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Computation of the circle polynomials of Zernike

The circle polynomials of Zernike are a vital tool in the analysis of optical systems. Decomposition of wavefronts into Zernike polynomials can be insightful. Computation in the Zernike basis, however, is quite cumbersome and inefficient. This paper will address how rational polynomials such as Zernike, Laguerre, Legendre and Chebyshev can be represented as affine combinations of a Taylor monomial set. This paper also demonstrates the efficiency of common routines using a Taylor basis as well as implementation and optimization issues.