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 under-constrained 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 [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 application of the solver, like 3D trisector curves or kinematic simulations in 3D are discussed.