Fast Detection of Degenerate Predicates in Free Space Construction

Abstract

An implementation of a computational geometry algorithm is robust if the combinatorial output is correct for every input. Robustness is achieved by ensuring that the predicates in the algorithm are evaluated correctly. A predicate is the sign of an algebraic expression whose variables are input parameters. The hardest case is detecting degenerate predicates where the value of the expression equals zero. We encounter this case in constructing the free space of a polyhedron that rotates around a fixed axis and translates freely relative to a stationary polyhedron. Each predicate involved in the construction is expressible as the sign of a univariate polynomial [Formula: see text] evaluated at a zero [Formula: see text] of a univariate polynomial [Formula: see text], where the coefficients of [Formula: see text] and [Formula: see text] are polynomials in the coordinates of the polyhedron vertices. A predicate is degenerate when [Formula: see text] is a zero of a common factor of [Formula: see text] and [Formula: see text]. We present an efficient degeneracy detection algorithm based on a one-time factoring of all the univariate polynomials over the ring of multivariate polynomials in the vertex coordinates. Our algorithm is 3500 times faster than the standard algorithm based on greatest common divisor computation. It reduces the share of degeneracy detection in our free space computations from 90% to 0.5% of the running time.