\begin{split}\begin{bmatrix}x' \\ y' \\ z' \\ h'\end{bmatrix} = @@ -511,8 +513,8 @@ columns filled with random numbers.

-
-rotate_y(theta)[source]
+
+rotate_y(theta)[source]

Rotation about the y dirction.

\[\begin{split}\begin{bmatrix}x' \\ y' \\ z' \\ h'\end{bmatrix} = @@ -526,8 +528,8 @@ columns filled with random numbers.

-
-rotate_z(theta)[source]
+
+rotate_z(theta)[source]

Rotation about the z dirction.

\[\begin{split}\begin{bmatrix}x' \\ y' \\ z' \\ h'\end{bmatrix} = @@ -541,8 +543,8 @@ columns filled with random numbers.

-
-scale(sx, sy=None, sz=None)[source]
+
+scale(sx, sy=None, sz=None)[source]

Scaling

uniform scaling if sx=sy=sz=s. Note that scaling happens around the origin, so objects not @@ -562,8 +564,8 @@ it should be.

-
-translate(tx, ty, tz)[source]
+
+translate(tx, ty, tz)[source]

Translation

\[\begin{split}\begin{bmatrix}x' \\ y' \\ z' \\ h'\end{bmatrix} = @@ -577,13 +579,13 @@ it should be.

-
-xyz()[source]
+
+xyz()[source]
-
-static zeros(length)[source]
+
+static zeros(length)[source]

Returns a zero vector of length m or matrix of size rows, n columns filled with zeros.

@@ -621,11 +623,24 @@ columns filled with zeros.

Navigation

+

Contents:

+

Related Topics

@@ -659,7 +674,7 @@ columns filled with zeros.

& Alabaster 0.7.12 | - Page source
diff --git a/docs/build/html/pylib.numerical.fit.html b/docs/build/html/pylib.numerical.fit.html new file mode 100644 index 0000000..d765b81 --- /dev/null +++ b/docs/build/html/pylib.numerical.fit.html @@ -0,0 +1,189 @@ + + + + + + + pylib.numerical.fit module — pylib 2019.12.21 documentation + + + + + + + + + + + + + + + + + + + + + + +
+
+
+ + +
+ +
+

pylib.numerical.fit module

+

Function and approximation.

+
+
Date
+

2019-10-15

+
+
+
+
+gauss(x, *p)[source]
+

Gauss distribution function.

+
+\[f(x)=ae^{-(x-b)^{2}/(2c^{2})}\]
+
+
Parameters
+
    +

  • x (int or float or list or numpy.ndarray) – positions where the gauss function will be calculated

    • +

  • p (list) –

    gauss parameters [a, b, c, d]:

    +
      +

    • a – amplitude (\(\int y \,\mathrm{d}x=1 \Leftrightarrow a=1/(c\sqrt{2\pi})\) )

      • +

    • b – expected value \(\mu\) (position of maximum, default = 0)

      • +

    • c – standard deviation \(\sigma\) (variance \(\sigma^2=c^2\))

      • +

    • d – vertical offset (default = 0)

      • +
    +

    • +
+
+
Returns
+

gauss values at given positions x

+
+
Return type
+

numpy.ndarray

+
+
+
+ +
+
+gauss_fit(x, y, e=None, x_fit=None, verbose=False)[source]
+

Fit Gauss distribution function to data.

+
+
Parameters
+
    +

  • x (int or float or list or numpy.ndarray) – positions

    • +

  • y (int or float or list or numpy.ndarray) – values

    • +

  • e (int or float or list or numpy.ndarray) – error values (default = None)

    • +

  • x_fit (int or float or list or numpy.ndarray) – positions of fitted function (default = None, if None then x +is used)

    • +

  • verbose (bool) – verbose information (default = False)

    • +
+
+
Returns
+

    +

  • numpy.ndarray – fitted values (y_fit)

    • +

  • numpy.ndarray – parameters of gauss distribution function (popt: +amplitude a, expected value \(\mu\), standard deviation +\(\sigma\), vertical offset d)

    • +

  • numpy.float64 – full width at half maximum (FWHM)

    • +
+

+
+
Return type
+

tuple

+
+
+
+

See also

+

gauss()

+
+
+ +
+ + +
+ +
+
+ +
+
+
+ ©2019, Daniel Weschke. + + | + Powered by Sphinx 2.2.1 + & Alabaster 0.7.12 + + | + Page source +
+ + + + + + \ No newline at end of file diff --git a/docs/build/html/pylib.numerical.html b/docs/build/html/pylib.numerical.html index 5e80ba4..687cd80 100644 --- a/docs/build/html/pylib.numerical.html +++ b/docs/build/html/pylib.numerical.html @@ -4,7 +4,7 @@ - pylib.numerical package — pylib 2019.5.19 documentation + pylib.numerical package — pylib 2019.12.21 documentation @@ -16,6 +16,8 @@ + + @@ -32,494 +34,18 @@
-
-

pylib.numerical package

+
+

pylib.numerical package

Submodules

-
-
-

pylib.numerical.fit module

-

Function and approximation.

-
-
Date
-

2019-10-15

-
-
-
-
-gauss(x, *p)[source]
-

Gauss distribution function.

-
-\[f(x)=ae^{-(x-b)^{2}/(2c^{2})}\]
-
-
Parameters
-
-
-
Returns
-

gauss values at given positions x

-
-
Return type
-

numpy.ndarray

-
-
-
- -
-
-gauss_fit(x, y, e=None, x_fit=None, verbose=False)[source]
-

Fit Gauss distribution function to data.

-
-
Parameters
-
    -

  • x (int or float or list or numpy.ndarray) – positions

    • -

  • y (int or float or list or numpy.ndarray) – values

    • -

  • e (int or float or list or numpy.ndarray) – error values (default = None)

    • -

  • x_fit (int or float or list or numpy.ndarray) – positions of fitted function (default = None, if None then x -is used)

    • -

  • verbose (bool) – verbose information (default = False)

    • -
-
-
Returns
-

    -

  • numpy.ndarray – fitted values (y_fit)

    • -

  • numpy.ndarray – parameters of gauss distribution function (popt: -amplitude a, expected value \(\mu\), standard deviation -\(\sigma\), vertical offset d)

    • -

  • numpy.float64 – full width at half maximum (FWHM)

    • -
-

-
-
Return type
-

tuple

-
-
-
-

See also

-

gauss()

-
- -
-
-

pylib.numerical.integration module

-

Numerical integration, numerical quadrature.

-

de: numerische Integration, numerische Quadratur.

-
-
Date
-

2015-10-15

-
-
-
-
-trapez(f, a=0, b=1, N=10, x=None, verbose=False, save_values=False)[source]
-

Integration of \(f(x)\) using the trapezoidal rule -(Simpson’s rule, Kepler’s rule).

-

de: Trapezregel, Simpsonregel (Thomas Simpson), Keplersche -Fassregel (Johannes Kepler)

-
-
Parameters
-
    -

  • f (function or list) – function to integrate.

    • -

  • a (float) – lower limit of integration (default = 0).

    • -

  • b (float) – upper limit of integration (default = 1).

    • -

  • N (int) – specify the number of subintervals.

    • -

  • x (list) – variable of integration, necessary if f is a list -(default = None).

    • -

  • verbose (bool) – print information (default = False)

    • -
-
-
Returns
-

the definite integral as approximated by trapezoidal -rule.

-
-
Return type
-

float

-
-
-

The trapezoidal rule approximates the integral by the area of a -trapezoid with base h=b-a and sides equal to the values of the -integrand at the two end points.

-
-\[f_n(x) = f(a)+\frac{f(b)-f(a)}{b-a}(x-a)\]
-
-\[\begin{split}I &= \int\limits_a^b f(x) \,\mathrm{d}x \\ -I &\approx \int\limits_a^b f_n(x) \,\mathrm{d}x \\ - &= \int\limits_a^b - \left( f(a)+\frac{f(b)-f(a)}{b-a}(x-a) \right) - \mathrm{d}x \\ - &= \left.\left( f(a)-a\frac{f(b)-f(a)}{b-a} \right) - x \right\vert_a^b + - \left. \frac{f(b)-f(a)}{b-a} \frac{x^2}{2} - \right\vert_a^b \\ - &= \frac{b-a}{2}\left[f(a)+f(b)\right]\end{split}\]
-

The composite trapezium rule. If the interval is divided into n -segments (not necessarily equal)

-
-\[a = x_0 \leq x_1 \leq x_2 \leq \ldots \leq x_n = b\]
-
-\[\begin{split}I &\approx \sum\limits_{i=0}^{n-1} \frac{1}{2} (x_{i+1}-x_i) -\left[f(x_{i+1})+f(x_i)\right] \\\end{split}\]
-

Special Case (Equaliy spaced base points)

-
-\[x_{i+1}-x_i = h \quad \forall i\]
-
-\[I \approx h \left\{ \frac{1}{2} \left[f(x_0)+f(x_n)\right] + - \sum\limits_{i=1}^{n-1} f(x_i) \right\}\]
-

Example

-
-\[\begin{split}I &= \int\limits_a^b f(x) \,\mathrm{d}x \\ -f(x) &= x^2 \\ -a &= 0 \\ -b &= 1\end{split}\]
-

analytical solution

-
-\[I = \int\limits_{0}^{1} x^2 \,\mathrm{d}x - = \left. \frac{1}{3} x^3 \right\vert_0^1 - = \frac{1}{3}\]
-

numerical solution

-
>>> f = lambda(x): x**2
->>> trapez(f, 0, 1, 1)
-0.5
->>> trapez(f, 0, 1, 10)
-0.3350000000000001
->>> trapez(f, 0, 1, 100)
-0.33335000000000004
-
-
-
- -
-
-

pylib.numerical.ode module

-

Numerical solver of ordinary differential equations.

-

Solves the initial value problem for systems of first order -ordinary differential equations.

-
-
Date
-

2015-09-21

-
-
-
-
-e1(f, x0, t, *p, verbose=False)[source]
-

Explicit first-order method / -(standard, or forward) Euler method / -Runge-Kutta 1st order method.

-

de: -Euler’sche Polygonzugverfahren / explizite Euler-Verfahren / -Euler-Cauchy-Verfahren / Euler-vorwärts-Verfahren

-
-
Parameters
-
    -

  • f (function) – the function to solve

    • -

  • x0 (list) – initial condition

    • -

  • t (list) – time

    • -

  • *p – parameters of the function (thickness, diameter, -…)

    • -

  • verbose (bool) – print information (default = False)

    • -
-
-
-

Approximate the solution of the initial value problem

-
-\[\begin{split}\dot{x} &= f(t,x) \\ -x(t_0) &= x_0\end{split}\]
-

Choose a value h for the size of every step and set

-
-\[t_i = t_0 + i h ~,\quad i=1,2,\ldots,n\]
-

The derivative of the solution is approximated as the forward -difference equation

-
-\[\dot{x}_i = f(t_i, x_i) = \frac{x_{i+1} - x_i}{t_{i+1}-t_i}\]
-

Therefore one step \(h\) of the Euler method from -\(t_i\) to \(t_{i+1}\) is

-
-\[\begin{split}x_{i+1} &= x_i + (t_{i+1}-t_i) f(t_i, x_i) \\ -x_{i+1} &= x_i + h f(t_i, x_i) \\\end{split}\]
-

Example 1:

-
-\[\begin{split}m\ddot{u} + d\dot{u} + ku = f(t) \\ -\ddot{u} = m^{-1}(f(t) - d\dot{u} - ku) \\\end{split}\]
-

with

-
-\[\begin{split}x_1 &= u &\quad \dot{x}_1 = \dot{u} = x_2 \\ -x_2 &= \dot{u} &\quad \dot{x}_2 = \ddot{u} \\\end{split}\]
-

becomes

-
-\[\begin{split}\dot{x}_1 &= x_2 \\ -\dot{x}_2 &= m^{-1}(f(t) - d x_2 - k x_1) \\\end{split}\]
-

or

-
-\[\begin{split}\dot{x} &= f(t,x) \\ -\begin{bmatrix} \dot{x}_1 \\ \dot{x}_2 \end{bmatrix} &= -\begin{bmatrix} x_2 \\ m^{-1}(f(t) - d x_2 - k x_1) -\end{bmatrix} \\ -&= -\begin{bmatrix} 0 \\ m^{-1} f(t) \end{bmatrix} + -\begin{bmatrix} 0 & 1 \\ -m^{-1} k & -m^{-1} d \end{bmatrix} -\begin{bmatrix} x_1 \\ x_2 \end{bmatrix}\end{split}\]
-

Example 2:

-
-\[\begin{split}m(u)\ddot{u} + d(u,\dot{u})\dot{u} + k(u)u = f(t) \\ -\ddot{u} = m^{-1}(u)(f(t) - d(u,\dot{u})\dot{u} - k(u)u) \\\end{split}\]
-

with

-
-\[\begin{split}x_1 &= u &\quad \dot{x}_1 = \dot{u} = x_2 \\ -x_2 &= \dot{u} &\quad \dot{x}_2 = \ddot{u} \\\end{split}\]
-

becomes

-
-\[\begin{split}\dot{x}_1 &= x_2 \\ -\dot{x}_2 &= - m^{-1}(x_1)(f(t) - d(x_1,x_2) x_2 - k(x_1) x_1) \\\end{split}\]
-

or

-
-\[\begin{split}\dot{x} &= f(t,x) \\ -\begin{bmatrix} \dot{x}_1 \\ \dot{x}_2 \end{bmatrix} &= -\begin{bmatrix} - x_2 \\ m^{-1}(x_1)(f(t) - d(x_1,x_2) x_2 - k(x_1) x_1) - \end{bmatrix} \\ -&= -\begin{bmatrix} 0 \\ m^{-1}(x_1) f(t) \end{bmatrix} + -\begin{bmatrix} - 0 & 1 \\ -m^{-1}(x_1) k(x_1) & -m^{-1} d(x_1,x_2) - \end{bmatrix} -\begin{bmatrix} x_1 \\ x_2 \end{bmatrix}\end{split}\]
-

The Euler method is a first-order method, which means that the -local error (error per step) is proportional to the square of -the step size, and the global error (error at a given time) is -proportional to the step size.

-
- -
-
-e2(f, x0, t, *p, verbose=False)[source]
-

Explicit second-order method / Runge-Kutta 2nd order method.

-
-
Parameters
-
    -

  • f (function) – the function to solve

    • -

  • x0 (list) – initial condition

    • -

  • t (list) – time

    • -

  • *p – parameters of the function (thickness, diameter, -…)

    • -

  • verbose (bool) – print information (default = False)

    • -
-
-
-
- -
-
-e4(f, x0, t, *p, verbose=False)[source]
-

Explicit fourth-order method / Runge-Kutta 4th order method.

-
-
Parameters
-
    -

  • f (function) – the function to solve

    • -

  • x0 (list) – initial condition

    • -

  • t (list) – time

    • -

  • *p – parameters of the function (thickness, diameter, -…)

    • -

  • verbose (bool) – print information (default = False)

    • -
-
-
-
- -
-
-fpi(f, xi, ti, ti1, *p, max_iterations=1000, tol=1e-09, verbose=False)[source]
-

Fixed-point iteration.

-
-
Parameters
-
    -

  • f (function) – the function to iterate \(f = \dot{x}(x,t)\)

    • -

  • xi (list) – initial condition \(x_i\)

    • -

  • ti (float) – time \(t_i\)

    • -

  • ti1 (float) – time \(t_{i+1}\)

    • -

  • *p – parameters of the function (thickness, diameter, -…)

    • -

  • max_iterations (int) – maximum number of iterations

    • -

  • tol (float) – tolerance against residuum \(\varepsilon\) -(default = 1e-9)

    • -

  • verbose (bool) – print information (default = False)

    • -
-
-
Returns
-

\(x_{i}\)

-
-
-
-\[x_{i,j=0} = x_{i}\]
-
-\[x_{i,j+1} = x_i + \dot{x}(x_{i,j}, t_{i+1})\cdot(t_{i+1}-t_i)\]
-
-\[\text{residuum} = \frac{\lVert x_{i,j+1}-x_{i,j}\rVert} - {\lVert x_{i,j+1} \rVert} < \varepsilon\]
-
-\[x_{i} = x_{i,j=\text{end}}\]
-
- -
-
-i1(f, x0, t, *p, max_iterations=1000, tol=1e-09, verbose=False)[source]
-

Implicite first-order method / backward Euler method.

-
-
Parameters
-
    -

  • f (function) – the function to solve

    • -

  • x0 (list) – initial condition

    • -

  • t (list) – time

    • -

  • *p – parameters of the function (thickness, diameter, -…)

    • -

  • max_iterations (int) – maximum number of iterations

    • -

  • tol (float) – tolerance against residuum (default = 1e-9)

    • -

  • verbose (bool) – print information (default = False)

    • -
-
-
-

The backward Euler method has order one and is A-stable.

-
- -
-
-newmark_newtonraphson(f, x0, xp0, xpp0, t, *p, gamma=0.5, beta=0.25, max_iterations=1000, tol=1e-09, verbose=False)[source]
-

Newmark method.

-
-
Parameters
-
    -

  • f (function) – the function to solve

    • -

  • x0 (list) – initial condition

    • -

  • xp0 (list) – initial condition

    • -

  • xpp0 (list) – initial condition

    • -

  • t (list) – time

    • -

  • *p – parameters of the function (thickness, diameter, -…)

    • -

  • gamma (float) – newmark parameter for velocity (default = 0.5)

    • -

  • beta (float) – newmark parameter for displacement (default = 0.25)

    • -

  • max_iterations (int) – maximum number of iterations

    • -

  • tol (float) – tolerance against residuum (default = 1e-9)

    • -

  • verbose (bool) – print information (default = False)

    • -
-
-
-
- -
-
-newmark_newtonraphson_rdk(fnm, x0, xp0, xpp0, t, *p, gamma=0.5, beta=0.25, max_iterations=1000, tol=1e-09, verbose=False)[source]
-

Newmark method.

-
-
Parameters
-
    -

  • f (function) – the function to solve

    • -

  • x0 (list) – initial condition

    • -

  • xp0 (list) – initial condition

    • -

  • xpp0 (list) – initial condition

    • -

  • t (list) – time

    • -

  • *p – parameters of the function (thickness, diameter, -…)

    • -

  • gamma (float) – newmark parameter for velocity (default = 0.5)

    • -

  • beta (float) – newmark parameter for displacement (default = 0.25)

    • -

  • max_iterations (int) – maximum number of iterations

    • -

  • tol (float) – tolerance against residuum (default = 1e-9)

    • -

  • verbose (bool) – print information (default = False)

    • -
-
-
-
- -
-
-

pylib.numerical.ode_model module

-

Mathmatical models governed by ordinary differential equations.

-

Describes initial value problems as systems of first order ordinary differential -equations.

-
-
Date
-

2019-05-25

-
-
-
-
-disk(x, t, *p)[source]
-

Rotation of an eccentric disk.

-
-
Parameters
-
    -

  • x (list) – values of the function

    • -

  • t (list) – time

    • -

  • *p –

    parameters of the function

    -
      -

    • diameter

      • -

    • eccentricity

      • -

    • torque

      • -
    -

    • -
-
-
-
- -
-
-disk_nm(xn, xpn, xppn, t, *p)[source]
-

Rotation of an eccentric disk.

-
-
Parameters
-
    -

  • xn (list) – values of the function

    • -

  • xpn (list) – first derivative values of the function

    • -

  • xppn (list) – second derivative values of the function

    • -

  • t (list) – time

    • -

  • *p –

    parameters of the function

    -
      -

    • diameter

      • -

    • eccentricity

      • -

    • torque

      • -
    -

    • -
-
-
-
- -
-
-disk_nmmdk(xn, xpn, xppn, t, *p)[source]
-

Rotation of an eccentric disk.

-
-
Parameters
-
    -

  • xn (list) – values of the function

    • -

  • xpn (list) – derivative values of the function

    • -

  • xppn (list) – second derivative values of the function

    • -

  • t (list) – time

    • -

  • *p –

    parameters of the function

    -
      -

    • diameter

      • -

    • eccentricity

      • -

    • torque

      • -
    -

    • -
-
-
-
- -
-
-

Module contents

@@ -540,11 +66,24 @@ equations.

Navigation

+

Contents:

+

Related Topics

diff --git a/docs/build/html/pylib.numerical.integration.html b/docs/build/html/pylib.numerical.integration.html new file mode 100644 index 0000000..bad7bc8 --- /dev/null +++ b/docs/build/html/pylib.numerical.integration.html @@ -0,0 +1,205 @@ + + + + + + + pylib.numerical.integration module — pylib 2019.12.21 documentation + + + + + + + + + + + + + + + + + + + + + + +
+
+
+ + +
+ +
+

pylib.numerical.integration module

+

Numerical integration, numerical quadrature.

+

de: numerische Integration, numerische Quadratur.

+
+
Date
+

2015-10-15

+
+
+
+
+trapez(f, a=0, b=1, N=10, x=None, verbose=False, save_values=False)[source]
+

Integration of \(f(x)\) using the trapezoidal rule +(Simpson’s rule, Kepler’s rule).

+

de: Trapezregel, Simpsonregel (Thomas Simpson), Keplersche +Fassregel (Johannes Kepler)

+
+
Parameters
+
    +

  • f (function or list) – function to integrate.

    • +

  • a (float) – lower limit of integration (default = 0).

    • +

  • b (float) – upper limit of integration (default = 1).

    • +

  • N (int) – specify the number of subintervals.

    • +

  • x (list) – variable of integration, necessary if f is a list +(default = None).

    • +

  • verbose (bool) – print information (default = False)

    • +
+
+
Returns
+

the definite integral as approximated by trapezoidal +rule.

+
+
Return type
+

float

+
+
+

The trapezoidal rule approximates the integral by the area of a +trapezoid with base h=b-a and sides equal to the values of the +integrand at the two end points.

+
+\[f_n(x) = f(a)+\frac{f(b)-f(a)}{b-a}(x-a)\]
+
+\[\begin{split}I &= \int\limits_a^b f(x) \,\mathrm{d}x \\ +I &\approx \int\limits_a^b f_n(x) \,\mathrm{d}x \\ + &= \int\limits_a^b + \left( f(a)+\frac{f(b)-f(a)}{b-a}(x-a) \right) + \mathrm{d}x \\ + &= \left.\left( f(a)-a\frac{f(b)-f(a)}{b-a} \right) + x \right\vert_a^b + + \left. \frac{f(b)-f(a)}{b-a} \frac{x^2}{2} + \right\vert_a^b \\ + &= \frac{b-a}{2}\left[f(a)+f(b)\right]\end{split}\]
+

The composite trapezium rule. If the interval is divided into n +segments (not necessarily equal)

+
+\[a = x_0 \leq x_1 \leq x_2 \leq \ldots \leq x_n = b\]
+
+\[\begin{split}I &\approx \sum\limits_{i=0}^{n-1} \frac{1}{2} (x_{i+1}-x_i) +\left[f(x_{i+1})+f(x_i)\right] \\\end{split}\]
+

Special Case (Equaliy spaced base points)

+
+\[x_{i+1}-x_i = h \quad \forall i\]
+
+\[I \approx h \left\{ \frac{1}{2} \left[f(x_0)+f(x_n)\right] + + \sum\limits_{i=1}^{n-1} f(x_i) \right\}\]
+

Example

+
+\[\begin{split}I &= \int\limits_a^b f(x) \,\mathrm{d}x \\ +f(x) &= x^2 \\ +a &= 0 \\ +b &= 1\end{split}\]
+

analytical solution

+
+\[I = \int\limits_{0}^{1} x^2 \,\mathrm{d}x + = \left. \frac{1}{3} x^3 \right\vert_0^1 + = \frac{1}{3}\]
+

numerical solution

+
>>> f = lambda(x): x**2
+>>> trapez(f, 0, 1, 1)
+0.5
+>>> trapez(f, 0, 1, 10)
+0.3350000000000001
+>>> trapez(f, 0, 1, 100)
+0.33335000000000004
+
+
+
+ +
+ + +
+ +
+
+ +
+
+
+ ©2019, Daniel Weschke. + + | + Powered by Sphinx 2.2.1 + & Alabaster 0.7.12 + + | + Page source +
+ + + + + + \ No newline at end of file diff --git a/docs/build/html/pylib.numerical.ode.html b/docs/build/html/pylib.numerical.ode.html new file mode 100644 index 0000000..93c6fa5 --- /dev/null +++ b/docs/build/html/pylib.numerical.ode.html @@ -0,0 +1,357 @@ + + + + + + + pylib.numerical.ode module — pylib 2019.12.21 documentation + + + + + + + + + + + + + + + + + + + + + + +
+
+
+ + +
+ +
+

pylib.numerical.ode module

+

Numerical solver of ordinary differential equations.

+

Solves the initial value problem for systems of first order +ordinary differential equations.

+
+
Date
+

2015-09-21

+
+
+
+
+e1(f, x0, t, *p, verbose=False)[source]
+

Explicit first-order method / +(standard, or forward) Euler method / +Runge-Kutta 1st order method.

+

de: +Euler’sche Polygonzugverfahren / explizite Euler-Verfahren / +Euler-Cauchy-Verfahren / Euler-vorwärts-Verfahren

+
+
Parameters
+
    +

  • f (function) – the function to solve

    • +

  • x0 (list) – initial condition

    • +

  • t (list) – time

    • +

  • *p – parameters of the function (thickness, diameter, +…)

    • +

  • verbose (bool) – print information (default = False)

    • +
+
+
+

Approximate the solution of the initial value problem

+
+\[\begin{split}\dot{x} &= f(t,x) \\ +x(t_0) &= x_0\end{split}\]
+

Choose a value h for the size of every step and set

+
+\[t_i = t_0 + i h ~,\quad i=1,2,\ldots,n\]
+

The derivative of the solution is approximated as the forward +difference equation

+
+\[\dot{x}_i = f(t_i, x_i) = \frac{x_{i+1} - x_i}{t_{i+1}-t_i}\]
+

Therefore one step \(h\) of the Euler method from +\(t_i\) to \(t_{i+1}\) is

+
+\[\begin{split}x_{i+1} &= x_i + (t_{i+1}-t_i) f(t_i, x_i) \\ +x_{i+1} &= x_i + h f(t_i, x_i) \\\end{split}\]
+

Example 1:

+
+\[\begin{split}m\ddot{u} + d\dot{u} + ku = f(t) \\ +\ddot{u} = m^{-1}(f(t) - d\dot{u} - ku) \\\end{split}\]
+

with

+
+\[\begin{split}x_1 &= u &\quad \dot{x}_1 = \dot{u} = x_2 \\ +x_2 &= \dot{u} &\quad \dot{x}_2 = \ddot{u} \\\end{split}\]
+

becomes

+
+\[\begin{split}\dot{x}_1 &= x_2 \\ +\dot{x}_2 &= m^{-1}(f(t) - d x_2 - k x_1) \\\end{split}\]
+

or

+
+\[\begin{split}\dot{x} &= f(t,x) \\ +\begin{bmatrix} \dot{x}_1 \\ \dot{x}_2 \end{bmatrix} &= +\begin{bmatrix} x_2 \\ m^{-1}(f(t) - d x_2 - k x_1) +\end{bmatrix} \\ +&= +\begin{bmatrix} 0 \\ m^{-1} f(t) \end{bmatrix} + +\begin{bmatrix} 0 & 1 \\ -m^{-1} k & -m^{-1} d \end{bmatrix} +\begin{bmatrix} x_1 \\ x_2 \end{bmatrix}\end{split}\]
+

Example 2:

+
+\[\begin{split}m(u)\ddot{u} + d(u,\dot{u})\dot{u} + k(u)u = f(t) \\ +\ddot{u} = m^{-1}(u)(f(t) - d(u,\dot{u})\dot{u} - k(u)u) \\\end{split}\]
+

with

+
+\[\begin{split}x_1 &= u &\quad \dot{x}_1 = \dot{u} = x_2 \\ +x_2 &= \dot{u} &\quad \dot{x}_2 = \ddot{u} \\\end{split}\]
+

becomes

+
+\[\begin{split}\dot{x}_1 &= x_2 \\ +\dot{x}_2 &= + m^{-1}(x_1)(f(t) - d(x_1,x_2) x_2 - k(x_1) x_1) \\\end{split}\]
+

or

+
+\[\begin{split}\dot{x} &= f(t,x) \\ +\begin{bmatrix} \dot{x}_1 \\ \dot{x}_2 \end{bmatrix} &= +\begin{bmatrix} + x_2 \\ m^{-1}(x_1)(f(t) - d(x_1,x_2) x_2 - k(x_1) x_1) + \end{bmatrix} \\ +&= +\begin{bmatrix} 0 \\ m^{-1}(x_1) f(t) \end{bmatrix} + +\begin{bmatrix} + 0 & 1 \\ -m^{-1}(x_1) k(x_1) & -m^{-1} d(x_1,x_2) + \end{bmatrix} +\begin{bmatrix} x_1 \\ x_2 \end{bmatrix}\end{split}\]
+

The Euler method is a first-order method, which means that the +local error (error per step) is proportional to the square of +the step size, and the global error (error at a given time) is +proportional to the step size.

+
+ +
+
+e2(f, x0, t, *p, verbose=False)[source]
+

Explicit second-order method / Runge-Kutta 2nd order method.

+
+
Parameters
+
    +

  • f (function) – the function to solve

    • +

  • x0 (list) – initial condition

    • +

  • t (list) – time

    • +

  • *p – parameters of the function (thickness, diameter, +…)

    • +

  • verbose (bool) – print information (default = False)

    • +
+
+
+
+ +
+
+e4(f, x0, t, *p, verbose=False)[source]
+

Explicit fourth-order method / Runge-Kutta 4th order method.

+
+
Parameters
+
    +

  • f (function) – the function to solve

    • +

  • x0 (list) – initial condition

    • +

  • t (list) – time

    • +

  • *p – parameters of the function (thickness, diameter, +…)

    • +

  • verbose (bool) – print information (default = False)

    • +
+
+
+
+ +
+
+fpi(f, xi, ti, ti1, *p, max_iterations=1000, tol=1e-09, verbose=False)[source]
+

Fixed-point iteration.

+
+
Parameters
+
    +

  • f (function) – the function to iterate \(f = \dot{x}(x,t)\)

    • +

  • xi (list) – initial condition \(x_i\)

    • +

  • ti (float) – time \(t_i\)

    • +

  • ti1 (float) – time \(t_{i+1}\)

    • +

  • *p – parameters of the function (thickness, diameter, +…)

    • +

  • max_iterations (int) – maximum number of iterations

    • +

  • tol (float) – tolerance against residuum \(\varepsilon\) +(default = 1e-9)

    • +

  • verbose (bool) – print information (default = False)

    • +
+
+
Returns
+

\(x_{i}\)

+
+
+
+\[x_{i,j=0} = x_{i}\]
+
+\[x_{i,j+1} = x_i + \dot{x}(x_{i,j}, t_{i+1})\cdot(t_{i+1}-t_i)\]
+
+\[\text{residuum} = \frac{\lVert x_{i,j+1}-x_{i,j}\rVert} + {\lVert x_{i,j+1} \rVert} < \varepsilon\]
+
+\[x_{i} = x_{i,j=\text{end}}\]
+
+ +
+
+i1(f, x0, t, *p, max_iterations=1000, tol=1e-09, verbose=False)[source]
+

Implicite first-order method / backward Euler method.

+
+
Parameters
+
    +

  • f (function) – the function to solve

    • +

  • x0 (list) – initial condition

    • +

  • t (list) – time

    • +

  • *p – parameters of the function (thickness, diameter, +…)

    • +

  • max_iterations (int) – maximum number of iterations

    • +

  • tol (float) – tolerance against residuum (default = 1e-9)

    • +

  • verbose (bool) – print information (default = False)

    • +
+
+
+

The backward Euler method has order one and is A-stable.

+
+ +
+
+newmark_newtonraphson(f, x0, xp0, xpp0, t, *p, gamma=0.5, beta=0.25, max_iterations=1000, tol=1e-09, verbose=False)[source]
+

Newmark method.

+
+
Parameters
+
    +

  • f (function) – the function to solve

    • +

  • x0 (list) – initial condition

    • +

  • xp0 (list) – initial condition

    • +

  • xpp0 (list) – initial condition

    • +

  • t (list) – time

    • +

  • *p – parameters of the function (thickness, diameter, +…)

    • +

  • gamma (float) – newmark parameter for velocity (default = 0.5)

    • +

  • beta (float) – newmark parameter for displacement (default = 0.25)

    • +

  • max_iterations (int) – maximum number of iterations

    • +

  • tol (float) – tolerance against residuum (default = 1e-9)

    • +

  • verbose (bool) – print information (default = False)

    • +
+
+
+
+ +
+
+newmark_newtonraphson_rdk(fnm, x0, xp0, xpp0, t, *p, gamma=0.5, beta=0.25, max_iterations=1000, tol=1e-09, verbose=False)[source]
+

Newmark method.

+
+
Parameters
+
    +

  • f (function) – the function to solve

    • +

  • x0 (list) – initial condition

    • +

  • xp0 (list) – initial condition

    • +

  • xpp0 (list) – initial condition

    • +

  • t (list) – time

    • +

  • *p – parameters of the function (thickness, diameter, +…)

    • +

  • gamma (float) – newmark parameter for velocity (default = 0.5)

    • +

  • beta (float) – newmark parameter for displacement (default = 0.25)

    • +

  • max_iterations (int) – maximum number of iterations

    • +

  • tol (float) – tolerance against residuum (default = 1e-9)

    • +

  • verbose (bool) – print information (default = False)

    • +
+
+
+
+ +
+ + +
+ +
+
+ +
+
+
+ ©2019, Daniel Weschke. + + | + Powered by Sphinx 2.2.1 + & Alabaster 0.7.12 + + | + Page source +
+ + + + + + \ No newline at end of file diff --git a/docs/build/html/pylib.numerical.ode_model.html b/docs/build/html/pylib.numerical.ode_model.html new file mode 100644 index 0000000..f03a722 --- /dev/null +++ b/docs/build/html/pylib.numerical.ode_model.html @@ -0,0 +1,194 @@ + + + + + + + pylib.numerical.ode_model module — pylib 2019.12.21 documentation + + + + + + + + + + + + + + + + + + + + + + +
+
+
+ + +
+ +
+

pylib.numerical.ode_model module

+

Mathmatical models governed by ordinary differential equations.

+

Describes initial value problems as systems of first order ordinary differential +equations.

+
+
Date
+

2019-05-25

+
+
+
+
+disk(x, t, *p)[source]
+

Rotation of an eccentric disk.

+
+
Parameters
+
    +

  • x (list) – values of the function

    • +

  • t (list) – time

    • +

  • *p –

    parameters of the function

    +
      +

    • diameter

      • +

    • eccentricity

      • +

    • torque

      • +
    +

    • +
+
+
+
+ +
+
+disk_nm(xn, xpn, xppn, t, *p)[source]
+

Rotation of an eccentric disk.

+
+
Parameters
+
    +

  • xn (list) – values of the function

    • +

  • xpn (list) – first derivative values of the function

    • +

  • xppn (list) – second derivative values of the function

    • +

  • t (list) – time

    • +

  • *p –

    parameters of the function

    +
      +

    • diameter

      • +

    • eccentricity

      • +

    • torque

      • +
    +

    • +
+
+
+
+ +
+
+disk_nmmdk(xn, xpn, xppn, t, *p)[source]
+

Rotation of an eccentric disk.

+
+
Parameters
+
    +

  • xn (list) – values of the function

    • +

  • xpn (list) – derivative values of the function

    • +

  • xppn (list) – second derivative values of the function

    • +

  • t (list) – time

    • +

  • *p –

    parameters of the function

    +
      +

    • diameter

      • +

    • eccentricity

      • +

    • torque

      • +
    +

    • +
+
+
+
+ +
+ + +
+ +
+
+ +
+
+
+ ©2019, Daniel Weschke. + + | + Powered by Sphinx 2.2.1 + & Alabaster 0.7.12 + + | + Page source +
+ + + + + + \ No newline at end of file diff --git a/docs/build/html/time_of_day.html b/docs/build/html/pylib.time_of_day.html similarity index 70% rename from docs/build/html/time_of_day.html rename to docs/build/html/pylib.time_of_day.html index 2a512ed..cc3de20 100644 --- a/docs/build/html/time_of_day.html +++ b/docs/build/html/pylib.time_of_day.html @@ -4,7 +4,7 @@ - time_of_day module — pylib 2019.5.19 documentation + pylib.time_of_day module — pylib 2019.12.21 documentation @@ -16,6 +16,7 @@ + @@ -32,8 +33,8 @@
-
-

time_of_day module

+
+

pylib.time_of_day module

Calculate time.

Date
@@ -41,8 +42,8 @@
-
-days(time)[source]
+
+days(time)[source]

The days of the time (year).

Parameters
@@ -58,8 +59,8 @@
-
-days_norm(time)[source]
+
+days_norm(time)[source]

The days normalized to 365.2425 (Gregorian, on average) days.

Parameters
@@ -75,8 +76,8 @@
-
-hours(time)[source]
+
+hours(time)[source]

The hours of the time.

Parameters
@@ -92,8 +93,8 @@
-
-hours_norm(time)[source]
+
+hours_norm(time)[source]

The hours normalized to 24 hours.

Parameters
@@ -109,8 +110,8 @@
-
-in_seconds(time)[source]
+
+in_seconds(time)[source]

If time is time.struct_time convert to float seconds.

Parameters
@@ -126,8 +127,8 @@
-
-minutes(time)[source]
+
+minutes(time)[source]

The minutes of the time.

Parameters
@@ -143,8 +144,8 @@
-
-minutes_norm(time)[source]
+
+minutes_norm(time)[source]

The minutes normalized to 60 minutes.

Parameters
@@ -160,8 +161,8 @@
-
-seconds(time)[source]
+
+seconds(time)[source]

The seconds of the time.

Parameters
@@ -177,8 +178,8 @@
-
-seconds_norm(time)[source]
+
+seconds_norm(time)[source]

The seconds normalized to 60 seconds.

Parameters
@@ -194,8 +195,8 @@
-
-transform(time_norm, length, offset=0)[source]
+
+transform(time_norm, length, offset=0)[source]

Transform normalized time value to new length.

Parameters
@@ -233,11 +234,23 @@

Navigation

+

Contents:

+

Related Topics

@@ -271,7 +284,7 @@ & Alabaster 0.7.12 | - Page source
diff --git a/docs/build/html/search.html b/docs/build/html/search.html index c56b910..a611c20 100644 --- a/docs/build/html/search.html +++ b/docs/build/html/search.html @@ -4,7 +4,7 @@ - Search — pylib 2019.5.19 documentation + Search — pylib 2019.12.21 documentation @@ -77,6 +77,10 @@

Navigation

+

Contents:

+

Related Topics

diff --git a/docs/build/html/searchindex.js b/docs/build/html/searchindex.js index 0d71a9b..2fec14d 100644 --- a/docs/build/html/searchindex.js +++ b/docs/build/html/searchindex.js @@ -1 +1 @@ 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to pylib\u2019s documentation!","pylib","pylib package","pylib.data module","pylib.date module","pylib.drawblock module","pylib.function module","pylib.geometry module","pylib.geometry2d module","pylib.geometry2d_plot module","pylib.mathematics module","pylib.numerical package","pylib.numerical.fit module","pylib.numerical.integration module","pylib.numerical.ode module","pylib.numerical.ode_model module","pylib.time_of_day module"],titleterms:{"function":6,data:3,date:4,document:0,drawblock:5,fit:12,geometri:7,geometry2d:8,geometry2d_plot:9,indic:0,integr:13,mathemat:10,modul:[3,4,5,6,7,8,9,10,12,13,14,15,16],numer:[11,12,13,14,15],ode:14,ode_model:15,packag:[2,11],pylib:[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],submodul:[2,11],subpackag:2,tabl:0,time_of_dai:16,welcom:0}}) \ No newline at end of file diff --git a/docs/source/conf.py b/docs/source/conf.py index 6f2afd2..77c126e 100644 --- a/docs/source/conf.py +++ b/docs/source/conf.py @@ -25,7 +25,7 @@ copyright = '2019, Daniel Weschke' author = 'Daniel Weschke' # The full version, including alpha/beta/rc tags -release = '2019.5.19' +release = '2019.12.21' # -- General configuration --------------------------------------------------- diff --git a/docs/source/data.rst b/docs/source/data.rst deleted file mode 100644 index a5dfafd..0000000 --- a/docs/source/data.rst +++ /dev/null @@ -1,7 +0,0 @@ -data module -=========== - -.. automodule:: data - :members: - :undoc-members: - :show-inheritance: diff --git a/docs/source/date.rst b/docs/source/date.rst deleted file mode 100644 index 67ad670..0000000 --- a/docs/source/date.rst +++ /dev/null @@ -1,7 +0,0 @@ -date module -=========== - -.. automodule:: date - :members: - :undoc-members: - :show-inheritance: diff --git a/docs/source/function.rst b/docs/source/function.rst deleted file mode 100644 index 0e0123f..0000000 --- a/docs/source/function.rst +++ /dev/null @@ -1,7 +0,0 @@ -function module -=============== - -.. automodule:: function - :members: - :undoc-members: - :show-inheritance: diff --git a/docs/source/geometry.rst b/docs/source/geometry.rst deleted file mode 100644 index 88d7478..0000000 --- a/docs/source/geometry.rst +++ /dev/null @@ -1,7 +0,0 @@ -geometry module -=============== - -.. automodule:: geometry - :members: - :undoc-members: - :show-inheritance: diff --git a/docs/source/index.rst b/docs/source/index.rst index df4eb59..e0ae45b 100644 --- a/docs/source/index.rst +++ b/docs/source/index.rst @@ -10,6 +10,8 @@ Welcome to pylib's documentation! :maxdepth: 2 :caption: Contents: + modules + Indices and tables diff --git a/docs/source/mathematics.rst b/docs/source/mathematics.rst deleted file mode 100644 index 8447d59..0000000 --- a/docs/source/mathematics.rst +++ /dev/null @@ -1,7 +0,0 @@ -mathematics module -================== - -.. automodule:: mathematics - :members: - :undoc-members: - :show-inheritance: diff --git a/docs/source/numerical.rst b/docs/source/numerical.rst deleted file mode 100644 index bdde3b2..0000000 --- a/docs/source/numerical.rst +++ /dev/null @@ -1,46 +0,0 @@ -numerical package -================= - -Submodules ----------- - -numerical.fit module --------------------- - -.. automodule:: numerical.fit - :members: - :undoc-members: - :show-inheritance: - -numerical.integration module ----------------------------- - -.. automodule:: numerical.integration - :members: - :undoc-members: - :show-inheritance: - -numerical.ode module --------------------- - -.. automodule:: numerical.ode - :members: - :undoc-members: - :show-inheritance: - -numerical.ode\_model module ---------------------------- - -.. automodule:: numerical.ode_model - :members: - :undoc-members: - :show-inheritance: - - -Module contents ---------------- - -.. automodule:: numerical - :members: - :undoc-members: - :show-inheritance: diff --git a/docs/source/pylib.data.rst b/docs/source/pylib.data.rst new file mode 100644 index 0000000..c742b93 --- /dev/null +++ b/docs/source/pylib.data.rst @@ -0,0 +1,7 @@ +pylib.data module +================= + +.. automodule:: pylib.data + :members: + :undoc-members: + :show-inheritance: diff --git a/docs/source/pylib.date.rst b/docs/source/pylib.date.rst new file mode 100644 index 0000000..df3342e --- /dev/null +++ b/docs/source/pylib.date.rst @@ -0,0 +1,7 @@ +pylib.date module +================= + +.. automodule:: pylib.date + :members: + :undoc-members: + :show-inheritance: diff --git a/docs/source/pylib.drawblock.rst b/docs/source/pylib.drawblock.rst new file mode 100644 index 0000000..9c98daf --- /dev/null +++ b/docs/source/pylib.drawblock.rst @@ -0,0 +1,7 @@ +pylib.drawblock module +====================== + +.. automodule:: pylib.drawblock + :members: + :undoc-members: + :show-inheritance: diff --git a/docs/source/pylib.function.rst b/docs/source/pylib.function.rst new file mode 100644 index 0000000..b29f9f3 --- /dev/null +++ b/docs/source/pylib.function.rst @@ -0,0 +1,7 @@ +pylib.function module +===================== + +.. automodule:: pylib.function + :members: + :undoc-members: + :show-inheritance: diff --git a/docs/source/pylib.geometry.rst b/docs/source/pylib.geometry.rst new file mode 100644 index 0000000..2914b7f --- /dev/null +++ b/docs/source/pylib.geometry.rst @@ -0,0 +1,7 @@ +pylib.geometry module +===================== + +.. automodule:: pylib.geometry + :members: + :undoc-members: + :show-inheritance: diff --git a/docs/source/pylib.geometry2d.rst b/docs/source/pylib.geometry2d.rst new file mode 100644 index 0000000..c356c63 --- /dev/null +++ b/docs/source/pylib.geometry2d.rst @@ -0,0 +1,7 @@ +pylib.geometry2d module +======================= + +.. automodule:: pylib.geometry2d + :members: + :undoc-members: + :show-inheritance: diff --git a/docs/source/pylib.geometry2d_plot.rst b/docs/source/pylib.geometry2d_plot.rst new file mode 100644 index 0000000..53dd963 --- /dev/null +++ b/docs/source/pylib.geometry2d_plot.rst @@ -0,0 +1,7 @@ +pylib.geometry2d\_plot module +============================= + +.. automodule:: pylib.geometry2d_plot + :members: + :undoc-members: + :show-inheritance: diff --git a/docs/source/pylib.mathematics.rst b/docs/source/pylib.mathematics.rst new file mode 100644 index 0000000..07741ee --- /dev/null +++ b/docs/source/pylib.mathematics.rst @@ -0,0 +1,7 @@ +pylib.mathematics module +======================== + +.. automodule:: pylib.mathematics + :members: + :undoc-members: + :show-inheritance: diff --git a/docs/source/pylib.numerical.fit.rst b/docs/source/pylib.numerical.fit.rst new file mode 100644 index 0000000..df29399 --- /dev/null +++ b/docs/source/pylib.numerical.fit.rst @@ -0,0 +1,7 @@ +pylib.numerical.fit module +========================== + +.. automodule:: pylib.numerical.fit + :members: + :undoc-members: + :show-inheritance: diff --git a/docs/source/pylib.numerical.integration.rst b/docs/source/pylib.numerical.integration.rst new file mode 100644 index 0000000..98c1f0e --- /dev/null +++ b/docs/source/pylib.numerical.integration.rst @@ -0,0 +1,7 @@ +pylib.numerical.integration module +================================== + +.. automodule:: pylib.numerical.integration + :members: + :undoc-members: + :show-inheritance: diff --git a/docs/source/pylib.numerical.ode.rst b/docs/source/pylib.numerical.ode.rst new file mode 100644 index 0000000..d4512f2 --- /dev/null +++ b/docs/source/pylib.numerical.ode.rst @@ -0,0 +1,7 @@ +pylib.numerical.ode module +========================== + +.. automodule:: pylib.numerical.ode + :members: + :undoc-members: + :show-inheritance: diff --git a/docs/source/pylib.numerical.ode_model.rst b/docs/source/pylib.numerical.ode_model.rst new file mode 100644 index 0000000..7926ffa --- /dev/null +++ b/docs/source/pylib.numerical.ode_model.rst @@ -0,0 +1,7 @@ +pylib.numerical.ode\_model module +================================= + +.. automodule:: pylib.numerical.ode_model + :members: + :undoc-members: + :show-inheritance: diff --git a/docs/source/pylib.numerical.rst b/docs/source/pylib.numerical.rst index 2222ee7..cb41396 100644 --- a/docs/source/pylib.numerical.rst +++ b/docs/source/pylib.numerical.rst @@ -1,46 +1,17 @@ pylib.numerical package ======================= -Submodules ----------- - -pylib.numerical.fit module --------------------------- - -.. automodule:: pylib.numerical.fit - :members: - :undoc-members: - :show-inheritance: - -pylib.numerical.integration module ----------------------------------- - -.. automodule:: pylib.numerical.integration - :members: - :undoc-members: - :show-inheritance: - -pylib.numerical.ode module --------------------------- - -.. automodule:: pylib.numerical.ode - :members: - :undoc-members: - :show-inheritance: - -pylib.numerical.ode\_model module ---------------------------------- - -.. automodule:: pylib.numerical.ode_model - :members: - :undoc-members: - :show-inheritance: - - -Module contents ---------------- - .. automodule:: pylib.numerical :members: :undoc-members: :show-inheritance: + +Submodules +---------- + +.. toctree:: + + pylib.numerical.fit + pylib.numerical.integration + pylib.numerical.ode + pylib.numerical.ode_model diff --git a/docs/source/pylib.rst b/docs/source/pylib.rst index af4bdd2..b66d511 100644 --- a/docs/source/pylib.rst +++ b/docs/source/pylib.rst @@ -1,6 +1,11 @@ pylib package ============= +.. automodule:: pylib + :members: + :undoc-members: + :show-inheritance: + Subpackages ----------- @@ -11,75 +16,14 @@ Subpackages Submodules ---------- -pylib.data module ------------------ +.. toctree:: -.. automodule:: pylib.data - :members: - :undoc-members: - :show-inheritance: - -pylib.date module ------------------ - -.. automodule:: pylib.date - :members: - :undoc-members: - :show-inheritance: - -pylib.drawblock module ----------------------- - -.. automodule:: pylib.drawblock - :members: - :undoc-members: - :show-inheritance: - -pylib.function module ---------------------- - -.. automodule:: pylib.function - :members: - :undoc-members: - :show-inheritance: - -pylib.geometry module ---------------------- - -.. automodule:: pylib.geometry - :members: - :undoc-members: - :show-inheritance: - -pylib.geometry\_plot module ---------------------------- - -.. automodule:: pylib.geometry_plot - :members: - :undoc-members: - :show-inheritance: - -pylib.mathematics module ------------------------- - -.. automodule:: pylib.mathematics - :members: - :undoc-members: - :show-inheritance: - -pylib.time\_of\_day module --------------------------- - -.. automodule:: pylib.time_of_day - :members: - :undoc-members: - :show-inheritance: - - -Module contents ---------------- - -.. automodule:: pylib - :members: - :undoc-members: - :show-inheritance: + pylib.data + pylib.date + pylib.drawblock + pylib.function + pylib.geometry + pylib.geometry2d + pylib.geometry2d_plot + pylib.mathematics + pylib.time_of_day diff --git a/docs/source/pylib.time_of_day.rst b/docs/source/pylib.time_of_day.rst new file mode 100644 index 0000000..a886eca --- /dev/null +++ b/docs/source/pylib.time_of_day.rst @@ -0,0 +1,7 @@ +pylib.time\_of\_day module +========================== + +.. automodule:: pylib.time_of_day + :members: + :undoc-members: + :show-inheritance: diff --git a/docs/source/time_of_day.rst b/docs/source/time_of_day.rst deleted file mode 100644 index 2b8922a..0000000 --- a/docs/source/time_of_day.rst +++ /dev/null @@ -1,7 +0,0 @@ -time\_of\_day module -==================== - -.. automodule:: time_of_day - :members: - :undoc-members: - :show-inheritance: diff --git a/pylib/drawblock.py b/pylib/drawblock.py index 30c0def..6146980 100644 --- a/pylib/drawblock.py +++ b/pylib/drawblock.py @@ -1,11 +1,22 @@ #!/usr/bin/env python # -*- coding: utf-8 -*- +"""Draw with block characters. +:Date: 2019-11-15 + +.. module:: drawblock + :platform: *nix, Windows + :synopsis: Draw with block characters. + +.. moduleauthor:: Daniel Weschke +""" def histogram(f, x=None): """Histogram chart with block symbols. - dots: + :: + + dots ,_, |8| |7| @@ -17,8 +28,8 @@ def histogram(f, x=None): |1| ``` - ▁▂▃▄▅▆▇█ - 12345678 + ▁▂▃▄▅▆▇█ + 12345678 """ from collections import defaultdict chars = defaultdict(lambda: defaultdict(int)) diff --git a/pylib/geometry.py b/pylib/geometry.py index ff457d7..41f1be8 100644 --- a/pylib/geometry.py +++ b/pylib/geometry.py @@ -2,7 +2,7 @@ # -*- coding: utf-8 -*- """2D geometry objects. -:Date: 2019-08-28 +:Date: 2019-12-21 .. module:: geometry :platform: *nix, Windows @@ -11,501 +11,215 @@ .. moduleauthor:: Daniel Weschke """ import math -import numpy as np +import copy +from mathematics import vector, matrix +class Direction(vector): + """Direction in local coordinate system""" + def __init__(self, x=1, y=0, z=0): + super().__init__([x, y, z, 0]) -def distance(point1, point2): - """Distance between two points (or length of a straight line). + def __str__(self): + return str(self.xyz()) - :param point1: first point (first end point of straight line) - :type point1: tuple - :param point2: second point (second end point of straight line) - :type point2: tuple +class Point(vector): + """Point in local coordinate system""" + def __init__(self, x=0, y=0, z=0): + super().__init__([x, y, z, 1]) - :returns: distance between the two points - :rtype: float + def __str__(self): + return str(self.xyz()) + +class CS(matrix): + """Coordinate system """ - return math.sqrt((point2[0]-point1[0])**2 + (point2[1]-point1[1])**2) + def __init__(self, x=[1, 0, 0], y=[0, 1, 0], z=[0, 0, 1]): + super().__init__([[*x, 0], [*y, 0], [*z, 0], [0, 0, 0, 1]]) + def __str__(self): + return '[' + ', '.join([str(i) for i in self.get_coordinates()]) + ']' -def angle(point1, point2=None): - """Angle of point or between two points. + @staticmethod + def x90(): + return CS((1, 0, 0), (0, 0, -1), (0, 1, 0)) - :param point1: (first) point - :type point1: tuple - :param point2: second point (default = None) - :type point2: tuple + @staticmethod + def xm90(): + return CS((1, 0, 0), (0, 0, 1), (0, -1, 0)) - :returns: angle of point or between two points - :rtype: float + @staticmethod + def y90(): + return CS((0, 0, 1), (0, 1, 0), (-1, 0, 0)) + + @staticmethod + def ym90(): + return CS((0, 0, -1), (0, 1, 0), (1, 0, 0)) + + def get_coordinates(self): + """Get coordinates in 3d space""" + return self[:3,:3] + +class World(): + """World-space with world-space coordinates """ - if point2 is None: - return math.atan2(point1[1], point1[0]) - return math.atan2(point2[1]-point1[1], point2[0]-point1[0]) + def __init__(self): + self._cs = CS() # Camera + self._objects = [] + self._store_init() + def __iter__(self): + """Returns the Iterator object""" + return iter(self.objects()) -def translate(vec, *pts): - """Translate a point or polygon by a given vector. + def _store_init(self): + """Initialize or reset calculated values, because a new object was added. + """ + self._bb = None + self._sd = None - :param vec: translation vector - :type vec: tuple - :param `*pts`: points to translate + def cs(self, cs=None): + if cs: + self._cs = cs + return self._cs - :returns: (point_x, point_y) or (point1, point2, ...) - :rtype: tuple + def ch_cs(self, cs): + self._cs = self._cs * cs + return self - .. seealso:: - :meth:`translate_xy` + def rotate_x(self, theta): + self._cs.rotate_x(theta) + return self + + def rotate_y(self, theta): + self._cs.rotate_y(theta) + return self + + def rotate_z(self, theta): + self._cs.rotate_z(theta) + return self + + def translate(self, tx, ty, tz): + self._cs.translate(tx, ty, tz) + return self + + def scale(self, sx, sy=None, sz=None): + self._cs.scale(sx, sy, sz) + return self + + def objects(self): + return [copy.deepcopy(i).ch_cs(self._cs) for i in self._objects] + + def add(self, *objects): + self._store_init() # calculated values are not correct anymore + [self._objects.append(i) for i in objects] + return self + + def bounding_box(self): + if self._bb is not None: + return self._bb + + xmin = math.inf + ymin = math.inf + zmin = math.inf + xmax = -math.inf + ymax = -math.inf + zmax = -math.inf + for i in self._objects: + xi, yi, zi = map(min, i.xyz()) + xs, ys, zs = map(max, i.xyz()) + #xmax = x if x > xmax: xmax = x + xmin = xi if xi < xmin else xmin + ymin = yi if yi < ymin else ymin + zmin = zi if zi < zmin else zmin + xmax = xs if xs > xmax else xmax + ymax = ys if ys > ymax else ymax + zmax = zs if zs > zmax else zmax + + self._bb = xmin, xmax, ymin, ymax, zmin, zmax + return self._bb + + def space_diagonal(self): + if self._sd is not None: + return self._sd + + bb = self.bounding_box() + a, b, c = bb[1]-bb[0], bb[3]-bb[2], bb[5]-bb[4] + return math.sqrt(a**2+b**2+c**2) + + def center(self): + bb = self.bounding_box() + self.ch_cs([[1,0,0,-(bb[1]-bb[0])/2],[0,1,0,-(bb[3]-bb[2])/2],[0,0,1,-(bb[5]-bb[4])/2],[0,0,0,1]]) + return self + +# TODO: Wireframe(list) or Wireframe(matrix) ? +# list of Points +class Wireframe(): + """Open and closed wireframe object in local coordinate system + + This class create its own points (copy). """ - vx, vy = vec - return tuple([(x+vx, y+vy) for (x, y) in pts]) - - -def translate_xy(vec, x, y): - """Translate a point or polygon by a given vector. - - :param vec: translation vector - :type vec: tuple - :param x: points to translate - :type x: int or float or list - :param y: points to translate - :type y: int or float or list - - :returns: (x', y') - :rtype: tuple - - .. seealso:: - :meth:`translate` - """ - vx, vy = vec - - if not hasattr(x, "__len__"): - x = [x] - - if not hasattr(y, "__len__"): - y = [y] - - xp = [xi+vx for xi in x] - yp = [yi+vy for yi in y] - - # no list if it is only one value - if len(xp) == 1: - return xp[0], yp[0] - - return xp, yp - - -def rotate(origin, angle, *pts, **kwargs): - """Rotate a point or polygon counterclockwise by a given angle - around a given origin. The angle should be given in radians. - - :param origin: the center of rotation - :type origin: tuple - :param angle: the rotation angle - :type angle: int or float - :param `*pts`: points to rotate - :param `**kwargs`: options - - :returns: (point_x, point_y) or (point1, point2, ...) - :rtype: tuple - - .. seealso:: - :meth:`rotate_xy` - """ - ox, oy = origin - - # add first point to the end - if "closed" in kwargs and kwargs["closed"] is True: - pts += (pts[0],) - - result = tuple([(ox + math.cos(angle) * (px - ox) - math.sin(angle) * (py - oy), - oy + math.sin(angle) * (px - ox) + math.cos(angle) * (py - oy)) - for (px, py) in pts]) - - # no tuple in tuple if it is only one point - if len(pts) == 1: - return result[0][0], result[0][1] - - return result - - -def rotate_deg(origin, angle, *pts, **kwargs): - """Rotate a point or polygon counterclockwise by a given angle - around a given origin. The angle should be given in degrees. - - :param origin: the center of rotation - :type origin: tuple - :param angle: the rotation angle - :type angle: int or float - :param `*pts`: points to rotate - :param `**kwargs`: options - - :returns: (point_x, point_y) or (point1, point2, ...) - :rtype: tuple - - .. seealso:: - :meth:`rotate` - """ - return rotate(origin, angle*math.pi/180, *pts, **kwargs) - - -def rotate_xy(origin, angle, x, y, **kwargs): - """Rotate x and y coordinates counterclockwise by a given angle - around a given origin. The angle should be given in radians. - - :param origin: the center of rotation - :type origin: tuple - :param angle: the rotation angle - :type angle: int or float - :param x: x coordinates - :type x: int or float or list - :param y: y coordinates - :type y: int or float or list - :param `**kwargs`: options - - .. seealso:: - :meth:`rotate` - """ - ox, oy = origin - - if not hasattr(x, "__len__"): - x = [x] - - if not hasattr(y, "__len__"): - y = [y] - - # add first point to the end - if "closed" in kwargs and kwargs["closed"] is True: - x.append(x[0]) - y.append(y[0]) - - x_result = [ox + math.cos(angle) * (xi - ox) - math.sin(angle) * (yi - oy) - for xi, yi in zip(x, y)] - - y_result = [oy + math.sin(angle) * (xi - ox) + math.cos(angle) * (yi - oy) - for xi, yi in zip(x, y)] - - # no list if it is only one value - if len(x_result) == 1: - return x_result[0], y_result[0] - - return x_result, y_result - - -def rectangle(width, height): - """\ - :param width: the width of the rectangle - :type width: int or float - :param height: the height of the rectangle - :type height: int or float - - :returns: (point1, point2, point3, point4) - :rtype: tuple - """ - pt1 = (-width/2, -height/2) - pt2 = (width/2, -height/2) - pt3 = (width/2, height/2) - pt4 = (-width/2, height/2) - return pt1, pt2, pt3, pt4, pt1 - - -def square(width): - """\ - :param width: the edge size of the square - :type width: int or float - - :returns: (point1, point2, point3, point4) - :rtype: tuple - - .. seealso:: - :meth:`rectangle` - """ - return rectangle(width, width) - - -def lines(pts, **kwargs): - """Lines defined by a list of end points. - - :param pts: list of points in absolute global coordinate system. If - keyword inc is given than the inc decides what the left and the - right end point of the line is, otherwise it is assumed that the - points build a solid line, that is lines between the given points - in given order. - :type pts: list - - :param `**kwargs`: options: - - * deformation -- list of points. Additional deformation - (translation) at point. - * factor -- factor of the deformation (default = 1). - * inc -- the incidence table, a list of 2 element lists. The inc - decides what the left and the right end point of the line is. - * index_offset -- starting index of lists (default = 0). - - :returns: list of endpoints for each line; - [((point1_x, point1_y), (point2_x, point2_y)), - (p1, p2), - ...] - :rtype: list - - .. seealso:: - :meth:`~geometry_plot.plot_lines` of the :mod:`geometry_plot` - module to plot the lines - """ - if 'index_offset' not in kwargs: - kwargs['index_offset'] = 0 - if 'deformation' in kwargs: - if 'factor' not in kwargs: - kwargs['factor'] = 1 - pts = [(p[0]+d[0]*kwargs['factor'], p[1]+d[1]*kwargs['factor']) for - p, d in zip(pts, kwargs['deformation'])] - if 'inc' in kwargs: - return [(pts[l-kwargs['index_offset']], - pts[r-kwargs['index_offset']]) for l, r in kwargs['inc']] - return list(zip(pts[:-1], pts[1:])) - - -def cubics(pts, **kwargs): - """Cubic lines defined by a list of two end points. The deformation - as displacement and rotation (radians) is defined element wise as - keyword argument deformation or global node wise as - global_deformation. The global coordinate system is xy. x in the - right direction and y in the top direction. - - :param pts: list of points in absolute global coordinate system. If - keyword inc is given than the inc decides what the left and the - right end point of the line is, otherwise it is assumed that the - points build a solid line, that is lines between the given points - in given order. - :type pts_rot: list - - :param `**kwargs`: options: - - * deformation -- list of deformation element wise. Additional - deformation (translation and rotation in radians) at element - left and right node. - * rotation_plane -- rotation plane of the element wise - deformation defined by a string; either 'xy' or 'xz' (default - = 'xy'). x in the right direction and y in the top direction - or z in the bottom direction. - * global_deformation -- list of deformation global node wise. - Additional deformation (horizontal translation, vertical - translation and rotation in radians) at node. - * factor -- factor of the derformation (default = 1). - * inc -- the incidence table, a list of 2 element lists. The inc - decides what the left and the right end point of the line is. - * index_offset -- starting index of lists (default = 0). - - :returns: list of endpoints for each line; - [(((point1_x, point1_y) angle1), ((point2_x, point2_y), angle2), - (p1, angle1, p2, angle2), - ...] - :rtype: list - """ - if 'index_offset' not in kwargs: - kwargs['index_offset'] = 0 - if 'deformation' in kwargs or 'global_deformation' in kwargs: - if 'factor' not in kwargs: - kwargs['factor'] = 1 - if 'inc' in kwargs: - if 'global_deformation' in kwargs: - lr = [(pts[l-kwargs['index_offset']], - pts[r-kwargs['index_offset']]) for l, r in kwargs['inc']] - ang = [angle(l, r) for l, r in lr] - # system deformation - U = kwargs['global_deformation'] - if 'rotation_plane' in kwargs and 'xz' == kwargs['rotation_plane']: - # system deformation left X Z RY right X Z RY element wise - # Z downwards -> convert to X Y RZ - Ue = [[ U[(l-kwargs['index_offset'])][0], - -U[(l-kwargs['index_offset'])][1], - U[(l-kwargs['index_offset'])][2], - U[(r-kwargs['index_offset'])][0], - -U[(r-kwargs['index_offset'])][1], - U[(r-kwargs['index_offset'])][2]] for l, r in kwargs['inc']] - else: - # system deformation left X Y RZ right X Y RZ element wise - Ue = [[U[(l-kwargs['index_offset'])][0], - U[(l-kwargs['index_offset'])][1], - U[(l-kwargs['index_offset'])][2], - U[(r-kwargs['index_offset'])][0], - U[(r-kwargs['index_offset'])][1], - U[(r-kwargs['index_offset'])][2]] for l, r in kwargs['inc']] - # element deformation, X Y RZ to x r rz - # back transformation T^T = [[c, s, 0], [-s, c, 0], [0, 0, 1]] - u = [[( math.cos(angi)*Uei[0]+math.sin(angi)*Uei[1])*kwargs['factor'], - (-math.sin(angi)*Uei[0]+math.cos(angi)*Uei[1])*kwargs['factor'], - Uei[2] *kwargs['factor'], - ( math.cos(angi)*Uei[3]+math.sin(angi)*Uei[4])*kwargs['factor'], - (-math.sin(angi)*Uei[3]+math.cos(angi)*Uei[4])*kwargs['factor'], - Uei[5] *kwargs['factor']] - for Uei, angi in zip(Ue, ang)] - else: # deformation - # the deformation is in element coordinate system, therefore the angle is needed - if 'rotation_plane' in kwargs and 'xz' == kwargs['rotation_plane']: - u = [[ ue[0]*kwargs['factor'], - -ue[1]*kwargs['factor'], - ue[2]*kwargs['factor'], - ue[3]*kwargs['factor'], - -ue[4]*kwargs['factor'], - ue[5]*kwargs['factor']] for ue in kwargs['deformation']] - else: - u = [[ui*kwargs['factor'] for ui in ue] for ue in kwargs['deformation']] - return [(pts[l-kwargs['index_offset']], - pts[r-kwargs['index_offset']], - d) for (l, r), d in zip(kwargs['inc'], u)] - return list(zip(pts[:-1], pts[1:])) - - -def interpolate_hermite(lvd, lr, rvd, rr, lhd=0, rhd=0, scale_x=1, scale_y=1, samples=10): - r"""Interpolate cubic line with hermite boundary conditions. - - :param lvd: left vertcal deflection - :type lvd: int or float - :param lr: left rotation - :type lr: int or float - :param rvd: right vertical deflection - :type rvd: int or float - :param rr: right rotation - :type rr: int or float - :param lhd: left horizontal deformation (default = 0) - :type lhd: int or float - :param rhd: right horizontal deformation (default = 0) - :type rhd: int or float - :param scale_x: length of element (default = 1) - :type scale_x: int or float - :param scale_y: factor of the deformation (default = 1). - This does not change the length. - :type scale_y: int or float - :param samples: number of sampling points (default = 10) - :type samples: int - - .. math:: - s = \frac{x - x_1}{L} \\ - x = s\,L + x_1 - - """ - L = scale_x - Lp = L + rhd - lhd - - # x=[0,1] in non-dimensional coordinates - x = np.linspace(0, 1, num=samples) - N1 = 1 - 3*x**2 + 2*x**3 - N2 = ( x - 2*x**2 + x**3)*Lp - N3 = 3*x**2 - 2*x**3 - N4 = ( - x**2 + x**3)*Lp - x = x*Lp - - - # x=[0,L] in global coordinates - x = np.linspace(0, Lp, num=samples) - N1 = 1 - 3*x**2/Lp**2 + 2*x**3/Lp**3 - N2 = x - 2*x**2/Lp + x**3/Lp**2 - N3 = 3*x**2/Lp**2 - 2*x**3/Lp**3 - N4 = - x**2/Lp + x**3/Lp**2 - - - v = N1*lvd + N2*lr + N3*rvd + N4*rr - x = x + lhd - y = v*scale_y - - return x, y - - -# -# matplotlib format, return lists for x and y -# - -def line(point1, point2, samples=2): - """Line defined by two end points. - - .. math:: - y = \\frac{y_2-y_1}{x_2-x_1}(x-x_1) + y_1 - - :param point1: one end point - :type point1: tuple - :param point2: other end point - :type point2: tuple - :param samples: number of sampling points (default = 2) - :type samples: int - - :returns: ((point1_x, point2_x), (points1_y, point2_y)) or - ([sample_point1_x, sample_point2_x, ...], - [sample_points1_y, sample_point2_y, ...]) - :rtype: tuple - - :Example: - - >>> x, y = line((0, 0), (1, 0)) - >>> print(x, y) - ((0, 1), (0, 0)) - """ - p1x, p1y = point1 - p2x, p2y = point2 - - denominator = (p1x - p2x) - - if samples > 2 and denominator > 0: - x = np.linspace(p1x, p2x, samples) - a = (p1y - p2y) / denominator - b = (p1x*p2y - p2x*p1y) / denominator - y = a*x + b - return x, y - return (p1x, p2x), (p1y, p2y) # matplotlib format - - -def cubic(point1, angle1, point2, angle2, samples=10): - """Cubic line defined by two end points and the rotation in radians - at the points. - - :param point1: one end point - :type point1: tuple - :param angle1: the slope at the one end point - :type angle1: int or float - :param point2: other end point - :type point2: tuple - :param angle2: the slope at the other end point - :type angle2: int or float - :param samples: number of sampling points (default = 10) - :type samples: int - - :returns: ([sample_point1_x, sample_point2_x, ...], - [sample_points1_y, sample_point2_y, ...]) - :rtype: tuple - """ - p1x, p1y = point1 - p2x, p2y = point2 - - x = np.linspace(p1x, p2x, num=samples) - - p1ys = math.tan(angle1) - p2ys = math.tan(angle2) - a = (p1x*p1ys + p1x*p2ys - p2x*p1ys - p2x*p2ys - 2*p1y + 2*p2y)/(p1x**3 - 3*p1x**2*p2x + 3*p1x*p2x**2 - p2x**3) - b = (- p1x**2*p1ys - 2*p1x**2*p2ys - p1x*p2x*p1ys + p1x*p2x*p2ys + 3*p1x*p1y - 3*p1x*p2y + 2*p2x**2*p1ys + p2x**2*p2ys + 3*p2x*p1y - 3*p2x*p2y)/(p1x**3 - 3*p1x**2*p2x + 3*p1x*p2x**2 - p2x**3) - c = (p1x**3*p2ys + 2*p1x**2*p2x*p1ys + p1x**2*p2x*p2ys - p1x*p2x**2*p1ys - 2*p1x*p2x**2*p2ys - 6*p1x*p2x*p1y + 6*p1x*p2x*p2y - p2x**3*p1ys)/(p1x**3 - 3*p1x**2*p2x + 3*p1x*p2x**2 - p2x**3) - d = (- p1x**3*p2x*p2ys + p1x**3*p2y - p1x**2*p2x**2*p1ys + p1x**2*p2x**2*p2ys - 3*p1x**2*p2x*p2y + p1x*p2x**3*p1ys + 3*p1x*p2x**2*p1y - p2x**3*p1y)/(p1x**3 - 3*p1x**2*p2x + 3*p1x*p2x**2 - p2x**3) - y = a*x**3 + b*x**2 + c*x + d - return x, y - - -def cubic_deg(point1, angle1, point2, angle2): - """Cubic line defined by two end points and the roation in degree - at the points. - - :param point1: one end point - :type point1: tuple - :param angle1: the slope at the one end point - :type angle1: int or float - :param point2: other end point - :type point2: tuple - :param angle2: the slope at the other end point - :type angle2: int or float - - :returns: ([sample_point1_x, sample_point2_x, ...], - [sample_points1_y, sample_point2_y, ...]) - :rtype: tuple - - .. seealso:: - :meth:`cubic` - """ - return cubic(point1, angle1 * math.pi/180, point2, angle2 * math.pi/180) + def __init__(self, *points, closed=False): + self._points = [copy.copy(i) for i in points] + self.closed = closed + + def __str__(self): + return '[' + ', '.join([str(point) for point in self._points]) + ']' + + def __iter__(self): + """Returns the Iterator object""" + return iter(self.points()) + + def points(self): + """Get coordinates in 3d space""" + result = [i for i in self._points] + return result if not self.closed else result + [result[0]] + + def xyz(self): + return zip(*[i.xyz() for i in self.points()]) + + def rotate_x(self, theta): + self._points = [point.rotate_x(theta) for point in self._points] + return self + + def rotate_y(self, theta): + self._points = [point.rotate_y(theta) for point in self._points] + return self + + def rotate_z(self, theta): + self._points = [point.rotate_z(theta) for point in self._points] + return self + + def translate(self, tx, ty, tz): + self._points = [point.translate(tx, ty, tz) for point in self._points] + return self + + def scale(self, sx, sy=None, sz=None): + if not sy: + sy = sx + sz = sx + self._points = [point.scale(sx, sy, sz) for point in self._points] + return self + + def ch_cs(self, cs): + self._points = [point.ch_cs(cs) for point in self._points] + return self + +class Line(Wireframe): + """Line a open wireframe object in local coordinate system""" + def __init__(self, point1=Point(-1, 0, 0), point2=Point(1, 0, 0)): + super().__init__(point1, point2) + +class Polygon(Wireframe): + """Polygon as closed wireframe object in local coordinate system""" + def __init__(self, *points): + super().__init__(*points, closed=True) + +class Circle(Polygon): + """Circle a closed wireframe object in local coordinate system""" + def __init__(self, radius=1, n=10): + points = [] + for i in range(n): + x = radius * math.sin(i*2*math.pi/n) + y = radius * math.cos(i*2*math.pi/n) + points.append(Point(x, y, 0)) + super().__init__(*points) diff --git a/pylib/geometry2d.py b/pylib/geometry2d.py new file mode 100644 index 0000000..7c733a9 --- /dev/null +++ b/pylib/geometry2d.py @@ -0,0 +1,511 @@ +#!/usr/bin/env python +# -*- coding: utf-8 -*- +"""2D geometry objects. + +:Date: 2019-08-28 + +.. module:: geometry2d + :platform: *nix, Windows + :synopsis: 2D geometry objects. + +.. moduleauthor:: Daniel Weschke +""" +import math +import numpy as np + + +def distance(point1, point2): + """Distance between two points (or length of a straight line). + + :param point1: first point (first end point of straight line) + :type point1: tuple + :param point2: second point (second end point of straight line) + :type point2: tuple + + :returns: distance between the two points + :rtype: float + """ + return math.sqrt((point2[0]-point1[0])**2 + (point2[1]-point1[1])**2) + + +def angle(point1, point2=None): + """Angle of point or between two points. + + :param point1: (first) point + :type point1: tuple + :param point2: second point (default = None) + :type point2: tuple + + :returns: angle of point or between two points + :rtype: float + """ + if point2 is None: + return math.atan2(point1[1], point1[0]) + return math.atan2(point2[1]-point1[1], point2[0]-point1[0]) + + +def translate(vec, *pts): + """Translate a point or polygon by a given vector. + + :param vec: translation vector + :type vec: tuple + :param `*pts`: points to translate + + :returns: (point_x, point_y) or (point1, point2, ...) + :rtype: tuple + + .. seealso:: + :meth:`translate_xy` + """ + vx, vy = vec + return tuple([(x+vx, y+vy) for (x, y) in pts]) + + +def translate_xy(vec, x, y): + """Translate a point or polygon by a given vector. + + :param vec: translation vector + :type vec: tuple + :param x: points to translate + :type x: int or float or list + :param y: points to translate + :type y: int or float or list + + :returns: (x', y') + :rtype: tuple + + .. seealso:: + :meth:`translate` + """ + vx, vy = vec + + if not hasattr(x, "__len__"): + x = [x] + + if not hasattr(y, "__len__"): + y = [y] + + xp = [xi+vx for xi in x] + yp = [yi+vy for yi in y] + + # no list if it is only one value + if len(xp) == 1: + return xp[0], yp[0] + + return xp, yp + + +def rotate(origin, angle, *pts, **kwargs): + """Rotate a point or polygon counterclockwise by a given angle + around a given origin. The angle should be given in radians. + + :param origin: the center of rotation + :type origin: tuple + :param angle: the rotation angle + :type angle: int or float + :param `*pts`: points to rotate + :param `**kwargs`: options + + :returns: (point_x, point_y) or (point1, point2, ...) + :rtype: tuple + + .. seealso:: + :meth:`rotate_xy` + """ + ox, oy = origin + + # add first point to the end + if "closed" in kwargs and kwargs["closed"] is True: + pts += (pts[0],) + + result = tuple([(ox + math.cos(angle) * (px - ox) - math.sin(angle) * (py - oy), + oy + math.sin(angle) * (px - ox) + math.cos(angle) * (py - oy)) + for (px, py) in pts]) + + # no tuple in tuple if it is only one point + if len(pts) == 1: + return result[0][0], result[0][1] + + return result + + +def rotate_deg(origin, angle, *pts, **kwargs): + """Rotate a point or polygon counterclockwise by a given angle + around a given origin. The angle should be given in degrees. + + :param origin: the center of rotation + :type origin: tuple + :param angle: the rotation angle + :type angle: int or float + :param `*pts`: points to rotate + :param `**kwargs`: options + + :returns: (point_x, point_y) or (point1, point2, ...) + :rtype: tuple + + .. seealso:: + :meth:`rotate` + """ + return rotate(origin, angle*math.pi/180, *pts, **kwargs) + + +def rotate_xy(origin, angle, x, y, **kwargs): + """Rotate x and y coordinates counterclockwise by a given angle + around a given origin. The angle should be given in radians. + + :param origin: the center of rotation + :type origin: tuple + :param angle: the rotation angle + :type angle: int or float + :param x: x coordinates + :type x: int or float or list + :param y: y coordinates + :type y: int or float or list + :param `**kwargs`: options + + .. seealso:: + :meth:`rotate` + """ + ox, oy = origin + + if not hasattr(x, "__len__"): + x = [x] + + if not hasattr(y, "__len__"): + y = [y] + + # add first point to the end + if "closed" in kwargs and kwargs["closed"] is True: + x.append(x[0]) + y.append(y[0]) + + x_result = [ox + math.cos(angle) * (xi - ox) - math.sin(angle) * (yi - oy) + for xi, yi in zip(x, y)] + + y_result = [oy + math.sin(angle) * (xi - ox) + math.cos(angle) * (yi - oy) + for xi, yi in zip(x, y)] + + # no list if it is only one value + if len(x_result) == 1: + return x_result[0], y_result[0] + + return x_result, y_result + + +def rectangle(width, height): + """\ + :param width: the width of the rectangle + :type width: int or float + :param height: the height of the rectangle + :type height: int or float + + :returns: (point1, point2, point3, point4) + :rtype: tuple + """ + pt1 = (-width/2, -height/2) + pt2 = (width/2, -height/2) + pt3 = (width/2, height/2) + pt4 = (-width/2, height/2) + return pt1, pt2, pt3, pt4, pt1 + + +def square(width): + """\ + :param width: the edge size of the square + :type width: int or float + + :returns: (point1, point2, point3, point4) + :rtype: tuple + + .. seealso:: + :meth:`rectangle` + """ + return rectangle(width, width) + + +def lines(pts, **kwargs): + """Lines defined by a list of end points. + + :param pts: list of points in absolute global coordinate system. If + keyword inc is given than the inc decides what the left and the + right end point of the line is, otherwise it is assumed that the + points build a solid line, that is lines between the given points + in given order. + :type pts: list + + :param `**kwargs`: options: + + * deformation -- list of points. Additional deformation + (translation) at point. + * factor -- factor of the deformation (default = 1). + * inc -- the incidence table, a list of 2 element lists. The inc + decides what the left and the right end point of the line is. + * index_offset -- starting index of lists (default = 0). + + :returns: list of endpoints for each line; + [((point1_x, point1_y), (point2_x, point2_y)), + (p1, p2), + ...] + :rtype: list + + .. seealso:: + :meth:`~geometry_plot.plot_lines` of the :mod:`geometry_plot` + module to plot the lines + """ + if 'index_offset' not in kwargs: + kwargs['index_offset'] = 0 + if 'deformation' in kwargs: + if 'factor' not in kwargs: + kwargs['factor'] = 1 + pts = [(p[0]+d[0]*kwargs['factor'], p[1]+d[1]*kwargs['factor']) for + p, d in zip(pts, kwargs['deformation'])] + if 'inc' in kwargs: + return [(pts[l-kwargs['index_offset']], + pts[r-kwargs['index_offset']]) for l, r in kwargs['inc']] + return list(zip(pts[:-1], pts[1:])) + + +def cubics(pts, **kwargs): + """Cubic lines defined by a list of two end points. The deformation + as displacement and rotation (radians) is defined element wise as + keyword argument deformation or global node wise as + global_deformation. The global coordinate system is xy. x in the + right direction and y in the top direction. + + :param pts: list of points in absolute global coordinate system. If + keyword inc is given than the inc decides what the left and the + right end point of the line is, otherwise it is assumed that the + points build a solid line, that is lines between the given points + in given order. + :type pts_rot: list + + :param `**kwargs`: options: + + * deformation -- list of deformation element wise. Additional + deformation (translation and rotation in radians) at element + left and right node. + * rotation_plane -- rotation plane of the element wise + deformation defined by a string; either 'xy' or 'xz' (default + = 'xy'). x in the right direction and y in the top direction + or z in the bottom direction. + * global_deformation -- list of deformation global node wise. + Additional deformation (horizontal translation, vertical + translation and rotation in radians) at node. + * factor -- factor of the derformation (default = 1). + * inc -- the incidence table, a list of 2 element lists. The inc + decides what the left and the right end point of the line is. + * index_offset -- starting index of lists (default = 0). + + :returns: list of endpoints for each line; + [(((point1_x, point1_y) angle1), ((point2_x, point2_y), angle2), + (p1, angle1, p2, angle2), + ...] + :rtype: list + """ + if 'index_offset' not in kwargs: + kwargs['index_offset'] = 0 + if 'deformation' in kwargs or 'global_deformation' in kwargs: + if 'factor' not in kwargs: + kwargs['factor'] = 1 + if 'inc' in kwargs: + if 'global_deformation' in kwargs: + lr = [(pts[l-kwargs['index_offset']], + pts[r-kwargs['index_offset']]) for l, r in kwargs['inc']] + ang = [angle(l, r) for l, r in lr] + # system deformation + U = kwargs['global_deformation'] + if 'rotation_plane' in kwargs and 'xz' == kwargs['rotation_plane']: + # system deformation left X Z RY right X Z RY element wise + # Z downwards -> convert to X Y RZ + Ue = [[ U[(l-kwargs['index_offset'])][0], + -U[(l-kwargs['index_offset'])][1], + U[(l-kwargs['index_offset'])][2], + U[(r-kwargs['index_offset'])][0], + -U[(r-kwargs['index_offset'])][1], + U[(r-kwargs['index_offset'])][2]] for l, r in kwargs['inc']] + else: + # system deformation left X Y RZ right X Y RZ element wise + Ue = [[U[(l-kwargs['index_offset'])][0], + U[(l-kwargs['index_offset'])][1], + U[(l-kwargs['index_offset'])][2], + U[(r-kwargs['index_offset'])][0], + U[(r-kwargs['index_offset'])][1], + U[(r-kwargs['index_offset'])][2]] for l, r in kwargs['inc']] + # element deformation, X Y RZ to x r rz + # back transformation T^T = [[c, s, 0], [-s, c, 0], [0, 0, 1]] + u = [[( math.cos(angi)*Uei[0]+math.sin(angi)*Uei[1])*kwargs['factor'], + (-math.sin(angi)*Uei[0]+math.cos(angi)*Uei[1])*kwargs['factor'], + Uei[2] *kwargs['factor'], + ( math.cos(angi)*Uei[3]+math.sin(angi)*Uei[4])*kwargs['factor'], + (-math.sin(angi)*Uei[3]+math.cos(angi)*Uei[4])*kwargs['factor'], + Uei[5] *kwargs['factor']] + for Uei, angi in zip(Ue, ang)] + else: # deformation + # the deformation is in element coordinate system, therefore the angle is needed + if 'rotation_plane' in kwargs and 'xz' == kwargs['rotation_plane']: + u = [[ ue[0]*kwargs['factor'], + -ue[1]*kwargs['factor'], + ue[2]*kwargs['factor'], + ue[3]*kwargs['factor'], + -ue[4]*kwargs['factor'], + ue[5]*kwargs['factor']] for ue in kwargs['deformation']] + else: + u = [[ui*kwargs['factor'] for ui in ue] for ue in kwargs['deformation']] + return [(pts[l-kwargs['index_offset']], + pts[r-kwargs['index_offset']], + d) for (l, r), d in zip(kwargs['inc'], u)] + return list(zip(pts[:-1], pts[1:])) + + +def interpolate_hermite(lvd, lr, rvd, rr, lhd=0, rhd=0, scale_x=1, scale_y=1, samples=10): + r"""Interpolate cubic line with hermite boundary conditions. + + :param lvd: left vertcal deflection + :type lvd: int or float + :param lr: left rotation + :type lr: int or float + :param rvd: right vertical deflection + :type rvd: int or float + :param rr: right rotation + :type rr: int or float + :param lhd: left horizontal deformation (default = 0) + :type lhd: int or float + :param rhd: right horizontal deformation (default = 0) + :type rhd: int or float + :param scale_x: length of element (default = 1) + :type scale_x: int or float + :param scale_y: factor of the deformation (default = 1). + This does not change the length. + :type scale_y: int or float + :param samples: number of sampling points (default = 10) + :type samples: int + + .. math:: + s = \frac{x - x_1}{L} \\ + x = s\,L + x_1 + + """ + L = scale_x + Lp = L + rhd - lhd + + # x=[0,1] in non-dimensional coordinates + x = np.linspace(0, 1, num=samples) + N1 = 1 - 3*x**2 + 2*x**3 + N2 = ( x - 2*x**2 + x**3)*Lp + N3 = 3*x**2 - 2*x**3 + N4 = ( - x**2 + x**3)*Lp + x = x*Lp + + + # x=[0,L] in global coordinates + x = np.linspace(0, Lp, num=samples) + N1 = 1 - 3*x**2/Lp**2 + 2*x**3/Lp**3 + N2 = x - 2*x**2/Lp + x**3/Lp**2 + N3 = 3*x**2/Lp**2 - 2*x**3/Lp**3 + N4 = - x**2/Lp + x**3/Lp**2 + + + v = N1*lvd + N2*lr + N3*rvd + N4*rr + x = x + lhd + y = v*scale_y + + return x, y + + +# +# matplotlib format, return lists for x and y +# + +def line(point1, point2, samples=2): + """Line defined by two end points. + + .. math:: + y = \\frac{y_2-y_1}{x_2-x_1}(x-x_1) + y_1 + + :param point1: one end point + :type point1: tuple + :param point2: other end point + :type point2: tuple + :param samples: number of sampling points (default = 2) + :type samples: int + + :returns: ((point1_x, point2_x), (points1_y, point2_y)) or + ([sample_point1_x, sample_point2_x, ...], + [sample_points1_y, sample_point2_y, ...]) + :rtype: tuple + + :Example: + + >>> x, y = line((0, 0), (1, 0)) + >>> print(x, y) + ((0, 1), (0, 0)) + """ + p1x, p1y = point1 + p2x, p2y = point2 + + denominator = (p1x - p2x) + + if samples > 2 and denominator > 0: + x = np.linspace(p1x, p2x, samples) + a = (p1y - p2y) / denominator + b = (p1x*p2y - p2x*p1y) / denominator + y = a*x + b + return x, y + return (p1x, p2x), (p1y, p2y) # matplotlib format + + +def cubic(point1, angle1, point2, angle2, samples=10): + """Cubic line defined by two end points and the rotation in radians + at the points. + + :param point1: one end point + :type point1: tuple + :param angle1: the slope at the one end point + :type angle1: int or float + :param point2: other end point + :type point2: tuple + :param angle2: the slope at the other end point + :type angle2: int or float + :param samples: number of sampling points (default = 10) + :type samples: int + + :returns: ([sample_point1_x, sample_point2_x, ...], + [sample_points1_y, sample_point2_y, ...]) + :rtype: tuple + """ + p1x, p1y = point1 + p2x, p2y = point2 + + x = np.linspace(p1x, p2x, num=samples) + + p1ys = math.tan(angle1) + p2ys = math.tan(angle2) + a = (p1x*p1ys + p1x*p2ys - p2x*p1ys - p2x*p2ys - 2*p1y + 2*p2y)/(p1x**3 - 3*p1x**2*p2x + 3*p1x*p2x**2 - p2x**3) + b = (- p1x**2*p1ys - 2*p1x**2*p2ys - p1x*p2x*p1ys + p1x*p2x*p2ys + 3*p1x*p1y - 3*p1x*p2y + 2*p2x**2*p1ys + p2x**2*p2ys + 3*p2x*p1y - 3*p2x*p2y)/(p1x**3 - 3*p1x**2*p2x + 3*p1x*p2x**2 - p2x**3) + c = (p1x**3*p2ys + 2*p1x**2*p2x*p1ys + p1x**2*p2x*p2ys - p1x*p2x**2*p1ys - 2*p1x*p2x**2*p2ys - 6*p1x*p2x*p1y + 6*p1x*p2x*p2y - p2x**3*p1ys)/(p1x**3 - 3*p1x**2*p2x + 3*p1x*p2x**2 - p2x**3) + d = (- p1x**3*p2x*p2ys + p1x**3*p2y - p1x**2*p2x**2*p1ys + p1x**2*p2x**2*p2ys - 3*p1x**2*p2x*p2y + p1x*p2x**3*p1ys + 3*p1x*p2x**2*p1y - p2x**3*p1y)/(p1x**3 - 3*p1x**2*p2x + 3*p1x*p2x**2 - p2x**3) + y = a*x**3 + b*x**2 + c*x + d + return x, y + + +def cubic_deg(point1, angle1, point2, angle2): + """Cubic line defined by two end points and the roation in degree + at the points. + + :param point1: one end point + :type point1: tuple + :param angle1: the slope at the one end point + :type angle1: int or float + :param point2: other end point + :type point2: tuple + :param angle2: the slope at the other end point + :type angle2: int or float + + :returns: ([sample_point1_x, sample_point2_x, ...], + [sample_points1_y, sample_point2_y, ...]) + :rtype: tuple + + .. seealso:: + :meth:`cubic` + """ + return cubic(point1, angle1 * math.pi/180, point2, angle2 * math.pi/180) diff --git a/pylib/geometry_plot.py b/pylib/geometry2d_plot.py similarity index 96% rename from pylib/geometry_plot.py rename to pylib/geometry2d_plot.py index 11c81bb..7f5aa65 100644 --- a/pylib/geometry_plot.py +++ b/pylib/geometry2d_plot.py @@ -4,15 +4,14 @@ :Date: 2019-08-20 -.. module:: geometry_plot +.. module:: geometry2d_plot :platform: *nix, Windows - :synopsis: Geometry plotting. + :synopsis: 2D geometry plotting. .. moduleauthor:: Daniel Weschke """ -import math import pylab -from geometry import ( +from geometry2d import ( distance, angle, line, interpolate_hermite, rotate_xy, translate_xy ) diff --git a/tests/test_geometry.py b/tests/test_geometry.py index d73c1da..df9cf09 100644 --- a/tests/test_geometry.py +++ b/tests/test_geometry.py @@ -1,6 +1,6 @@ """Test of geometry module. -:Date: 2019-08-20 +:Date: 2019-12-21 .. module:: test_geometry :platform: *nix, Windows @@ -13,247 +13,95 @@ import unittest import os import sys import math -from numpy import allclose sys.path.insert(0, os.path.abspath('../pylib')) -from data import fold_list -from geometry import translate_xy, rotate_xy, interpolate_hermite, lines, cubics -from geometry_plot import plot_lines, plot_cubic_lines - +from mathematics import vector +from geometry import Direction, Point, CS, Wireframe, Polygon class TestGeometry(unittest.TestCase): - global_points = [(0, 0), - (0, 3), - (3, 3), - (3, 0), - (3, 6), - (5, 6), - (5, 3), - (5, 0), - (7, 0)] - inc = [[1, 2], - [2, 3], - [4, 3], - [3, 5], - [5, 6], - [6, 7], - [7, 8], - [9, 7]] - # global deformation, horizontal right, vertical down, rotation - U = [ - 0.00000, 0.00000, -5.02012, - 12.77012, 0.06333, -2.72988, - 12.69350, 0.08599, 3.14685, - 0.00000, 0.00000, -7.92017, - -12.39817, 0.17450, 4.14446, - -12.43490, 0.32001, -3.10697, - -26.77684, 0.16528, 2.98214, - 0.00000, 0.00000, 0.00000, - -26.50991, 0.00000, -1.36053, - ] - U = [Ui/1000 for Ui in U] + def test_Direction(self): + p = Direction() + self.assertIsInstance( + p, list) + self.assertEqual( + p, [1, 0, 0, 0]) + self.assertEqual( + Direction(2, 3, 4), [2, 3, 4, 0]) - # global deformation, only horizontal right and vertical down - U2 = [ui for i, ui in enumerate(U) if i%3 != 2] + self.assertIsInstance( + p, vector) + self.assertEqual( + p, vector([1, 0, 0, 0])) + self.assertEqual( + Direction(2, 3, 4), vector([2, 3, 4, 0])) - # global deformation, horizontal right, vertical up, rotation - UN = [Ui if i%3 != 1 else -Ui for i, Ui in enumerate(U)] + def test_Direction_xyz(self): + self.assertEqual( + Direction().xyz(), [1, 0, 0]) + self.assertEqual( + Direction(1, -1, 1).xyz(), [1, -1, 1]) - # local deformation, horizontal right, vertical down, rotation - u = [ - [ 0.000000, 0.000000, -0.005020, -0.000063, 0.012770, -0.002730], - [ 0.012770, 0.000063, -0.002730, 0.012693, 0.000086, 0.003147], - [ 0.000000, 0.000000, -0.007920, -0.000086, 0.012693, 0.003147], - [-0.000086, 0.012693, 0.003147, -0.000175, -0.012398, 0.004144], - [-0.012398, 0.000175, 0.004144, -0.012435, 0.000320, -0.003107], - [ 0.000320, 0.012435, -0.003107, 0.000165, 0.026777, 0.002982], - [ 0.000165, 0.026777, 0.002982, 0.000000, 0.000000, 0.000000], - [ 0.014705, -0.022058, -0.001361, 0.014716, -0.022371, 0.002982], - ] + def test_Point(self): + p = Point() + self.assertIsInstance( + p, list) + self.assertEqual( + p, [0, 0, 0, 1]) + self.assertEqual( + Point(2, 3, 4), [2, 3, 4, 1]) - # local deformation, horizontal right, vertical up, rotation - uN = [[uii if i%3 != 1 else -uii for i, uii in enumerate(ui)] for ui in u] + self.assertIsInstance( + p, vector) + self.assertEqual( + p, vector([0, 0, 0, 1])) + self.assertEqual( + Point(2, 3, 4), vector([2, 3, 4, 1])) - # exaggeration factor for the deformation - factor = 20 + def test_Point_xyz(self): + self.assertEqual( + Point().xyz(), [0, 0, 0]) + self.assertEqual( + Point(1, -1, 1).xyz(), [1, -1, 1]) - # test of cubic lines - test_lns_l = [((0, 0), (0, 3), [0.0, -0.0, -0.1004, -0.00126, -0.2554, -0.0546]), - ((0, 3), (3, 3), [0.2554, -0.00126, -0.0546, 0.25386, -0.00172, 0.06294]), - ((3, 0), (3, 3), [0.0, -0.0, -0.1584, -0.00172, -0.25386, 0.06294]), - ((3, 3), (3, 6), [-0.00172, -0.25386, 0.06294, -0.0035, 0.24796, 0.08288]), - ((3, 6), (5, 6), [-0.24796, -0.0035, 0.08288, -0.2487, -0.0064, -0.06214]), - ((5, 6), (5, 3), [0.0064, -0.2487, -0.06214, 0.0033, -0.53554, 0.05964]), - ((5, 3), (5, 0), [0.0033, -0.53554, 0.05964, 0.0, -0.0, 0.0]), - ((7, 0), (5, 3), [0.2941, 0.44116, -0.02722, 0.29432, 0.44742, 0.05964])] - test_lns_g = [((0, 0), (0, 3), [0.0, -0.0, -0.1004024, -0.0012666, -0.2554024, -0.0545976]), - ((0, 3), (3, 3), [0.2554024, -0.0012666, -0.0545976, 0.25387, -0.0017198, 0.062937]), - ((3, 0), (3, 3), [0.0, -0.0, -0.1584034, -0.0017198, -0.25387, 0.062937]), - ((3, 3), (3, 6), [-0.0017198, -0.25387, 0.062937, -0.00349, 0.2479634, 0.0828892]), - ((3, 6), (5, 6), [-0.2479634, -0.00349, 0.0828892, -0.248698, -0.0064002, -0.0621394]), - ((5, 6), (5, 3), [0.0064002, -0.248698, -0.0621394, 0.0033056, -0.5355368, 0.0596428]), - ((5, 3), (5, 0), [0.0033056, -0.5355368, 0.0596428, 0.0, 0.0, 0.0]), - ((7, 0), (5, 3), [0.2941010455782632, 0.441151568367395, -0.0272106, 0.29431194259286797, 0.4474271690373891, 0.0596428])] + def test_Wireframe(self): + self.assertEqual( + Wireframe().points(), []) - def test_lines(self): - """test lines function (undeformed structure)""" - lns = lines(self.global_points, inc=self.inc, index_offset=1) - test = [((0, 0), (0, 3)), ((0, 3), (3, 3)), ((3, 0), (3, 3)), - ((3, 3), (3, 6)), ((3, 6), (5, 6)), ((5, 6), (5, 3)), - ((5, 3), (5, 0)), ((7, 0), (5, 3))] + # it is possible to assign other objects then Points but/and the + # class will not test if Points are used, this is just assumed + self.assertEqual( + Wireframe([]).points(), [[]]) - self.assertEqual(lns, test) - plot_lines(lns, linestyle=':') + self.assertEqual( + Wireframe(Point()).points(), [[0, 0, 0, 1]]) + self.assertEqual( + Wireframe(Point()).points(), [vector([0, 0, 0, 1])]) - def test_lines_deformation(self): - """test lines function with deformation (deformed structure)""" - lns = lines(self.global_points, deformation=fold_list(self.U2, 2), factor=self.factor, inc=self.inc, index_offset=1) - test = [((0.0, 0.0), (0.25540240000000003, 3.0012666)), ((0.25540240000000003, 3.0012666), (3.25387, 3.0017198)), ((3.0, 0.0), (3.25387, 3.0017198)), ((3.25387, 3.0017198), (2.7520366, 6.00349)), ((2.7520366, 6.00349), (4.751302, 6.0064002)), ((4.751302, 6.0064002), (4.4644632, 3.0033056)), ((4.4644632, 3.0033056), (5.0, 0.0)), ((6.4698018, 0.0), (4.4644632, 3.0033056))] - self.assertEqual(lns, test) - plot_lines(lns, color='C2') + def test_Polygon(self): + p0 = Point() + p1 = Point(1, 0, 0) + p2 = Point(1, 1, 1) + p3 = Point(0, 1, 1) + pg = Polygon(p0, p1, p2, p3) + self.assertEqual( + pg.points(), + [[0, 0, 0, 1], [1, 0 , 0, 1], [1, 1, 1, 1], [0, 1, 1, 1], [0, 0, 0, 1]]) - def test_translate_xy(self): - """test translate_xy function with single values""" - x = 2 - y = 4 - xp, yp = translate_xy((0.5, 1), x, y) - xt = 2.5 - yt = 5 - self.assertEqual(xp, xt) - self.assertEqual(yp, yt) - - def test_translate_xy_list(self): - """test translate_xy function""" - x = [ 0. , 0.33333333, 0.66666667, 1. , 1.33333333, - 1.66666667, 2. , 2.33333333, 2.66666667, 3. ] - y = [ 6.30000000e-05, -7.58828532e-04, -1.39770233e-03, - -1.84370370e-03, -2.08691495e-03, -2.11741838e-03, - -1.92529630e-03, -1.50063100e-03, -8.33504801e-04, - 8.60000000e-05] - xp, yp = translate_xy((1, 0.001), x, y) - xt = [1.0, - 1.3333333333333333, - 1.6666666666666665, - 2.0, - 2.333333333333333, - 2.666666666666667, - 3.0, - 3.333333333333333, - 3.6666666666666665, - 4.0] - yt = [0.0010629999999999999, - 0.00024117146776406044, - -0.00039770233196159102, - -0.0008437037037037035, - -0.0010869149519890263, - -0.0011174183813443071, - -0.00092529629629629624, - -0.00050063100137174286, - 0.0001664951989026062, - 0.0010859999999999999] - self.assertTrue(allclose(xp, xt)) - self.assertTrue(allclose(yp, yt)) - - def test_rotate_xy(self): - """test translate_xy function with single values""" - x = 2 - y = 4 - xp, yp = rotate_xy((0.5, 1), math.pi/2, x, y) - xt = -2.5 - yt = 2.5 - self.assertEqual(xp, xt) - self.assertEqual(yp, yt) - - def test_rotate_xy_list(self): - """test translate_xy function""" - x = [ 0. , 0.33333333, 0.66666667, 1. , 1.33333333, - 1.66666667, 2. , 2.33333333, 2.66666667, 3. ] - y = [ 6.30000000e-05, -7.58828532e-04, -1.39770233e-03, - -1.84370370e-03, -2.08691495e-03, -2.11741838e-03, - -1.92529630e-03, -1.50063100e-03, -8.33504801e-04, - 8.60000000e-05] - xp, yp = rotate_xy((0, 0), math.pi/2, x, y) - xt = [-6.3e-05, - 0.00075882853223595997, - 0.0013977023319616318, - 0.0018437037037037647, - 0.0020869149519891078, - 0.002117418381344409, - 0.0019252962962964188, - 0.0015006310013718858, - 0.0008335048010975571, - -8.5999999999816312e-05] - yt = [3.8576374173141622e-21, - 0.33333333333333331, - 0.66666666666666663, - 1.0, - 1.3333333333333333, - 1.6666666666666667, - 2.0, - 2.333333333333333, - 2.6666666666666665, - 3.0] - self.assertTrue(allclose(xp, xt)) - self.assertTrue(allclose(yp, yt)) - - def test_interpolate_hermite(self): - """test translate_xy function""" - # u: - # - left side displacement to right - # - left side rotation counterclockwise - # - right side displacement to right - # - right side rotation counterclockwise - u = [ 0.012770, 0.000063, -0.002730, 0.012693, 0.000086, 0.003147] - x, y = interpolate_hermite(u[1], u[2], u[4], u[5], scale_x=3) - xt = [ 0. , 0.33333333, 0.66666667, 1. , 1.33333333, - 1.66666667, 2. , 2.33333333, 2.66666667, 3. ] - yt = [ 6.30000000e-05, -7.58828532e-04, -1.39770233e-03, - -1.84370370e-03, -2.08691495e-03, -2.11741838e-03, - -1.92529630e-03, -1.50063100e-03, -8.33504801e-04, - 8.60000000e-05] - self.assertTrue(allclose(x, xt)) - self.assertTrue(allclose(y, yt)) - - def test_cubics_deformation_xz(self): - """test cubics function with (local) deformation""" - lns = cubics(self.global_points, deformation=self.u, rotation_plane='xz', factor=self.factor, inc=self.inc, index_offset=1) - for l, t in zip(lns, self.test_lns_l): - self.assertEqual(l[0], t[0]) # left point - self.assertEqual(l[1], t[1]) # right point - #self.assertEqual(l[2], t[2]) # deformation - self.assertTrue(allclose(l[2], t[2])) - plot_cubic_lines(lns, color='C3', samples=40) - - def test_cubics_deformation_xy(self): - """test cubics function with (local) deformation""" - lns = cubics(self.global_points, deformation=self.uN, rotation_plane='xy', factor=self.factor, inc=self.inc, index_offset=1) - for l, t in zip(lns, self.test_lns_l): - self.assertEqual(l[0], t[0]) # left point - self.assertEqual(l[1], t[1]) # right point - #self.assertEqual(l[2], t[2]) # deformation - self.assertTrue(allclose(l[2], t[2])) - plot_cubic_lines(lns, color='C3', samples=40) - - def test_cubics_global_deformation_xz(self): - """test cubics function with global_deformation""" - lns = cubics(self.global_points, global_deformation=fold_list(self.U, 3), rotation_plane='xz', factor=self.factor, inc=self.inc, index_offset=1) - for l, t in zip(lns, self.test_lns_g): - self.assertEqual(l[0], t[0]) # left point - self.assertEqual(l[1], t[1]) # right point - #self.assertEqual(l[2], t[2]) # deformation - self.assertTrue(allclose(l[2], t[2])) - plot_cubic_lines(lns, color='C4', samples=40) - - def test_cubics_global_deformation_xy(self): - """test cubics function with global_deformation""" - lns = cubics(self.global_points, global_deformation=fold_list(self.UN, 3), rotation_plane='xy', factor=self.factor, inc=self.inc, index_offset=1) - for l, t in zip(lns, self.test_lns_g): - self.assertEqual(l[0], t[0]) # left point - self.assertEqual(l[1], t[1]) # right point - #self.assertEqual(l[2], t[2]) # deformation - self.assertTrue(allclose(l[2], t[2])) - plot_cubic_lines(lns, color='C4', samples=40) + def test_Polygon_ch_cs(self): + # example object to rotate + p0 = Point() + p1 = Point(1, 0, 0) + p2 = Point(1, 1, 1) + p3 = Point(0, 1, 1) + theta = math.pi/2 + test = [[0.0, 0.0, 0.0, 1], [0, 0.0, -1.0, 1], [1.0, 1.0, -1, 1], + [1.0, 1.0, 0, 1], [0.0, 0.0, 0.0, 1]] + # using a coordinate system object + pg = Polygon(p0, p1, p2, p3) + cs = CS().rotate_y(theta) + pg.ch_cs(cs) + [self.assertAlmostEqual(i, j) for i, j in zip(pg.points(), test)] if __name__ == '__main__': - unittest.main() + unittest.main(verbosity=2) diff --git a/tests/test_geometry2d.py b/tests/test_geometry2d.py new file mode 100644 index 0000000..444c289 --- /dev/null +++ b/tests/test_geometry2d.py @@ -0,0 +1,259 @@ +"""Test of geometry2d module. + +:Date: 2019-08-20 + +.. module:: test_geometry2d + :platform: *nix, Windows + :synopsis: Test of geometry2d module. + +.. moduleauthor:: Daniel Weschke +""" +import unittest + +import os +import sys +import math +from numpy import allclose +sys.path.insert(0, os.path.abspath('../pylib')) +from data import fold_list +from geometry2d import translate_xy, rotate_xy, interpolate_hermite, lines, cubics +from geometry2d_plot import plot_lines, plot_cubic_lines + + +class TestGeometry2d(unittest.TestCase): + global_points = [(0, 0), + (0, 3), + (3, 3), + (3, 0), + (3, 6), + (5, 6), + (5, 3), + (5, 0), + (7, 0)] + inc = [[1, 2], + [2, 3], + [4, 3], + [3, 5], + [5, 6], + [6, 7], + [7, 8], + [9, 7]] + + # global deformation, horizontal right, vertical down, rotation + U = [ + 0.00000, 0.00000, -5.02012, + 12.77012, 0.06333, -2.72988, + 12.69350, 0.08599, 3.14685, + 0.00000, 0.00000, -7.92017, + -12.39817, 0.17450, 4.14446, + -12.43490, 0.32001, -3.10697, + -26.77684, 0.16528, 2.98214, + 0.00000, 0.00000, 0.00000, + -26.50991, 0.00000, -1.36053, + ] + U = [Ui/1000 for Ui in U] + + # global deformation, only horizontal right and vertical down + U2 = [ui for i, ui in enumerate(U) if i%3 != 2] + + # global deformation, horizontal right, vertical up, rotation + UN = [Ui if i%3 != 1 else -Ui for i, Ui in enumerate(U)] + + # local deformation, horizontal right, vertical down, rotation + u = [ + [ 0.000000, 0.000000, -0.005020, -0.000063, 0.012770, -0.002730], + [ 0.012770, 0.000063, -0.002730, 0.012693, 0.000086, 0.003147], + [ 0.000000, 0.000000, -0.007920, -0.000086, 0.012693, 0.003147], + [-0.000086, 0.012693, 0.003147, -0.000175, -0.012398, 0.004144], + [-0.012398, 0.000175, 0.004144, -0.012435, 0.000320, -0.003107], + [ 0.000320, 0.012435, -0.003107, 0.000165, 0.026777, 0.002982], + [ 0.000165, 0.026777, 0.002982, 0.000000, 0.000000, 0.000000], + [ 0.014705, -0.022058, -0.001361, 0.014716, -0.022371, 0.002982], + ] + + # local deformation, horizontal right, vertical up, rotation + uN = [[uii if i%3 != 1 else -uii for i, uii in enumerate(ui)] for ui in u] + + # exaggeration factor for the deformation + factor = 20 + + # test of cubic lines + test_lns_l = [((0, 0), (0, 3), [0.0, -0.0, -0.1004, -0.00126, -0.2554, -0.0546]), + ((0, 3), (3, 3), [0.2554, -0.00126, -0.0546, 0.25386, -0.00172, 0.06294]), + ((3, 0), (3, 3), [0.0, -0.0, -0.1584, -0.00172, -0.25386, 0.06294]), + ((3, 3), (3, 6), [-0.00172, -0.25386, 0.06294, -0.0035, 0.24796, 0.08288]), + ((3, 6), (5, 6), [-0.24796, -0.0035, 0.08288, -0.2487, -0.0064, -0.06214]), + ((5, 6), (5, 3), [0.0064, -0.2487, -0.06214, 0.0033, -0.53554, 0.05964]), + ((5, 3), (5, 0), [0.0033, -0.53554, 0.05964, 0.0, -0.0, 0.0]), + ((7, 0), (5, 3), [0.2941, 0.44116, -0.02722, 0.29432, 0.44742, 0.05964])] + test_lns_g = [((0, 0), (0, 3), [0.0, -0.0, -0.1004024, -0.0012666, -0.2554024, -0.0545976]), + ((0, 3), (3, 3), [0.2554024, -0.0012666, -0.0545976, 0.25387, -0.0017198, 0.062937]), + ((3, 0), (3, 3), [0.0, -0.0, -0.1584034, -0.0017198, -0.25387, 0.062937]), + ((3, 3), (3, 6), [-0.0017198, -0.25387, 0.062937, -0.00349, 0.2479634, 0.0828892]), + ((3, 6), (5, 6), [-0.2479634, -0.00349, 0.0828892, -0.248698, -0.0064002, -0.0621394]), + ((5, 6), (5, 3), [0.0064002, -0.248698, -0.0621394, 0.0033056, -0.5355368, 0.0596428]), + ((5, 3), (5, 0), [0.0033056, -0.5355368, 0.0596428, 0.0, 0.0, 0.0]), + ((7, 0), (5, 3), [0.2941010455782632, 0.441151568367395, -0.0272106, 0.29431194259286797, 0.4474271690373891, 0.0596428])] + + def test_lines(self): + """test lines function (undeformed structure)""" + lns = lines(self.global_points, inc=self.inc, index_offset=1) + test = [((0, 0), (0, 3)), ((0, 3), (3, 3)), ((3, 0), (3, 3)), + ((3, 3), (3, 6)), ((3, 6), (5, 6)), ((5, 6), (5, 3)), + ((5, 3), (5, 0)), ((7, 0), (5, 3))] + + self.assertEqual(lns, test) + plot_lines(lns, linestyle=':') + + def test_lines_deformation(self): + """test lines function with deformation (deformed structure)""" + lns = lines(self.global_points, deformation=fold_list(self.U2, 2), factor=self.factor, inc=self.inc, index_offset=1) + test = [((0.0, 0.0), (0.25540240000000003, 3.0012666)), ((0.25540240000000003, 3.0012666), (3.25387, 3.0017198)), ((3.0, 0.0), (3.25387, 3.0017198)), ((3.25387, 3.0017198), (2.7520366, 6.00349)), ((2.7520366, 6.00349), (4.751302, 6.0064002)), ((4.751302, 6.0064002), (4.4644632, 3.0033056)), ((4.4644632, 3.0033056), (5.0, 0.0)), ((6.4698018, 0.0), (4.4644632, 3.0033056))] + self.assertEqual(lns, test) + plot_lines(lns, color='C2') + + def test_translate_xy(self): + """test translate_xy function with single values""" + x = 2 + y = 4 + xp, yp = translate_xy((0.5, 1), x, y) + xt = 2.5 + yt = 5 + self.assertEqual(xp, xt) + self.assertEqual(yp, yt) + + def test_translate_xy_list(self): + """test translate_xy function""" + x = [ 0. , 0.33333333, 0.66666667, 1. , 1.33333333, + 1.66666667, 2. , 2.33333333, 2.66666667, 3. ] + y = [ 6.30000000e-05, -7.58828532e-04, -1.39770233e-03, + -1.84370370e-03, -2.08691495e-03, -2.11741838e-03, + -1.92529630e-03, -1.50063100e-03, -8.33504801e-04, + 8.60000000e-05] + xp, yp = translate_xy((1, 0.001), x, y) + xt = [1.0, + 1.3333333333333333, + 1.6666666666666665, + 2.0, + 2.333333333333333, + 2.666666666666667, + 3.0, + 3.333333333333333, + 3.6666666666666665, + 4.0] + yt = [0.0010629999999999999, + 0.00024117146776406044, + -0.00039770233196159102, + -0.0008437037037037035, + -0.0010869149519890263, + -0.0011174183813443071, + -0.00092529629629629624, + -0.00050063100137174286, + 0.0001664951989026062, + 0.0010859999999999999] + self.assertTrue(allclose(xp, xt)) + self.assertTrue(allclose(yp, yt)) + + def test_rotate_xy(self): + """test translate_xy function with single values""" + x = 2 + y = 4 + xp, yp = rotate_xy((0.5, 1), math.pi/2, x, y) + xt = -2.5 + yt = 2.5 + self.assertEqual(xp, xt) + self.assertEqual(yp, yt) + + def test_rotate_xy_list(self): + """test translate_xy function""" + x = [ 0. , 0.33333333, 0.66666667, 1. , 1.33333333, + 1.66666667, 2. , 2.33333333, 2.66666667, 3. ] + y = [ 6.30000000e-05, -7.58828532e-04, -1.39770233e-03, + -1.84370370e-03, -2.08691495e-03, -2.11741838e-03, + -1.92529630e-03, -1.50063100e-03, -8.33504801e-04, + 8.60000000e-05] + xp, yp = rotate_xy((0, 0), math.pi/2, x, y) + xt = [-6.3e-05, + 0.00075882853223595997, + 0.0013977023319616318, + 0.0018437037037037647, + 0.0020869149519891078, + 0.002117418381344409, + 0.0019252962962964188, + 0.0015006310013718858, + 0.0008335048010975571, + -8.5999999999816312e-05] + yt = [3.8576374173141622e-21, + 0.33333333333333331, + 0.66666666666666663, + 1.0, + 1.3333333333333333, + 1.6666666666666667, + 2.0, + 2.333333333333333, + 2.6666666666666665, + 3.0] + self.assertTrue(allclose(xp, xt)) + self.assertTrue(allclose(yp, yt)) + + def test_interpolate_hermite(self): + """test translate_xy function""" + # u: + # - left side displacement to right + # - left side rotation counterclockwise + # - right side displacement to right + # - right side rotation counterclockwise + u = [ 0.012770, 0.000063, -0.002730, 0.012693, 0.000086, 0.003147] + x, y = interpolate_hermite(u[1], u[2], u[4], u[5], scale_x=3) + xt = [ 0. , 0.33333333, 0.66666667, 1. , 1.33333333, + 1.66666667, 2. , 2.33333333, 2.66666667, 3. ] + yt = [ 6.30000000e-05, -7.58828532e-04, -1.39770233e-03, + -1.84370370e-03, -2.08691495e-03, -2.11741838e-03, + -1.92529630e-03, -1.50063100e-03, -8.33504801e-04, + 8.60000000e-05] + self.assertTrue(allclose(x, xt)) + self.assertTrue(allclose(y, yt)) + + def test_cubics_deformation_xz(self): + """test cubics function with (local) deformation""" + lns = cubics(self.global_points, deformation=self.u, rotation_plane='xz', factor=self.factor, inc=self.inc, index_offset=1) + for l, t in zip(lns, self.test_lns_l): + self.assertEqual(l[0], t[0]) # left point + self.assertEqual(l[1], t[1]) # right point + #self.assertEqual(l[2], t[2]) # deformation + self.assertTrue(allclose(l[2], t[2])) + plot_cubic_lines(lns, color='C3', samples=40) + + def test_cubics_deformation_xy(self): + """test cubics function with (local) deformation""" + lns = cubics(self.global_points, deformation=self.uN, rotation_plane='xy', factor=self.factor, inc=self.inc, index_offset=1) + for l, t in zip(lns, self.test_lns_l): + self.assertEqual(l[0], t[0]) # left point + self.assertEqual(l[1], t[1]) # right point + #self.assertEqual(l[2], t[2]) # deformation + self.assertTrue(allclose(l[2], t[2])) + plot_cubic_lines(lns, color='C3', samples=40) + + def test_cubics_global_deformation_xz(self): + """test cubics function with global_deformation""" + lns = cubics(self.global_points, global_deformation=fold_list(self.U, 3), rotation_plane='xz', factor=self.factor, inc=self.inc, index_offset=1) + for l, t in zip(lns, self.test_lns_g): + self.assertEqual(l[0], t[0]) # left point + self.assertEqual(l[1], t[1]) # right point + #self.assertEqual(l[2], t[2]) # deformation + self.assertTrue(allclose(l[2], t[2])) + plot_cubic_lines(lns, color='C4', samples=40) + + def test_cubics_global_deformation_xy(self): + """test cubics function with global_deformation""" + lns = cubics(self.global_points, global_deformation=fold_list(self.UN, 3), rotation_plane='xy', factor=self.factor, inc=self.inc, index_offset=1) + for l, t in zip(lns, self.test_lns_g): + self.assertEqual(l[0], t[0]) # left point + self.assertEqual(l[1], t[1]) # right point + #self.assertEqual(l[2], t[2]) # deformation + self.assertTrue(allclose(l[2], t[2])) + plot_cubic_lines(lns, color='C4', samples=40) + + +if __name__ == '__main__': + unittest.main()