function module¶

Mathematical equations.

Date: 2019-11-04

cosine_wave(A=1, k=1, f=1, phi=0, D=0, degree=False)[source]¶

A cosine wave is said to be sinusoidal, because, \(\cos(x) = \sin(x + \pi/2)\), 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.

Parameters

A (float or int) – amplitude
k (float or int) – (angular) wave number
f (float or int) – ordinary frequency
phi (float or int) – phase
D (float or int) – non-zero center amplitude
degree (bool) – boolean to switch between radians and degree. If False phi is interpreted in radians and if True then phi is interpreted in degrees.

Results

sine wave function of spatial variable x and optional time t

See also

function_sine_wave_degree()

epitrochoid(R, r, d)[source]¶

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.

Parameters

R (float) – radius of the fixed interior circle
r – radius of the rolling circle outside of the fixed circle
d (float) – distance from the center of the exterior circle

Typre r

float

Results

functions for x of theta and y of theta

Return type

tuple

\[\begin{split}x(\theta) = (R + r)\cos\theta - d\cos\left(\frac{R+r}{r}\theta\right) \\ y(\theta) = (R + r)\sin\theta - d\sin\left(\frac{R+r}{r}\theta\right) \\ \theta = \left[0, 2\pi\right]\end{split}\]

       *  *  *
    * R         *
  *               *
 *                 *     *  *
*                   * * r      *
*                   ** ....     *
*                   **   d      *
*                   * *        *
 *                 *     *  *
  *               *
    *          *
       *  *  *

>>> x, y = epitrochoid(3, 1, 0.5)[:1]
>>> x, y, theta_end = epitrochoid(3, 1, 0.5)

hypotrochoid(R, r, d)[source]¶

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.

Parameters

R (float) – radius of the fixed exterior circle
r – radius of the rolling circle inside of the fixed circle
d (float) – distance from the center of the interior circle

Typre r

float

Results

functions for x of theta and y of theta

Return type

tuple

\[\begin{split}x(\theta) = (R - r)\cos\theta + d\cos\left(\frac{R-r}{r}\theta\right) \\ y(\theta) = (R - r)\sin\theta - d\sin\left(\frac{R-r}{r}\theta\right) \\ \theta = \left[0, 2\pi\frac{\mathrm{lcm}(r, R)}{R}\right]\end{split}\]

       *  *  *
    * R         *
  *               *
 *           *  *  *
*         * r      **
*        *     .... *
*        *      d   *
*         *        **
 *           *  *  *
  *               *
    *          *
       *  *  *

>>> x, y = hyotrochoid(20, 6, 6)[:1]
>>> x, y, theta_end = hyotrochoid(20, 6, 6)

See also

mathematics.lcm()

sine_wave(A=1, k=1, f=1, phi=0, D=0, degree=False)[source]¶

A sine wave or sinusoid is a mathematical curve that describes a smooth periodic oscillation.

Parameters

A (float or int) – amplitude
k (float or int) – (angular) wave number
f (float or int) – ordinary frequency
phi (float or int) – phase
D (float or int) – non-zero center amplitude
degree (bool) – boolean to switch between radians and degree. If False phi is interpreted in radians and if True then phi is interpreted in degrees.

Results

sine wave function of spatial variable x and optional time t

In general, the function is:

\[\begin{split}y(x,t) = A\sin(kx + 2\pi f t + \varphi) + D \\ y(x,t) = A\sin(kx + \omega t + \varphi) + D\end{split}\]

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:

\[k={\omega \over v}={2\pi f \over v}={2\pi \over \lambda }\]

where λ (lambda) is the wavelength, f is the frequency, and v is the linear speed.

function module¶

pylib

Navigation

Related Topics