Topologically guaranteed solution of surface–surface intersections via no–loop and single–component tests

We present an algorithm which computes the intersection curve(s) of two free-form surfaces in 3D. The surface-surface intersection (SSI) problem is equivalent to solving an underconstrained polynomial system of three equations with four unknowns, over some domain D ∈ R. The solution of such a system is a curve in 4-space. This work extends the single solution test for a set of algebraic constraints from zero dimensional solutions to univariate solutions, in R. 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 Newton-Raphson 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. This extension is mostly discussed in R for n equations with n+1 unknowns while examples are provided for the SSI problem.