Volume of Intersection of a Cone with a Sphere

TL;DR

This paper derives formulas for calculating the volume of the intersection between a cone and a sphere, accommodating various spatial configurations using advanced integral techniques involving elliptic integrals.

Contribution

It introduces a novel integral approach to compute the intersection volume for arbitrary cone-sphere arrangements, extending previous simpler cases.

Findings

Derived formulas for intersection volume in general configurations

Expressed volume integrals using elliptic integrals

Validated formulas with specific geometric cases

Abstract

The manuscript provides formulas for the volume of a body defined by the intersection of a solid cone and a solid sphere as a function of the sphere radius, of the distance between cone apex and sphere center, and of the cone aperture angle. If the sphere center lies on the (extended) cone axis the analysis may be based on cylinder coordinates fixed at the cone axis, and the volume is the sum of the well-known volumes of finite cones and sphere caps. At the general geometry the sphere center is not on the (extended) cone axis. Our approach calculates the volume by slicing space perpendicular to the cone axis and by integrating the lens areas defined by the sphere-cone intersection. These volume integrals are rephrased with the aid of the Byrd-Friedmann tables to Elliptic Integrals of the First, Second and Third Kind.