On the number of pseudo-triangulations of certain point sets
Journal of Combinatorial Theory Series A
Proceedings of the twenty-fourth annual symposium on Computational geometry
Morphing polyhedra with parallel faces: Counterexamples
Computational Geometry: Theory and Applications
Pointed binary encompassing trees: Simple and optimal
Computational Geometry: Theory and Applications
WADS '09 Proceedings of the 11th International Symposium on Algorithms and Data Structures
Resolving Loads with Positive Interior Stresses
WADS '09 Proceedings of the 11th International Symposium on Algorithms and Data Structures
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Preprocessing Imprecise Points and Splitting Triangulations
SIAM Journal on Computing
Convex partitions with 2-edge connected dual graphs
Journal of Combinatorial Optimization
Visibility-preserving convexifications using single-vertex moves
Information Processing Letters
Enumerating non-crossing minimally rigid frameworks
COCOON'06 Proceedings of the 12th annual international conference on Computing and Combinatorics
Maximizing maximal angles for plane straight-line graphs
WADS'07 Proceedings of the 10th international conference on Algorithms and Data Structures
| Hi-index | 0.00 |
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.