Volume and Surface Area of two Orthogonal, Partially Intersecting Cylinders: A Generalization of the Steinmetz Solid

TL;DR

This paper derives exact integral formulas and empirical approximations for the volume and surface area of partially intersecting orthogonal cylinders, extending classical solutions to more general intersection scenarios with validated accuracy.

Contribution

It introduces general integral expressions and empirical approximation functions for the intersection volume and surface area of partially intersecting cylinders, broadening the classical Steinmetz solid analysis.

Findings

Exact integral formulas for intersection volume and surface area.

Empirical approximation functions with less than 15% error.

Validation confirms high accuracy of solutions.

Abstract

The intersection of two orthogonal cylinders represents a classical problem in computational geometry with direct applications to engineering design, manufacturing, and numerical simulation. While analytical solutions exist for the fully intersecting case, the Steinmetz solid, partial intersections with arbitrary depth ratios require numerical methods or approximations. This work presents general integral expressions for both the intersection volume and surface area as explicit functions of the intersection depth. Accompanying these exact formulations are empirical approximation functions, which provide closed-form evaluations with relative errors below 15% across the full range of intersection depth. Validation against Quasi-Monte Carlo simulation confirms the accuracy of both the analytical and approximate solutions.