Compressed algebraic cubature over polygons with applications to optical design

Abstract

In this paper we propose an algorithm to determine cubature rules of algebraic degree of exactness δ on general polygons P, by means of Matlab polyshape objects and near minimal rules on triangles, obtaining by Caratheodory–Tchakaloff subsampling a PI (Positive Interior) final formula with cardinality at most Nδ=(δ+1)(δ+2)∕2. We test our algorithm on polygons with different shape, and we also discuss an application to the computation of the RMSWE (Root Mean Square Wavefront Error) on obscured and vignetted pupils, in the framework of optical design by numerical ray tracing for the LSST (Large Synoptic Survey Telescope).