Excluded volume of polytopes and the geometry of inaccessible configurations

Determining the region of space that one object can't access due to the presence of another is a ubiquitous problem in physics and engineering, finding applications ranging from describing the ordering transitions of liquid crystals to performing path planning in robotics. Of particular interest in many domains is not only the region of inaccessible space but also its volume. For example, our understanding of emergent entropic transitions such as the crystallization of hard particles and the depletion interaction relies heavily on understanding how much space particles can and cannot occupy. Using the relevant theoretical tools from convex geometry we describe the excluded region and excluded volume concepts through the elementary yet practical perspective of convex polytopes. Formulas are derived for the excluded volume between two polytopes at fixed relative orientation, and the topology of the excluded region is explained using both mathematical arguments and the implementation of constructive algorithms. We utilize this theory to efficiently calculate the excluded volume of two convex polygons in 2D and polyhedra in 3D and describe its variation under translation, rotation, inversion, and dilation of one of the bodies. The theory developed in this work facilitates the investigation of the phase behavior of complex materials in which steric interactions play a dominating role.