Zernike Basis to Cartesian Transformations

TL;DR

This paper provides tabulations and transformations between Zernike basis functions and Cartesian coordinates, facilitating their use in expansions and product calculations in 2D and 3D applications.

Contribution

It introduces new tabulations of Zernike radial polynomials and derives transformations to and from Cartesian coordinates for 2D and 3D Zernike functions.

Findings

Tabulated radial polynomial expansions for 2D and 3D Zernike functions.

Derived and provided transformation formulas between Zernike functions and Cartesian coordinates.

Examples demonstrating the expansion of polynomial products using the tabulations.

Abstract

The radial polynomials of the 2D (circular) and 3D (spherical) Zernike functions are tabulated as powers of the radial distance. The reciprocal tabulation of powers of the radial distance in series of radial polynomials is also given, based on projections that take advantage of the orthogonality of the polynomials over the unit interval. They may play a role in the expansion of products of the polynomials into sums, which is demonstrated by some examples. Multiplication of the polynomials by the angular bases (azimuth, polar angle) defines the Zernike functions, for which we derive and tabulate transformations to and from the Cartesian coordinate system centered at the middle of the circle or sphere.