Towards exact geometric computation
Computational Geometry: Theory and Applications - Special issue: computational geometry, theory and applications
Improved construction of vertical decompositions of three-dimensional arrangements
Proceedings of the eighteenth annual symposium on Computational geometry
Hauptvortrag: Quantifier elimination for real closed fields by cylindrical algebraic decomposition
Proceedings of the 2nd GI Conference on Automata Theory and Formal Languages
On the computation of an arrangement of quadrics in 3D
Computational Geometry: Theory and Applications - Special issue on the 19th European workshop on computational geometry - EuroCG 03
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)
Advanced programming techniques applied to Cgal's arrangement package
Computational Geometry: Theory and Applications
Robust, generic and efficient construction of envelopes of surfaces in three-dimensional spaces
ESA'06 Proceedings of the 14th conference on Annual European Symposium - Volume 14
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
Exact geometric-topological analysis of algebraic surfaces
Proceedings of the twenty-fourth annual symposium on Computational geometry
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
Determining the topology of real algebraic surfaces
IMA'05 Proceedings of the 11th IMA international conference on Mathematics of Surfaces
A descartes algorithm for polynomials with bit-stream coefficients
CASC'05 Proceedings of the 8th international conference on Computer Algebra in Scientific Computing
Topology of families of implicit algebraic surfaces depending on a parameter
CASC'11 Proceedings of the 13th international conference on Computer algebra in scientific computing
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We present a generic framework on a set of surfaces S in R^3 that provides their geometric and topological analysis in order to support various algorithms and applications in computational geometry. Our implementation follows the generic programming paradigm, that is, to support a certain family of surfaces, we require a small set of types and some basic operations on them, all collected in a model of the newly presented SurfaceTraits_3 concept. The framework obtains geometric and topological information on a non-empty set of surfaces in two steps. First, important 0- and 1-dimensional features are projected onto the xy-plane, obtaining an arrangement A"S with certain properties. Second, for each of its components, a sample point is lifted back to R^3 while detecting intersections with the given surfaces. For the projection we rely on Cgal's Arrangement_2 package as basic tool. Anyhow, the complexity of the output is high, and thus, we particularly regard the framework as key ingredient for querying information on and constructing geometric objects from a small set of surfaces. Examples are meshing of single surfaces, the computation of space-curves defined by two surfaces, to compute lower envelopes of surfaces, or as a basic step to compute an efficient representation of a three-dimensional arrangement. We show that the well-known family of (semi-)algebraic surfaces fulfills the framework's requirements. As robust implementations on these surfaces are lacking these days, we consider the framework to be an important step to fill this gap. In particular, we instantiate the framework by a fully-fledged model for special algebraic surfaces, namely quadrics. This instantiation already supports main tasks demanded from rotational robot motion planning, for example, as expected to compute a Piano Mover's instance.