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.

翻译：两个正交圆柱体的相交问题代表了计算几何中的一个经典问题，在工程设计、制造和数值模拟中具有直接应用。虽然对于完全相交的情况（即斯坦梅茨立体）存在解析解，但对于具有任意深度比的部分相交情况，则需要数值方法或近似解。本文提出了关于相交体积和表面积的一般积分表达式，它们作为相交深度的显式函数。伴随这些精确公式的是经验近似函数，这些函数提供了封闭形式的评估，在整个相交深度范围内相对误差低于15%。通过与拟蒙特卡洛模拟的对比验证，确认了解析解与近似解的准确性。