Download 2-D Shapes Are Behind the Drapes! by Tracy Kompelien PDF

By Tracy Kompelien

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Title: 2-D Shapes Are at the back of the Drapes!
Author: Kompelien, Tracy
Publisher: Abdo Group
Publication Date: 2006/09/01
Number of Pages: 24
Binding kind: LIBRARY
Library of Congress: 2006012570

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Finally, let the number of pieces k be n. Reduction from Max 5-Partition to Min Piece Dissection. The optimization problem uses the same reduction as the decision problem, except that we do not specify k for the optimization problem. 3 When δ is small enough, the resulting packing problem is a direct simulation of 5-Partition. Intuitively, each dissected piece should contain only one element rectangle. Our reduction sets ds large enough that any piece containing parts of two element ds a1 a2 an ··· p dt p p ··· Fig.

For any convex polyhedron P , there exists an infinity of pairs of non self-overlapping nets of P that are reversible. Proof. Choose an arbitrary point s on the surface of P , but not on a vertex. The cut locus of s is the set of all points t on the surface of P such that the 20 J. Akiyama et al. Fig. 10. A lobster transforms into a fish; The separating cycle C is the hem of a pentagonal dihedron. shortest path from s to t is not unique. It is well known that the cut locus of s is a tree that spans all vertices of P .

Definition 1. k-Piece Dissection is the following decision problem: Input: two polygons P and Q of equal area, and a positive integer k. Output: whether P can be cut into k pieces such that these k pieces can be packed into Q (via translation, optional rotation, and optional reflection). To prevent ill-behaved cuts, we require every piece to be a Jordan region (with holes): the set of points interior to a Jordan curve e and exterior to k ≥ 0 Jordan curves h1 , h2 , . . , hk , such that e, h1 , h2 , .

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