An Automatic Method for Generating Symbolic Expressions of Zernike Circular Polynomials
Abstract
Zernike circular polynomials (ZCP) play a significant role in optics engineering. The symbolic expressions for ZCP are valuable for theoretic analysis and engineering designs. However, there are still two problems which remain open: firstly, there is a lack of sufficient mathematical formulas of the ZCP for optics designers; secondly the formulas for inter-conversion of Noll's single index and Born-Wolf's double indices of ZCP are neither uniquely determinate nor satisfactory. An automatic method for generating symbolic expressions for ZCP is proposed based on five essential factors: the new theorems for converting the single/double indices of the ZCP, the robust and effective numeric algorithms for computing key parameters of ZCP, the symbolic algorithms for generating mathematical expressions of ZCP, and meta-programming \& \LaTeX{} programming for generating the table of ZCP. The theorems, method, algorithms and system architecture proposed are beneficial to both optics design process, optics software, computer-output typesetting in publishing industry as well as STEM education.
Keywords
Cite
@article{arxiv.2301.01859,
title = {An Automatic Method for Generating Symbolic Expressions of Zernike Circular Polynomials},
author = {Hong-Yan Zhang and Yu Zhou and Fu-Yun Li},
journal= {arXiv preprint arXiv:2301.01859},
year = {2023}
}
Comments
This paper has proposed a method for creating mathematical table as demonstrated by generating Zenike circular polynomials. The code is available on GitHub site, please see: https://github.com/GrAbsRD/ZernikeSymbolicExpression