Computing the Closest Approach Distance of Two Ellipsoids

This paper presents two practical methods for computing the closest approach distance of two ellipsoids in their inter-center direction. The closest approach distance is crucial for collision handling in the dynamic simulation of rigid and deformable bodies approximated with ellipsoids. To find the closest approach distance, we formulate a set of equations for two ellipsoids contacting each other externally in terms of the inter-center distance, contact point, and normal vector. The equations are solved robustly and efficiently using a hybrid of the fixed-point iteration method and bisection method with root bracketing, and a hybrid of Newton's method and the bisection method. In addition to a stopping criterion expressed with the progress of the solution, we introduce a novel criterion expressed in terms of the error in distance. This criterion can be effectively employed in real-time applications such as computer games by allowing an unnoticeable error. Experimental results demonstrate the robustness and efficiency of the proposed methods in various experiments.