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

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.

Figures (6)

• Figure 1: The intersection depth is measured by $\delta = H/D$, which is indicated above each volume. The intersection volume of fully-intersecting cylinders is called a Steinmetz solid ($\delta=1)$.
• Figure 2: At any given height, the intersection shape has a rectangular cross section. Its width and length are highlighted by the green and red lines.
• Figure 3: Cross-sectional cuts illustrating the relation of width and height with the intersection volume. The origin of the coordinate system is placed in the center-of-mass of the intersection volume.
• Figure 4: Intersecting cylinders, illustrating geometric features for calculating the gray shell sections. The cylindrical coordinate system is placed in the bottom cylinder. Left: $\delta \leq 1/2$, Right: $\delta > 1/2$
• Figure 5: Reduced (volume|surface area) against intersection depth, displaying the integral solution, its empirical approximations, and two Monte Carlo approximations with different number of sample points. Left: reduced volume, Right: reduced surface area.