Zernike Basis to Cartesian Transformations

Details Zernike Basis to Cartesian Transformations Zernike Basis to Cartesian Transformations

2008-09-13

arxiv.org/abs/0809.2368v1

Serb. Astr. J. 179 (2009) 107-120.

Physics

Mathematical Physics

31 pages. RevTex4

Scientific paper

10.2298/SAJ0979107M

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.

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