Solid angle measure of polyhedral cones

TL;DR

This paper introduces new decomposition methods for polyhedral cones that enable efficient computation of their solid angle measures in high-dimensional spaces using hypergeometric series, with improved convergence properties.

Contribution

The paper presents two novel decompositions of simplicial cones that facilitate the use of hypergeometric series for solid angle computation in higher dimensions.

Findings

Decomposition methods enable computation in higher dimensions.

Tridiagonal structure reduces coordinate requirements.

Analysis of hypergeometric series convergence.

Abstract

This paper addresses the computation of normalized solid angle measure of polyhedral cones. This is well understood in dimensions two and three. For higher dimensions, assuming that a positive-definite criterion is met, the measure can be computed via a multivariable hypergeometric series. We present two decompositions of full-dimensional simplicial cones into finite families of cones satisfying the positive-definite criterion, enabling the use of the hypergeometric series to compute the solid angle measure of any polyhedral cone. Additionally, our second decomposition method yields cones with a special tridiagonal structure, reducing the number of required coordinates for the hypergeometric series formula. Furthermore, we investigate the convergence of the hypergeometric series for this case. Our findings provide a powerful tool for computing solid angle measures in high-dimensional spaces.