Die Bestimmung der Vorform- und Zwischenformgeometrien nimmt bei der Auslegung der Stadienfolge für das Gesenkschmieden eine zentrale Stellung ein. Sie erfolgt bislang iterativ und zum größten Teil wissensbasiert. In diesem Beitrag wird ein Ansatz vorgestellt, mit dem das Ziel verfolgt wird, durch die zeitlich rückwärtsschreitende Betrachtung des Schmiedeprozesses die Einlegegeometrie der jeweiligen Vorstufe zu berechnen. Es wird hierfür ein Materialflussmodell angewendet, welches auf der Ermittlung des geometrischen Widerstandes im Schmiedeteil basiert und schnelle Rechenzeiten ermöglicht.

We shall describe how concepts from differential geometry, especially from Riemannian geometry have been providing powerful tools creating major advances in geometric modeling, geometry processing and image analysis dealing with the topics presented in the title of this address. This talk includes a retrospective compiling contributions of the author's works showing how concepts from local and global differential geometry have introduced new methods into geometric modeling and shape interrogation and classification finally ending with modern state of the art research on geometry processing and image processing.

A major part of this seminar is dedicated to discussing how "efficient finger prints" useful for indexing and clustering digital data collections can be derived from spectra of Laplace operators. Laplace operators can be naturally associated with geometric objects such as surfaces and solids as well as (colored) images - including 2d- and 3d-image data. Recently the latter works obtained particular attention in the area of medical imaging.

Next we focus on cut loci, the medial axis and its inverse in Euclidean and Riemannian worlds. This work starts with basic medial axis results presented by the author in the early nineties. Those results state: The Medial Axis Transform can be used to reconstruct, modify and design a given shape ("Shape Reconstruction Theorem"). Under some weak assumptions, the medial axis contains the essence of the topological shape of the geometric object as it is a deformation retract of the given shape ("Topological Shape Theorem"). Therefore the medial axis contains the homotopy type of the given shape. We present recent results showing how geodesic Voronoi diagrams, geodesic medial axis and its inverse can be computed in 3d- or higher dimensional Riemannian spaces. The "medial axis inverse" allows to construct a medial modeler providing efficient features for shape optimization with respect to shape dependent mechanical properties.

This talk includes a retrospective compiling contributions of the author`s works showing how concepts from local and global differential geometry have introduced new methods into CAGD. The seminar is essentially covering two areas surface contact and applications and distance computations and applications.

We start sketching early contributions on curvature analysis in the context of surface contact. Here we are developing criteria for second order surface contact in two different situations, i.e., in case both surfaces have tangential contact along a curve and in case they have tangential contact in a single point. The second result can be used for computing curvature entities of surfaces with a degenerate parametrization (Wolter, 1992). The first result has been used in the context of the constructing curvature continuous blend surfaces. The results of both contact cases have been generalized by the speaker to surface contact of arbitrary order.

The second part of the talk dealing with distance computations is looking into the problems of computing points nearest to a given surface as well as tracing on a surface a curve of foot point being nearest to a point moving on a given space curve. The latter problems offer a natural setting for discussing in the given context:

1. Local computational methods such as tensorial differential equations for computing the orthogonal projection curve (J. Pegna, F.-E. Wolter (1989).
2. (Semi-) global methods, e.g., Vectorfield Index - Method, (Kriezis, Patrikalakis, Wolter, 1991).
3. Global differential geometric methods naturally yielding the concept of Cut Locus / Medial Axis, (e.g. Wolter, 1985,1992).

The aforementioned distance geometric concepts yield applications useful for finding small loops in surface intersections and have been used also in the context of quality control of manufacturing, e.g., submarine propulsors and turbine blades.