Cut Locus Medial Axis in Euclidean Space on Surfaces Franz-Erich Wolter Currently visiting professor at MIT Institute of Computer Science, University of Hannover February 11, 2000, 2pm - 3h30 Lubrano, CIT bldg., 4th floor, Brown University Joint Engineering/LEMS - Computer Science/Graphics Seminar Abstract The first part of the lecture gives an overview and new results on the Cut Locus and Medial Axis in the Euclidean Space. We show that Cut Locus and Medial Axis are natural tools to be used in Global Shape Interrogation and Representation. The second part of the lecture explains how these concepts can be generalized to (curved) surfaces. The Cut Locus, C_{A}, of a closed set, A, in the Euclidean space, E, is defined as the closure of the set containing all points, p, that have at least two shortest (straight line) segments to A. The Medial Axis of a solid D in E is a subset of D containing all points being center of a disc of maximal size that fits in the solid D. The Medial Axis of a solid D in E is a subset of D containing all points being center of a disc of maximal size that fits in the solid D. The Medial Axis with its related maximal disc radius function can be used to reconstruct its reference solid D because D is the union all maximal discs that fit in D. Keeping the medial axis of a reference solid D fixed and modifying the associated disc radius function, e.g. by shrinking or expanding the maximal disc radius function for some subsets of the medial axis, yields a natural design tool allowing in a simple way global shape modifications like thinning or fattening the shape. We also show how geodesic medial curves on surfaces can be computed efficiently by using methods from Riemannian geometry. These methods can be applied to compute efficiently Geodesic Voronoi Diagrams on surfaces and to compute the Geodesic Medial Axis for a surface the boundary of which is given by a finite union of piecewise curvature continuous arcs. -----------------