Sampling patterns for Zernike-like bases in non-standard geometries

Zernike polynomials are widely used in optics and ophthalmology due to their direct connection to classical optical aberrations. While orthogonal on the unit disk, their application to discrete data or non-circular domains–such as ellipses, annuli, and hexagons–presents challenges in terms of numerical stability and accuracy. In this work, we extend Zernike-like orthogonal functions to these non-standard geometries using diffeomorphic mappings and construct sampling patterns that preserve favorable numerical conditioning. We provide theoretical bounds for the condition numbers of the resulting collocation matrices and validate them through extensive numerical experiments. As a practical application, we demonstrate accurate wavefront interpolation and reconstruction in segmented mirror telescopes composed of hexagonal facets. Our results show that appropriately transferred sampling configurations, especially Optimal Concentric Sampling and Lebesgue points, allow stable high-order interpolation and effective wavefront modeling in complex optical systems. Moreover, the Optimal Concentric Samplings can be computed with an explicit expression, which is a significant advantage in practice.