Rote, Günter, Javno predavanje februar 2006

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Pseudotriangulations and the Expansion Polytope

Günter ROTE
Freie Universität Berlin
Torek 21.2.2006 ob 10:15 v Plemljevem seminarju

A pointed pseudotriangulation of a set of points in the plane is a partition of the convex hull into pseudotriangles: polygons with three convex corners and an arbitrary number of reflex vertices. This geometric structure arises naturally in the context of rigidity of frameworks and expansive motions: motions of points in the plane where no pairwise distance decreases. The set of expansive infinitesimal motions is a polyhedron. By perturbing its facets, one arrives at a polytope whose vertices are in one-to-one correspondence with the pointed pseudotriangulations. The expansion polytope can also be considered in one dimension. It leads to the well-known associahedron in this case.

The expansion polytope provides an indirect existence proof of infinitesimal expansive motions for a polygonal chain, which is a crucial step in the solution of the Carpenter's Rule Problem: Every planar polygonal chain can be straightened without self-intersections.

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