Adaptive Isotopic Approximation of Nonsingular Curves: the Parameterizability and Nonlocal Isotopy Approach |
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Authors: | Long Lin Chee Yap |
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Institution: | 1.Courant Institute of Mathematical Sciences,New York University,New York,USA |
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Abstract: | We consider domain subdivision algorithms for computing isotopic approximations of a nonsingular algebraic curve. The curve
is given by a polynomial equation f(X,Y)=0. Two algorithms in this area are from Snyder (1992) SIGGRAPH Comput. Graphics, 26(2), 121 and Plantinga and Vegter (2004) In Proc. Eurographics Symposium on Geometry Processing, pp. 245–254. We introduce a new algorithm that combines the advantages
of these two algorithms: like Snyder, we use the parameterizability criterion for subdivision, and like Plantinga and Vegter,
we exploit nonlocal isotopy. We further extend our algorithm in two important and practical directions: first, we allow subdivision
cells to be rectangles with arbitrary but bounded aspect ratios. Second, we extend the input domains to be regions R
0 with arbitrary geometry and which might not be simply connected. Our algorithm halts as long as the curve has no singularities
in the region, and intersects the boundary of R
0 transversally. Our algorithm is practical and easy to implement exactly. We report some very encouraging experimental results,
showing that our algorithms can be much more efficient than the algorithms of Plantinga–Vegter and Snyder. |
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