The complexity of a single face of a minkowski sum

Abstract

This note considers the complexity of a free region in the con guration space of a polygonal robot translating amidst polygonal obstacles in the plane Speci cally given polygonal sets P and Q with k and n vertices respectively k n the number of edges and vertices bounding a single face of the complement of the Minkowski sum P Q is nk k in the worst case The lower bound comes from a construction based on lower envelopes of line segments the upper bound comes from a combinatorial bound on Davenport Schinzel sequences that satisfy two alternation conditions Introduction and Background Let A and B be two sets in IR The Minkowski sum or vector sum of A and B denoted A B is the set fa b j a A b Bg The Minkowski sum is a useful concept in robot motion planning and related areas For example consider an obstacle A and a robot B that moves by translation We can choose a reference point r rigidly attached to B and suppose that B is placed such that the reference point coincides with the origin If we let B denote a copy of B rotated by then A B is the locus of placements of the reference point where A B This sum is often called a con guration space obstacle or C obstacle because B collides with A under rigid motion along a path exactly when the reference point r moved along intersects A B We con ne ourselves to the Minkowski sum of polygonal sets which is a polygonal set Let P and Q be two polygonal sets not necessarily connected with k and n vertices respectively The boundary of P Q comes from an arrangement of O nk line segments which has complexity bounded by O n k and this bound is tight in the worst case In applications such as motion planning and assembly planning however we only need to know the face complexity the number of segments that bound a single face of the complement of the Minkowski sum P Q in the worst case Figure depicts the outer face of a sum P Q Davenport Schinzel sequence analysis which is described in section shows that the face complexity is O nk nk where is the functional inverse of Ackermann s function School of Mathematical Sciences Tel Aviv University Tel Aviv Israel Department of Computer Science University of British Columbia Vancouver Canada Supported in part by an NSERC Postgraduate fellowship Department of Computer Science Polytechnic University New York USA Supported by NSF Grant CCR Robotics Lab Dept of Comp Sci Stanford University California USA Supported by NSF ARPA Grant IRI and by a grant from the Stanford Integrated Manufacturing Association SIMA Department of Computer Science University of British Columbia Vancouver Canada Supported in part by an NSERC Research Grant and a fellowship from the B C Advanced Systems Institute P Q P ⊕ Q outer boundary of P ⊕ Q v1

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Cite this paper

@inproceedings{HarPeled1995TheCO, title={The complexity of a single face of a minkowski sum}, author={Sariel Har-Peled and Timothy M. Chan and Boris Aronov and Dan Halperin and Jack Snoeyink}, booktitle={CCCG}, year={1995} }