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Complete, Exact and Efficient Computations with Cubic Curves

MPS-Authors
http://pubman.mpdl.mpg.de/cone/persons/resource/persons44369

Eigenwillig,  Arno
Algorithms and Complexity, MPI for Informatics, Max Planck Society;

http://pubman.mpdl.mpg.de/cone/persons/resource/persons44766

Kettner,  Lutz
Algorithms and Complexity, MPI for Informatics, Max Planck Society;

http://pubman.mpdl.mpg.de/cone/persons/resource/persons45414

Schömer,  Elmar
Algorithms and Complexity, MPI for Informatics, Max Planck Society;

http://pubman.mpdl.mpg.de/cone/persons/resource/persons45758

Wolpert,  Nicola
Algorithms and Complexity, MPI for Informatics, Max Planck Society;

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Citation

Eigenwillig, A., Kettner, L., Schömer, E., & Wolpert, N. (2004). Complete, Exact and Efficient Computations with Cubic Curves. In Proceedings of the Twentieth Annual Symposium on Computational Geometry: (SCG'04) (pp. 409-418). New York, USA: ACM.


Cite as: http://hdl.handle.net/11858/00-001M-0000-000F-2A58-9
Abstract
The Bentley-Ottmann sweep-line method can be used to compute the arrangement of planar curves provided a number of geometric primitives operating on the curves are available. We discuss the mathematics of the primitives for planar algebraic curves of degree three or less and derive efficient realizations. As a result, we obtain a complete, exact, and efficient algorithm for computing arrangements of cubic curves. Conics and cubic splines are special cases of cubic curves. The algorithm is complete in that it handles all possible degeneracies including singularities. It is exact in that it provides the mathematically correct result. It is efficient in that it can handle hundreds of curves with a quarter million of segments in the final arrangement.