Ridges and Ravines on a Surface and Related Geometry of Skeletons, Caustics, and Wavefronts

Publisher Summary This chapter investigates mutual relations between ridges, ravines, their characteristic points, such as end points, extreme points, and the accompanying geometry of the singularities of caustics and wavefronts. It is noted that random surfaces, landscapes, surfaces of organs, free-form surfaces, and other complicated surfaces can be considered as generic. Typical caustic singularities consist of cuspidal edges, swallowtails, pyramids, and purses . Singularity and corresponding bifurcation on a moving wavefront are also discussed. It is found that singularity theory provides very useful tools for investigation of the surfaces of objects whose topology is not a priori known, and which can be arbitrarily complicated. Algorithms for the extraction of the considered surface line and point features were developed and implemented on an IRIS Onyx workstation. Mathematical graphics was obtained by the package Maple for symbolic computations. The results so obtained are extremely promising in analyzing CT images, terrain feature recognition, fashion design, the quality control of free-form surfaces, and in other applications where objects having complex geometry are dealt with.