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lec14 - Geometric Constraint Lecture(Apr 26 May 3 2006 1...

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Geometric Constraint Lecture(Apr 26) May 3, 2006 1 Owen’s theorem: If the underlying abstract graph of the configuration is 3-connected or if any of the graphs in a certain recursive decomposition of the graph is 3-connected then the configuration is not RS (thus not QS), otherwise it is QS. 1.1 No 3-connected subgraphs => QS The Algorithm to compute a graph to see if it is QS: Input: graph for a well-constrained system Algorithm: The graph for a well-constrained system is 2-connected, otherwise it cannot be well- constrained. A 2-connected graph can be (1) containing no 3-connected subgraphs or (2) containing 3-connected subgraphs In the first case, it can be split into edges and triangles. The algorithms splits 2-connected graph into components and repeated the splitting for the split components and until no more splits are possible. And if there are some 3- connected components in the end, then it is not QS, otherwise it is QS. 1.2 3-connected graph => not QS 1.2.1 Three equivalent statements: 1. The geometry can be constructed in principle to satisfy the dimensions using a straight edge and
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