Marching cubes: A high resolution 3D surface construction algorithm
SIGGRAPH '87 Proceedings of the 14th annual conference on Computer graphics and interactive techniques
Algorithm for algebraic curve intersection
Computer-Aided Design
Calculating approximate curve arrangements using rounded arithmetic
SCG '89 Proceedings of the fifth annual symposium on Computational geometry
Curves and surfaces for computer aided geometric design
Curves and surfaces for computer aided geometric design
Computation of the solutions of nonlinear polynomial systems
Computer Aided Geometric Design
Design patterns: elements of reusable object-oriented software
Design patterns: elements of reusable object-oriented software
Arrangements and their applications in robotics: recent developments
WAFR Proceedings of the workshop on Algorithmic foundations of robotics
A new approach to the surface intersection problem
Computer Aided Geometric Design
Algorithmic geometry
Modern computer algebra
Geometric constraint solver using multivariate rational spline functions
Proceedings of the sixth ACM symposium on Solid modeling and applications
Efficient topology determination of implicitly defined algebraic plane curves
Computer Aided Geometric Design
Hauptvortrag: Quantifier elimination for real closed fields by cylindrical algebraic decomposition
Proceedings of the 2nd GI Conference on Automata Theory and Formal Languages
Interval Methods in Geometric Modeling
GMP '00 Proceedings of the Geometric Modeling and Processing 2000
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
Isotopic approximation of implicit curves and surfaces
Proceedings of the 2004 Eurographics/ACM SIGGRAPH symposium on Geometry processing
Contouring 1- and 2-Manifolds in Arbitrary Dimensions
SMI '05 Proceedings of the International Conference on Shape Modeling and Applications 2005
An approximate arrangement algorithm for semi-algebraic curves
Proceedings of the twenty-second annual symposium on Computational geometry
Almost tight recursion tree bounds for the Descartes method
Proceedings of the 2006 international symposium on Symbolic and algebraic computation
An exact, complete and efficient computation of arrangements of Bézier curves
Proceedings of the 2007 ACM symposium on Solid and physical modeling
Algorithms for Reporting and Counting Geometric Intersections
IEEE Transactions on Computers
Visualisation of Implicit Algebraic Curves
PG '07 Proceedings of the 15th Pacific Conference on Computer Graphics and Applications
A Subdivision Arrangement Algorithm for Semi-Algebraic Curves: An Overview
PG '07 Proceedings of the 15th Pacific Conference on Computer Graphics and Applications
Determining the topology of real algebraic surfaces
IMA'05 Proceedings of the 11th IMA international conference on Mathematics of Surfaces
On the topology of planar algebraic curves
Proceedings of the twenty-fifth annual symposium on Computational geometry
Deflation and certified isolation of singular zeros of polynomial systems
Proceedings of the 36th international symposium on Symbolic and algebraic computation
A subdivision approach to planar semi-algebraic sets
GMP'10 Proceedings of the 6th international conference on Advances in Geometric Modeling and Processing
On the isotopic meshing of an algebraic implicit surface
Journal of Symbolic Computation
Arrangement computation for planar algebraic curves
Proceedings of the 2011 International Workshop on Symbolic-Numeric Computation
Journal of Symbolic Computation
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We describe a new subdivision method to efficiently compute the topology and the arrangement of implicit planar curves. We emphasize that the output topology and arrangement are guaranteed to be correct. Although we focus on the implicit case, the algorithm can also treat parametric or piecewise linear curves without much additional work and no theoretical difficulties. The method isolates singular points from regular parts and deals with them independently. The topology near singular points is guaranteed through topological degree computation. In either case the topology inside regions is recovered from information on the boundary of a cell of the subdivision. Obtained regions are segmented to provide an efficient insertion operation while dynamically maintaining an arrangement structure. We use enveloping techniques of the polynomial represented in the Bernstein basis to achieve both efficiency and certification. It is finally shown on examples that this algorithm is able to handle curves defined by high degree polynomials with large coefficients, to identify regions of interest and use the resulting structure for either efficient rendering of implicit curves, point localization or boolean operation computation.