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Recent documents in Computer Science Faculty Publicationsen-usWed, 15 Jan 2014 17:23:35 PST3600Two approximate Minkowski sum algorithms
http://docs.lib.purdue.edu/cspubs/2
http://docs.lib.purdue.edu/cspubs/2Fri, 07 Oct 2011 06:27:42 PDT
We present two approximate Minkowski sum algorithms for planar regions bounded by line and circle segments. Both algorithms form a convolution curve, construct its arrangement, and use winding numbers to identify sum cells. The first uses the kinetic convolution and the second uses our monotonic convolution. The asymptotic running times of the exact algorithms are increased by kmlogm with m the number of segments in the convolution and with k the number of segment triples that are in cyclic vertical order due to approximate segment intersection. The approximate Minkowski sum is close to the exact sum of perturbation regions that are close to the input regions. We validate both algorithms on part packing tasks with industrial part shapes. The accuracy is near the floating point accuracy even after multiple iterated sums. The programs are 2% slower than direct floating point implementations of the exact algorithms. The monotonic algorithm is 42% faster than the kinetic algorithm.
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Victor Milenkovic et al.Controlled linear perturbation
http://docs.lib.purdue.edu/cspubs/1
http://docs.lib.purdue.edu/cspubs/1Fri, 23 Sep 2011 08:14:35 PDT
We present an algorithmic solution to the robustness problem in computational geometry, called controlled linear perturbation, and demonstrate it on Minkowski sums of polyhedra. The robustness problem is how to implement real RAM algorithms accurately and efficiently using computer arithmetic. Approximate computation in floating point arithmetic is efficient but can assign incorrect signs to geometric predicates, which can cause combinatorial errors in the algorithm output. We make approximate computation accurate by performing small input perturbations, which we compute using differential calculus. This strategy supports fast, accurate Minkowski sum computation. The only prior robust implementation uses a less efficient algorithm, requires exact algebraic computation, and is far slower based on our extensive testing.
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Elisha P. Sacks et al.