function module

Mathematical equations.

Date

2019-11-01

cosine_wave(A=1, k=1, f=1, phi=0, D=0, degree=False)

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()

hypotrochoid(R, r, d)

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)

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.

See also

function_cosine_wave_degree()