The theory and computation of solid angles - Santos Neto, Gervásio Protásio dos (2021)
Solid angles are higher-dimensional analogues of traditional, two-dimensional angles. They represent the same fundamental concept of how much of space is encompassed by a polyhedral cone. This generalization is useful in order to study geometric cones and polytopes in higher dimensions. Calculating these quantities in dimensions two and three is straightforward, due to known closed formulas in terms of the extremal vectors of the cone. However, for dimension four and greater, no nite numerical formula is known. This thesis is devoted to analyzing solid angles and computing them in dimensions greater than three. We present two new proofs of upper and lower bounds for solid angles of polyhedral cones based on their extremal vectors and the spectrum of their de ning matrix. One of these novel proofs explores the connection between Discrete Geometry and Theta Functions, a classic concept in Number Theory, Physics and Partial Di erential Equations, while the other is more combinatorial in nature. The fact that two new proofs of a known bound, with di erent approaches from one another, converge to the same result indicates that perhaps no better bound can be found. Onthe computational side, we discuss approximation methods for high dimensional angles based on hypergeometric series and on probability. We empirically analyze their performance and compare them. Furthermore, we describe the implementation of a Python package implementing these algorithms, the rst such open source package.