#!/usr/bin/env python
# -*- coding: utf-8 -*-
"""Mathematical equations.
:Date: 2019-11-02
.. module:: function
:platform: *nix, Windows
:synopsis: Mathematical equations.
.. moduleauthor:: Daniel Weschke <daniel.weschke@directbox.de>
"""
import math
from pylib.mathematics import lcm
[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:`function_cosine_wave_degree`
"""
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:`function_sine_wave_degree`
"""
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