Tools for analyzing the intersection curve between two quadrics through projection and lifting
Determining the topology of and parametric equations for the intersection curve of two quadrics are two well-studied topics in computational geometry. Nonetheless, details of these tasks can be quite involved, in particular if an accurate and robust treatment of special cases is required. This article proposes a concept based on 1) projecting the intersection curve onto a plane, 2) analyzing this projection and 3) lifting back correct curve parts onto the quadrics. The projection step via resultants is straightforward but yields an algebraic curve while the actual projection of real points of the intersection curve is bounded by the two quadrics' silhouettes, thus producing a semi-algebraic set, consisting of a quartic curve that is bounded by two conics (the ``cutcurve''). The authors exploit special properties of the cutcurve for analyzing its topology and for gathering information on diverse singularities that allow to conveniently lift back portions of the cutcurve onto the quadrics. All of this results in an (as expected) rather involved algorithm to determine the intersection curve's topology and square root parametrizations. The authors argue that it is of moderate algebraic complexity and compares favourable with more general curve plotting methods. They also illustrate it at hand of some examples. While the computations work best with rational input, they can be adapted to floating point numbers. The authors give some hints on that.
scientific article; zbMATH DE number 7337489
scientific article