R

pyoof.zernike.R(n, m, rho)

Radial Zernike polynomials generator (\(R^m_n(\varrho)\) from Born & Wolf definition). The \(m\), \(n\) are integers, \(n\geqslant 0\) and \(n - m\) even. Only used to compute the general expression for the Zernike circle polynomials, U.

Parameters

nint

It is \(n \geqslant 0\). Order of the radial component.

mint

Positive number, relative to the angle component.

rhondarray

Values for the radial component, \(\varrho = \sqrt{x^2 + y^2}\).

Returns

radial_polyndarray

Radial Zernike polynomial already evaluated, \(R^m_n(\varrho)\).

Notes

The original generating formula for the radial polynomials is given by,

\[R^{\pm m}_n (\varrho) = \frac{1}{\left(\frac{n-m}{2}\right)! \cdot \varrho^m}\left\{\frac{\mathrm{d}}{\mathrm{d}\left(\ varrho^2 \right)} \right\}^{\frac{n-m}{2}} \left\{ \left( \ varrho^2 \right)^{\frac{n+m}{2}} \cdot \left( \varrho^2 -1 \right)^{\frac{n-m}{2}} \right\},\]

Which can also be expressed as a polynomial sum.

Examples

To start using the radial polynomials simply call the package.

>>> import numpy as np
>>> from astropy import units as u
>>> from pyoof import zernike
>>> # only orthogonal under unitary circle
>>> r = np.linspace(-10, 10, 5) * u.m
>>> zernike.R(n=4, m=2, rho=r / r.max())
<Quantity [ 1. , -0.5,  0. , -0.5,  1. ]>