Document Type
Article
Publication Date
1-1-2005
Publication Title
Discrete and Computational Geometry
Abstract
This paper proposes a combinatorial approach to planning non-colliding trajectories for a polygonal bar-and-joint framework with n vertices. It is based on a new class of simple motions induced by expansive one-degree-of-freedom mechanisms, which guarantee noncollisions by moving all points away from each other. Their combinatorial structure is captured by pointed pseudo-triangulations, a class of embedded planar graphs for which we give several equivalent characterizations and exhibit rich rigidity theoretic properties. The main application is an efficient algorithm for the Carpenter's Rule Problem: convexify a simple bar-and-joint planar polygonal linkage using only non-self-intersecting planar motions. A step of the algorithm consists in moving a pseudo-triangulation-based mechanism along its unique trajectory in configuration space until two adjacent edges align. At the alignment event, a local alteration restores the pseudo-triangulation. The motion continues for O(n3) steps until all the points are in convex position. © 2005 Springer Science+Business Media, Inc.
Volume
34
Issue
4
First Page
587
Last Page
635
DOI
10.1007/s00454-005-1184-0
ISSN
01795376
Recommended Citation
Streinu, Ileana, "Pseudo-Triangulations, Rigidity and Motion Planning" (2005). Computer Science: Faculty Publications, Smith College, Northampton, MA.
https://scholarworks.smith.edu/csc_facpubs/314
Comments
Archived as published.