Cylindrical algebraic decomposition I: the basic algorithm
SIAM Journal on Computing
Planning, geometry, and complexity of robot motion
Planning, geometry, and complexity of robot motion
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
Robot Motion Planning
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
An efficient algorithm for the stratification and triangulation of an algebraic surface
Computational Geometry: Theory and Applications
Journal of Symbolic Computation
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We present a generic framework on a set of surfaces S in R3 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, i.e., 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 As with certain properties. Second, for each of its components, a sample point is lifted back to R3 while detecting intersections with the given surfaces. This idea is similar to Collins' cylindrical algebraic decomposition (cad). In contrast, we reduce the number of liftings using CGAL'S Arrangement_2 package as a basic tool. Properly instantiated, the framework provides main functionality required to support the computation of a Piano Mover's instance. On the other hand, 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 also inspirit the framework in two steps. First, we show that the well-known family of algebraic surfaces fulfils 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. Second, 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 [Latombe 1993]. How to provide a model for algebraic surfaces of arbitrary degree, is partly discussed in [Berberich et al. 2008].