Isotopic Arrangement of Simple Curves: An Exact Numerical Approach Based on Subdivision

This paper presents the first purely numerical (i.e., non-algebraic) subdivision algorithm for the iso-topic approximation of a simple arrangement of curves. The arrangement is “simple” in the sense thatany three curves have no common intersection, any two curves intersect transversally, and each curve isnon-singular. A curve is given as the zero set of an analytic function f : R → R, and effective intervalforms of f, ∂f∂x , ∂f∂y are available. Our solution generalizes the isotopic curve approximation algorithmsof Plantinga-Vegter (2004) and Lin-Yap (2009).We use certified numerical primitives based on interval methods. Such algorithms have many favor-able properties: they are practical, easy to implement, suffer no implementation gaps, integrate topolog-ical with geometric computation, and have adaptive as well as local complexity.