Heping Gao
ABSTRACT
We define and study exact, efficient representations of realization spaces of a natural class of underconstrained 2D Euclidean Distance constraint systems (Linkages or Frameworks) based on 1-dof Henneberg-I graphs. Each representation corresponds to a choice of parameters and yields a different parametrized configuration space. Our notion of efficiency is based on the algebraic complexities of sampling the configuration space and of obtaining a realization from the sample (parametrized) configuration. Significantly, we give purely combinatorial characterizations that capture (i) the class of graphs that have efficient configuration spaces and (ii) the possible choices of representation parameters that yield efficient configuration spaces for a given graph. Our results automatically yield an efficient algorithm for sampling realizations, without missing extreme or boundary realizations. In addition, our results formally show that our definition of efficient configuration space is robust and that our characterizations are tight. We choose the class of 1-dof Henneberg-I graphs in order to take the next step in a systematic and graded program of combinatorial characterizations of efficient configuration spaces. In particular, the results presented here are the first characterizations that go beyond graphs that have connected and convex configuration spaces.
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Digital Library
- H. Gao. Geometric Under-Constraints. Ph.D. thesis, University of Florida, 2008. Google ScholarDigital Library
- H. Gao, and M. Sitharam. Combinatorial Classification of 2D Underconstrained Sytems. Proceedings of the Seventh Asian Symposium on Computer Mathematics (ASCM 2005), Sung-il. Pae. and Hyungju. Park., Eds., 2005, Page 118--127.Google Scholar
- H. Gao and M. Sitharam. Characterizing 1-Dof Henneberg-I graphs with efficient configuration spaces. arXiv: 0810.1997 {cs.CG}.Google Scholar
- H. Gao and M. Sitharam. Characterizing 1-Dof Tree-Decomposable graphs with efficient configuration space. In preparation.Google Scholar
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- John C. Owen and Steve C. Power. The nonsolvability by radicals of generic 3-connected planar graphs. Transactions of AMS, 359(5): 2269--2303, 2006.Google ScholarCross Ref
- M. Sitharam. Graph based geometric constraint solving: problems, progress and directions. AMS-DIMACS volume on Computer Aided Design, D. Dutta, R. Janardhan, and M. Smid, Eds., 2005.Google Scholar
- M. Sitharam and H. Gao. Characterizing graphs with convex and connected configuration spaces. arXiv: 0809.3935 {cs.CG}.Google Scholar
- G. F. Zhang and X. S. Gao. Well-constrained Completion and Decomposition for Under-constrained Geometric Constraint Problems. International Journal of Computational Geometry and Applications, pages 461--478, 2006.Google ScholarCross Ref
Index Terms
- Characterizing 1-dof Henneberg-I graphs with efficient configuration spaces
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