Computing the volume of three intersecting cylinders with rotational symmetry

Abstract

This paper addresses the geometric problem of finding the intersection volume of three cylinders of equal radius, two of which are perpendicular to one another and the third of which forms two arbitrary angles with the other two. The article generalizes the classical Steinmetz solid to rotated, non-orthogonal cylinders, a topic that has minimal treatment in existing literature. The computation is done in two steps. In the first section, we deal with the case of rotation about a single axis. We employ analytical geometry and coordinate transformations by means of rotation matrices to obtain an exact formula for the intersection volume by strict integration. In the second section, we generalize the problem to the case of rotation around two axes. Due to the complexity of the resulting integrals, we compute the volume by combining analytical methods with Taylor series expansions for the more troublesome terms. Comparisons are drawn with standard numerical methods, and it is shown that even the approximate solutions are more accurate and computationally effective. This paper gives a sound and efficient method for the calculation of intersection volumes of rotated cylinders, which can be used in CAD modeling, engineering design, and physical simulations.