Computational geometry: an introduction
Computational geometry: an introduction
Comparison of three curve intersection algorithms
Computer-Aided Design
Loop detection in surface patch intersections
Computer Aided Geometric Design
Sweeping arrangements of curves
SCG '89 Proceedings of the fifth annual symposium on Computational geometry
Curve intersection using Be´zier clipping
Computer-Aided Design - Special Issue: Be´zier Techniques
Handbook of discrete and computational geometry
Generic programming and the STL: using and extending the C++ Standard Template Library
Generic programming and the STL: using and extending the C++ Standard Template Library
A core library for robust numeric and geometric computation
SCG '99 Proceedings of the fifteenth annual symposium on Computational geometry
LEDA: a platform for combinatorial and geometric computing
LEDA: a platform for combinatorial and geometric computing
A new constructive root bound for algebraic expressions
SODA '01 Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms
Geometric modeling with splines: an introduction
Geometric modeling with splines: an introduction
Complete, exact, and efficient computations with cubic curves
SCG '04 Proceedings of the twentieth annual symposium on Computational geometry
SCG '05 Proceedings of the twenty-first annual symposium on Computational geometry
Complete subdivision algorithms, I: intersection of Bezier curves
Proceedings of the twenty-second annual symposium on Computational geometry
Advanced programming techniques applied to Cgal's arrangement package
Computational Geometry: Theory and Applications
EXACUS: efficient and exact algorithms for curves and surfaces
ESA'05 Proceedings of the 13th annual European conference on Algorithms
Exact and efficient 2D-arrangements of arbitrary algebraic curves
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Computing the Voronoi cells of planes, spheres and cylinders in R3
Proceedings of the 2008 ACM symposium on Solid and physical modeling
Topology and arrangement computation of semi-algebraic planar curves
Computer Aided Geometric Design
Exact geometric and algebraic computations in CGAL
ICMS'10 Proceedings of the Third international congress conference on Mathematical software
A generic algebraic kernel for non-linear geometric applications
Proceedings of the twenty-seventh annual symposium on Computational geometry
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Arrangements of planar curves are fundamental structures in computational geometry. The arrangement package of CGAL can construct and maintain arrangements of various families of curves, when provided with the representation of the curves and some basic geometric functionality on them. It employs the exact computation paradigm in order to handle all degenerate cases in a robust manner. We present the representations and algorithms that are needed for implementing arrangements of Bézier curves using exact arithmetic. The implementation is efficient and complete, handling all degenerate cases. In order to avoid the prohibitive running times incurred by an indiscriminate usage of exact arithmetic, we make extensive use of the geometric properties of Bézier curves for filtering. As a result, most operations are carried out using fast approximate methods, and only in degenerate (or near-degenerate) cases do we resort to the exact, and more computationally demanding, procedures. To the best of our knowledge this is the first complete implementation that can construct arrangements of Bézier curves of any degree, and handle all degenerate cases in a robust manner.