Polynomial integration on regions defined by a triangle and a conic

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.

💡 Research Summary

The paper addresses a very specific yet practically important numerical integration problem that arises in optimization and simulation workflows: given a planar triangle (T) and a quadratic polynomial (f(x,y)), evaluate the integral of the product of two other quadratic polynomials (\phi_1(x,y)) and (\phi_2(x,y)) over the region where (f) is non‑negative, i.e.
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