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

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.