Empirical study of an evaluation-based subdivision algorithm for complex root isolation
Proceedings of the 2011 International Workshop on Symbolic-Numeric Computation
Non-local isotopic approximation of nonsingular surfaces
Computer-Aided Design
Near optimal tree size bounds on a simple real root isolation algorithm
Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation
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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.