figshare
Browse

Efficient and robust recurrence relations for the Zernike circle polynomials and their derivatives in Cartesian coordinates

Posted on 2018-07-10 - 20:44
For some time it has been known and recommended that the calculation of Zernike polynomials to radial orders higher than 8 to 10 should be performed using recurrence relations rather than explicit expressions due increasingly large cancellation errors. This paper presents a set of simple recurrence relations that can be used for the unit-normalized Zernike polynomials in polar coordinates and easily adapted to Cartesian coordinates as well. The recurrence relations are also well suited for the calculation of the Cartesian derivatives of the Zernike polynomials. The recurrence relations are easily extended to arbitrarily high orders. Assessments of the precision achievable with standard 64-bit floating point arithmetic show that Zernike polynomials up to radial order 30 can be calculated over the unit circle with errors not exceeding 5E-14, and up to radial order 50 with errors not exceeding 1.2E-13. Comparison with the Zernike capability in OpticStudio® (Zemax) shows that the recurrence relations are superior in performance (both speed and precision) over the existing algorithm implemented in the software. General pseudo-code for the calculation of Zernike polynomials and their derivatives is also presented.

CITE THIS COLLECTION

DataCite
3 Biotech
3D Printing in Medicine
3D Research
3D-Printed Materials and Systems
4OR
AAPG Bulletin
AAPS Open
AAPS PharmSciTech
Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg
ABI Technik (German)
Academic Medicine
Academic Pediatrics
Academic Psychiatry
Academic Questions
Academy of Management Discoveries
Academy of Management Journal
Academy of Management Learning and Education
Academy of Management Perspectives
Academy of Management Proceedings
Academy of Management Review
or
Select your citation style and then place your mouse over the citation text to select it.

SHARE

email
need help?