Given a real valued function f (X, Y), a box region B-0 subset of R-2 and epsilon > 0, we want to compute an E-isotopic polygonal approximation to the restriction of the curve S = f(-1) (0) = {p is an element of R-...
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Given a real valued function f (X, Y), a box region B-0 subset of R-2 and epsilon > 0, we want to compute an E-isotopic polygonal approximation to the restriction of the curve S = f(-1) (0) = {p is an element of R-2 : f (p) = 0} to B-0. We focus on subdivision algorithms because of their adaptive complexity and ease of implementation. Plantinga & Vegter gave a numerical subdivision algorithm that is exact when the curve S is bounded and non-singular. They used a computational model that relied only on function evaluation and interval arithmetic. We generalize their algorithm to any bounded (but possibly non-simply connected) region that does not contain singularities of S. With this generalization as a subroutine, we provide a method to detect isolated algebraic singularities and their branching degree. This appears to be the first complete purely numerical method to compute isotopic approximations of algebraic curves with isolated singularities. (C) 2011 Elsevier Ltd. All rights reserved.
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