Isotopic Arrangement of Simple Curves: an Exact Numerical Approach based on Subdivision

09/02/2020
by   Jyh-Ming Lien, et al.
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This paper presents the first purely numerical (i.e., non-algebraic) subdivision algorithm for the isotopic approximation of a simple arrangement of curves. The arrangement is "simple" in the sense that any three curves have no common intersection, any two curves intersect transversally, and each curve is non-singular. A curve is given as the zero set of an analytic function f:ℝ^2→ℝ^2, and effective interval forms of f, ∂f/∂x, ∂f/∂y are available. Our solution generalizes the isotopic curve approximation algorithms of Plantinga-Vegter (2004) and Lin-Yap (2009). We use certified numerical primitives based on interval methods. Such algorithms have many favorable properties: they are practical, easy to implement, suffer no implementation gaps, integrate topological with geometric computation, and have adaptive as well as local complexity. A version of this paper without the appendices appeared in Lien et al. (2014).

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