Oriented bounding surfaces with at most six common normals

Abstract

Abstract-We present a new type of oriented bounding surfaces, which is particularly well suited for shortest distance computations. The bounding surfaces are obtained by considering surfaces whose support functions are restrictions of quadratic polynomials to the unit sphere. We show that the common normals of two surfaces of this type -and hence their shortest distance -can be computed by solving a polynomial of degree six. This compares favorably with other existing bounding surfaces, such as quadric surfaces, where the computation of the common normals is known to lead to a polynomial of degree 24.

Fast Distance Computation Using Quadratically Supported Surfaces

“…As another advantage, the computational costs of computing the shortest distance between two of our primitives is much smaller than in the case of ellipsoids; it can be formulated as a root finding problem for a polynomial of degree 6. This fact has been established recently [14]. In the present paper we use this observation in order to formulate an algorithm for shortest distance computation between enclosures of objects by the new geometric primitives.…”

“…Using (3) and the corresponding equation for g, this condition leads to a system of three quadratic equations for n 0 , which have to be satisfied by a unit vector n 0 . As shown in [14], the three quadric surfaces (12) (where n 0 = (x, y, z) ⊤ ) intersect in a rational cubic curve, which can be parameterized easily. The unit normals that correspond to a point-pair with common normals are then found by intersecting this rational cubic curve with the unit sphere, leading to a polynomial equation of degree 6.…”