Item

Abstract

We present two approaches to the problem of calculating a cell in a 3-dimensional arrangement of quadrics. The first approach solves the problem using rational arithmetic. It works with reductions to planar arrangements of algebraic curves. Degenerate situations such as tangential intersections and self-intersections of curves are intrinsic to the planar arrangements we obtain. The coordinates of the intersection points are given by the roots of univariate polynomials. We succeed in locating all intersection points either by extended local box hit counting arguments or by globally characterizing them with simple square root expressions. The latter is realized by a clever factorization of the univariate polynomials. Only the combination of these two results facilitates a practical and implementable algorithm. The second approach operates directly in 3-space by applying classical solid modeling techniques. Whereas the first approach guarantees a correct solution in every case the second one may fail in some degenerate situations. But with the help of verified floating point arithmetic it can detect these critical cases and is faster if the quadrics are in general position.