Bokowski, Jürgen; matematični kolokvij maj 2001

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From a Convex Polytope to an Invariant of Matrices

Jürgen Bokowski

Tehniška Univerza Darmstadt, Nemčija

10. maj 2001


Consider the end points of a line segment and the vertices of a triangle in space. Does the line segment pierce the triangle? This question is fundamental and decisive in computer graphics, in robotics, and in many other geometric problems. For an answer one can calculate the point of intersection of the line with the plane, and decide whether it lies within the triangle. In applications, decisions of this type are performed on a large scale. The end points of the line segment and the vertices of the triangle in 3-space can be given in homogeneous coordinates by a 5x4 matrix M. The intersection property is invariant under rigid motions, and it can be expressed by forgetting the concept of matrix, which does change under rigid motions, by using only the so-called oriented matroid of M. Based on the theory of oriented matroids, we can compute the answer to the intersection problem based on just 5 determinant signs, the orientations of certain quadruples of points. The generalization to oriented matroids allows the computer generation of a complete set of geometric objects with given combinatorial properties. This fact has lead to a solution of a longstanding open problem in geometry concerning certain triangulated surfaces and their flat embeddings in 3-space. The talk tries to avoid technical difficulties and shows these intuitive ideas via a series of geometric models. Convex polytopes play a decisive central role.

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