A Subdivision Approach to Planar Semi-algebraic Sets [chapter]

Angelos Mantzaflaris, Bernard Mourrain
2010 Lecture Notes in Computer Science  
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
more » ... 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.
doi:10.1007/978-3-642-13411-1_8 fatcat:ggqtdkjvfzccpgkrgukbdarlfm