Largest Placements and Motion Planning of a Convex Polygon

We study two problems involving collision-free placements of a convex m-gon P in a planar polygonal environment: (i) We rst show that the largest similar copy of P inside another convex polygon Q with n edges can be computed in O(mn 2 log n) time. We also show that the combinatorial complexity of the space of all similar copies of P inside Q is O(mn 2), and that it can also be computed in O(mn 2 log n) time. (ii) We then consider the case where Q is an arbitrary polyg-onal environment with n edges. We give the rst (and relatively simple) algorithm that constructs the entire free connguration space (the 3-dimensional space of all free placements of P in Q) in time that is near-0 quadratic in mn, which is nearly optimal in the worst case. Previous solutions of the second problem were either incomplete, more expensive, or produced only part of the free connguration space. Combining our solution with parametric searching, we obtain an algorithm that nds the largest placement of P in Q in time that is also near-quadratic in mn. In addition, we describe an algorithm that preprocesses the computed free connguration space so that`reachability' queries can be answered in polylogarithmic time.

[1]  Sivan Toledo,et al.  Applications of parametric searching in geometric optimization , 1992, SODA '92.

[2]  Sivan Toledo,et al.  On critical orientations in the kedem-sharir motion planning algorithm , 1997, CCCG.

[3]  Leslie G. Valiant,et al.  Parallelism in Comparison Problems , 1975, SIAM J. Comput..

[4]  Klara Kedem,et al.  A Convex Polygon Among Polygonal Obstacles: Placement and High-clearance Motion , 1993, Comput. Geom..

[5]  Sivan Toledo,et al.  Extremal polygon containment problems , 1991, SCG '91.

[6]  Micha Sharir,et al.  An efficient motion-planning algorithm for a convex polygonal object in two-dimensional polygonal space , 1990, Discret. Comput. Geom..

[7]  Henry Spalding Baird.  Model-based image matching using location (pattern, recognition) , 1984 .

[8]  Micha Sharir,et al.  Planning a purely translational motion for a convex object in two-dimensional space using generalized Voronoi diagrams , 2016, Discret. Comput. Geom..

[9]  Bernard Chazelle,et al.  A deterministic view of random sampling and its use in geometry , 1990, Comb..

[10]  J. T. Shwartz,et al.  On the Piano Movers' Problem : III , 1983 .

[11]  Micha Sharir,et al.  The union of convex polyhedra in three dimensions , 1993, Proceedings of 1993 IEEE 34th Annual Foundations of Computer Science.

[12]  Leonidas J. Guibas,et al.  A kinetic framework for computational geometry , 1983, 24th Annual Symposium on Foundations of Computer Science (sfcs 1983).

[13]  Micha Sharir,et al.  Davenport-Schinzel sequences and their geometric applications , 1995, Handbook of Computational Geometry.

[14]  Nimrod Megiddo,et al.  Applying parallel computation algorithms in the design of serial algorithms , 1981, 22nd Annual Symposium on Foundations of Computer Science (sfcs 1981).

[15]  Mark de Berg,et al.  On lazy randomized incremental construction , 1994, STOC '94.

[16]  Micha Sharir,et al.  An Efficient and Simple Motion Planning Algorithm for a Ladder Amidst Polygonal Barriers , 1987, J. Algorithms.

[17]  Michael Ian Shamos,et al.  Computational geometry: an introduction , 1985 .

[18]  J. Schwartz,et al.  On the “piano movers'” problem I. The case of a two‐dimensional rigid polygonal body moving amidst polygonal barriers , 1983 .

[19]  Micha Sharir,et al.  Computing envelopes in four dimensions with applications , 1994, SCG '94.

[20]  Micha Sharir,et al.  On the number of critical free contacts of a convex polygonal object moving in two-dimensional polygonal space , 1987, Discret. Comput. Geom..

[21]  Roberto Tamassia,et al.  Efficient Point Location in a Convex Spatial Cell-Complex , 1989, SIAM J. Comput..