ISSAC '98 Proceedings of the 1998 international symposium on Symbolic and algebraic computation
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Computer Aided Geometric Design
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2009 SIAM/ACM Joint Conference on Geometric and Physical Modeling
Yet another algorithm for generalized Voronoï Diagrams
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Computer Aided Geometric Design
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Computer-Aided Design
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Semi-algebraic sets occur naturally when dealing with implicit models and boolean operations between them. In this work we present an algorithm to efficiently and in a certified way compute the connected components of semi-algebraic sets given by intersection or union of conjunctions of bi-variate equalities and inequalities. For any given precision, this algorithm can also provide a polygonal and isotopic approximation of the exact set. The idea is to localize the boundary curves by subdividing the space and then deduce their shape within small enough cells using only boundary information. Then a systematic traversal of the boundary curve graph yields polygonal regions isotopic to the connected components of the semi-algebraic set. Space subdivision is supported by a kd-tree structure and localization is done using Bernstein representation. We conclude by demonstrating our C++ implementation in the CAS Mathemagix.