September 4, 2008 8:26 WSPC/Guidelines paper3 International Journal of Computational Geometry & Applications c  World Scientific Publishing Company TWO APPROXIMATE MINKOWSKI SUM ALGORITHMS Victor Milenkovic Department of Computer Science, University of Miami Coral Gables, FL 33124-4245, USA vjm@cs.miami.edu Elisha Sacks Computer Science Department, Purdue University West Lafayette, IN 47907-2066, USA eps@cs.purdue.edu 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 km log m 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 10% faster than direct floating point implementations of the exact algorithms. The monotonic algorithm is 42% faster than the kinetic algorithm. Keywords : Minkowski sum; kinetic framework; robust computational geometry. 1. Introduction We present two approximate Minkowski sum algorithms for planar regions bounded by line and circle segments. Minkowski sums are an important computational ge- ometry concept whose applications include robot path planning, part layout, mech- anism design, and computer graphics. Prior algorithms apply to polygonal regions. The extension to circle segments is of theoretical and practical interest because line and circle segments are closed under Minkowski sums, so other algorithms can iter- ate these primitives. Moreover, curved shapes are approximated to a given accuracy with quadratically fewer circle segments than line segments. Applications typically model curves with 4–6 decimal digits accuracy, so employing circles reduces the model size by a factor of 100–1000. Although spline models are even more compact, they are not closed under Minkowski sums. The standard Minkowski sum algorithm 1 forms the kinetic convolution curve of the input regions, constructs its arrangement, and selects the cells with positive 1