Mathematics – Metric Geometry
Scientific paper
2009-11-02
Mathematics
Metric Geometry
36 pages, 9 figures
Scientific paper
A significant range of geometric structures whose rigidity is explored for both practical and theoretical purposes are formed by modifying generically isostatic triangulated spheres. In the block and hole structures (P, p), some edges are removed to make holes, and other edges are added to create rigid sub-structures called blocks. Previous work noted a combinatorial analogy in which blocks and holes played equivalent roles. In this paper, we connect stresses in such a structure (P, p) to first-order motions in a swapped structure (P', p), where holes become blocks and blocks become holes. When the initial structure is geometrically isostatic, this shows that the swapped structure is also geometrically isostatic, giving the strongest possible correspondence. We use a projective geometric presentation of the statics and the motions, to make the key underlying correspondences transparent.
Finbow Wendy
Ross Elissa
Whiteley Walter
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