Cyclides in computer aided geometric design
Computer Aided Geometric Design
On cyclides in geometric modeling
Computer Aided Geometric Design
A new intersection algorithm for cyclides and swept surfaces using circle decomposition
Computer Aided Geometric Design
Cyclides in computer aided geometric design II
Computer Aided Geometric Design
Handbook of discrete and computational geometry
SCG '05 Proceedings of the twenty-first annual symposium on Computational geometry
Algorithms in Real Algebraic Geometry (Algorithms and Computation in Mathematics)
Algorithms in Real Algebraic Geometry (Algorithms and Computation in Mathematics)
Fast and exact geometric analysis of real algebraic plane curves
Proceedings of the 2007 international symposium on Symbolic and algebraic computation
Algorithms for Reporting and Counting Geometric Intersections
IEEE Transactions on Computers
Exact and efficient 2D-arrangements of arbitrary algebraic curves
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Rational maximal parametrisations of dupin cyclides
Proceedings of the 12th IMA international conference on Mathematics of surfaces XII
ESA'07 Proceedings of the 15th annual European conference on Algorithms
Sweeping and maintaining two-dimensional arrangements on surfaces: a first step
ESA'07 Proceedings of the 15th annual European conference on Algorithms
EXACUS: efficient and exact algorithms for curves and surfaces
ESA'05 Proceedings of the 13th annual European conference on Algorithms
A descartes algorithm for polynomials with bit-stream coefficients
CASC'05 Proceedings of the 8th international conference on Computer Algebra in Scientific Computing
Arrangements of geodesic arcs on the sphere
Proceedings of the twenty-fourth annual symposium on Computational geometry
Visualizing Arcs of Implicit Algebraic Curves, Exactly and Fast
ISVC '09 Proceedings of the 5th International Symposium on Advances in Visual Computing: Part I
Constructing two-dimensional Voronoi diagrams via divide-and-conquer of envelopes in space
Transactions on computational science IX
Constructing two-dimensional Voronoi diagrams via divide-and-conquer of envelopes in space
Transactions on computational science IX
A generic algebraic kernel for non-linear geometric applications
Proceedings of the twenty-seventh annual symposium on Computational geometry
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An algorithm and implementation is presented to compute the exact arrangement induced by arbitrary algebraic surfaces on a parametrized ring dupin cyclide. The family of dupin cyclides contains as a special case the torus. The intersection of an algebraic surface of degree n with a reference cyclide is represented as a real algebraic curve of bi-degree (2n, 2n) in the two-dimensional parameter space of the cyclide. We use eigenwillig and kerber: "exact and efficient 2D-Arrangements of arbitrary algebraic Curves", SODA 2008, to compute a planar arrangement of such curves and extend their approach to obtain more asymptotic information about curves approaching the boundary of the cyclide's parameter space. With that, we can base our implementation on the general software framework by berberich et. al.: "sweeping and maintaining two-dimensional arrangements on surfaces: A first Step", ESA 2007. Our contribution provides the demanded techniques to model the special geometry of surfaces intersecting a cyclide and the special topology of the reference surface of genus one. The contained implementation is complete and does not assume generic position. Our experiments show that the combinatorial overhead of the framework does not harm the efficiency of the method. Our experiments show that the overall performance is strongly coupled to the efficiency of the implementation for arrangements of algebraic plane curves.