Simulation of simplicity: a technique to cope with degenerate cases in geometric algorithms
ACM Transactions on Graphics (TOG)
A geometric consistency theorem for a symbolic perturbation scheme
Journal of Computer and System Sciences
Efficient Delaunay triangulation using rational arithmetic
ACM Transactions on Graphics (TOG)
Efficient exact arithmetic for computational geometry
SCG '93 Proceedings of the ninth annual symposium on Computational geometry
Towards exact geometric computation
Computational Geometry: Theory and Applications - Special issue: computational geometry, theory and applications
A perturbation scheme for spherical arrangements with application to molecular modeling
Computational Geometry: Theory and Applications - special issue on applied computational geometry
Controlled perturbation for arrangements of polyhedral surfaces with application to swept volumes
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
Controlled perturbation for Delaunay triangulations
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Classroom examples of robustness problems in geometric computations
Computational Geometry: Theory and Applications
A simple but exact and efficient algorithm for complex root isolation
Proceedings of the 36th international symposium on Symbolic and algebraic computation
Reliable and efficient computational geometry via controlled perturbation
ICALP'06 Proceedings of the 33rd international conference on Automata, Languages and Programming - Volume Part I
Efficient real root approximation
Proceedings of the 36th international symposium on Symbolic and algebraic computation
A simple but exact and efficient algorithm for complex root isolation
Proceedings of the 36th international symposium on Symbolic and algebraic computation
Analytical aspects of tie breaking
Theoretical Computer Science
On the complexity of solving a bivariate polynomial system
Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation
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Controlled Perturbation (CP, for short) is an approach to obtaining efficient and robust implementations of a large class of geometric algorithms using the computational speed of multiple precision floating point arithmetic (compared to exact arithmetic), while bypassing the precision problems by perturbation. It also allows algorithms to be written without consideration of degenerate cases. CP replaces the input objects by a set of randomly perturbed (moved, scaled, stretched, etc.) objects and protects the evaluation of geometric predicates by guards. The execution is aborted if a guard indicates that the evaluation of a predicate with floating point arithmetic may return an incorrect result. If the execution is aborted, the algorithm is rerun on a new perturbation and maybe with a higher precision of the floating point arithmetic. If the algorithm runs to completion, it returns the correct output for the perturbed input. The analysis of CP algorithms relates various parameters: the perturbation amount, the arithmetic precision, the range of input values, and the number of input objects. We present a general methodology for analyzing CP algorithms. It is powerful enough to analyze all geometric predicates that are formulated as signs of polynomials.