NDLI: On the topology of planar algebraic curves

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On the topology of planar algebraic curves

Content Provider	ACM Digital Library
Author	Pouget, Marc Rouillier, Fabrice Peñaranda, Luis Cheng, Jinsan Lazard, Sylvain Tsigaridas, Elias
Abstract	We revisit the problem of computing the topology and geometry of a real algebraic plane curve. The topology is of prime interest but geometric information, such as the position of singular and critical points, is also relevant. A challenge is to compute efficiently this information for the given coordinate system even if the curve is not in generic position. Previous methods based on the cylindrical algebraic decomposition (CAD) use sub-resultant sequences and computations with polynomials with algebraic coefficients. A novelty of our approach is to replace these tools by Gröbner basis computations and isolation with rational univariate representations. This has the advantage of avoiding computations with polynomials with algebraic coefficients, even in non-generic positions. Our algorithm isolates critical points in boxes and computes a decomposition of the plane by rectangular boxes. This decomposition also induces a new approach for computing an arrangement of polylines isotopic to the input curve. We also present an analysis of the complexity of our algorithm. An implementation of our algorithm demonstrates its efficiency, in particular on high-degree non-generic curves.
Starting Page	361
Ending Page	370
Page Count	10
File Format	PDF
ISBN	9781605585017
DOI	10.1145/1542362.1542424
Language	English
Publisher	Association for Computing Machinery (ACM)
Publisher Date	2009-06-08
Publisher Place	New York
Access Restriction	Subscribed
Subject Keyword	Effective algebraic geometry Exact geometric computation Topology of algebraic curves
Content Type	Text
Resource Type	Article

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