Sobre as singularidades de contatos de superfícies genéricas em R^3

Abstract

In this work, we investigate the differential geometry of a smooth surface M immersed in R^3 through the study of its contact with planes, spheres, and lines from the singularity theory viewpoint. Each type of contact is described by the singularities of three associated families of functions: the family of height functions, the family of distance squared functions, and the family of orthogonal projections, respectively. We show that R- (or A-) singularities of these families determine special points and curves on the surface, called robust features. We present the characterization and description of the parabolic curve, the flecnodal curve, the ridge curve, and the cusp of Gauss. In particular, we show that the parabolic curve and the flecnodal curve have second-order contact at the cusp of Gauss. Finally, by analyzing the apparent contour of the surface, we establish an alternative proof of Koenderink’s Theorem.