A New Quantitative Measure for Separation and Penetration Between Convex Primitives and a Point Cloud or a Triangle Mesh

This article presents a new efficient way to quantitatively measure separation and penetration between a collection of convex primitives (including ellipsoids, capsules, cylinders, convex polyhedra, and triangles) and a point cloud or a triangle mesh. First, the minimum scaling factor of a convex primitive with respect to its centroid to contact a point or a triangle is proposed as a new distance metrics, which can be greater than, equal to, or less than one, implying that the point or the triangle is separated from, just contacts, or penetrates into the convex primitive. It can be computed mostly in closed form or occasionally with a 1-D gradient descent search, which is much faster than computing the Euclidean distance. Furthermore, an efficient algorithm is proposed to compute the smallest minimum scaling factor of convex primitives in a collection to a point cloud or a triangle mesh. It is based on the discovery that computing the minimum scaling factor of a convex primitive to a point or a triangle yields a plane separating more points or triangles from this or other convex primitives. Then, the overall smallest scaling factor can be found by checking only a few pairs of primitives and points or triangles, being significantly faster than the exhaustive search. In various numerical examples and comparison with the existing algorithms, the proposed metrics and algorithm show superior or comparable efficiency.