First moments of a polyhedron clipped by a paraboloid

TL;DR

This paper derives closed-form formulas for calculating the volume and centroid of polyhedra intersected by paraboloids, enabling efficient and accurate geometric computations crucial for various scientific and engineering applications.

Contribution

It introduces novel closed-form expressions for first moments of paraboloid-clipped polyhedra, along with methods to handle intersection ambiguities and validate robustness.

Findings

Provides robust, machine-accurate estimates of first moments.

Achieves computational efficiency comparable to existing algorithms.

Demonstrates effectiveness through extensive testing with diverse configurations.

Abstract

We provide closed-form expressions for the first moments (i.e., the volume and volume-weighted centroid) of a polyhedron clipped by a paraboloid, that is, of a polyhedron intersected with the subset of the three-dimensional real space located on one side of a paraboloid. These closed-form expressions are derived following successive applications of the divergence theorem and the judicious parametrization of the intersection of the polyhedron's faces with the paraboloid. We provide means for identifying ambiguous discrete intersection topologies, and propose a corrective procedure for preventing their occurence. Finally, we put our proposed closed-form expressions and numerical approach to the test with millions of random and manually engineered polyhedron/paraboloid intersection configurations. The results of these tests show that we are able to provide robust machine-accurate estimates of the first moments at a computational cost that is within one order of magnitude of that of state-of-the-art half-space clipping algorithms.