Topologically guaranteed univariate solutions of underconstrained polynomial systems via no-loop and single-component tests

We present an algorithm which robustly computes the intersection curve(s) of an underconstrained piecewise polynomial system consisting of n equations with n+1 unknowns. The solution of such a system is typically a curve in Rn+¹. This work extends the single solution test of Hanniel and Elber (2007) [6] for a set of algebraic constraints from zero-dimensional solutions to univariate solutions, in Rn+¹. Our method exploits two tests: a no-loop test (NLT) and a single-component test (SCT) that together isolate and separate domains D where the solution curve consists of just one single component. For such domains, a numerical curve tracing is applied. If one of those tests fails, D is subdivided. Finally, the single components are merged together and, consequently, the topological configuration of the resulting curve is guaranteed. Several possible applications of the solver, namely solving the surfacesurface intersection problem, computing 3D trisector curves, flecnodal curves or kinematic simulations in 3D are also discussed.