Polynomial integration on regions defined by a triangle and a conic

TL;DR

This paper introduces an efficient method for computing integrals over regions defined by a triangle and a conic, crucial for numerical optimization involving quadratic polynomial constraints.

Contribution

It proposes a novel partitioning approach to simplify integration over complex regions bounded by a triangle and a conic, improving computational efficiency.

Findings

Partitioning reduces integration complexity

Method is applicable to quadratic polynomial regions

Enhances numerical optimization techniques

Abstract

We present an efficient solution to the following problem, of relevance in a numerical optimization scheme: calculation of integrals of the type \[\iint_{T \cap \{f\ge0\}} \phi_1\phi_2 \, dx\,dy\] for quadratic polynomials $f,\phi_1,\phi_2$ on a plane triangle $T$. The naive approach would involve consideration of the many possible shapes of $T\cap\{f\geq0\}$ (possibly after a convenient transformation) and parameterizing its border, in order to integrate the variables separately. Our solution involves partitioning the triangle into smaller triangles on which integration is much simpler.