Local and global geometric methods for analysis, interrogation, reconstruction, modification and design of shape

Gives an overview of some recent methods useful for local and global shape analysis and for the design of solids. These methods include, as new tools for global and local shape analysis, the spectra of the Laplace and Laplace-Beltrami operators and the concept of stable umbilical points, i.e. stable singularities of the "principal curvature line" wire-frame model of the solid's boundary surface. Most of the material in this paper deals with the medial axis transform as a tool for shape interrogation, reconstruction, modification and design. We show that it appears to be possible to construct an intuitive user interface that allows one to mould shapes by employing the medial axis transform. We also explain that the medial axis and Voronoi diagram can also be defined and computed on free surfaces in a setting where the geodesic distance between two points p and q on a surface S is defined by the shortest surface path on S joining the two points p, q. This leads to the natural and computable generalized concepts of the geodesic medial axis and geodesic Voronoi diagram on free-form surfaces. Both can be computed with a reasonable speed and with high accuracy (of about 12 digits when double floating-point arithmetic is used for the computations).