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Families of surfaces: focal sets, ridges and umbilics

Published online by Cambridge University Press:  01 January 1999

J. W. BRUCE
Affiliation:
Department of Pure Mathematics, The University of Liverpool, PO Box 147, Liverpool L69 3BX, UK
P. J. GIBLIN
Affiliation:
Department of Pure Mathematics, The University of Liverpool, PO Box 147, Liverpool L69 3BX, UK
F. TARI
Affiliation:
Universidade de São Paulo, Instituto de Ciêcias Matemáticas de São Carlos, Caixa Postal 668, CEP 13560-970, São Carlos (SP), Brasil

Abstract

The contact between a surface in Euclidean space and the family of spheres carries a good deal of useful geometrical information about the surface. The centres of those spheres having degenerate tangency with the surface (in Arnold's notation of type A2) form the focal set, and the set of points on the surface where there is a contact of type A3, the so-called ridge curve, is of some interest in the field of computer vision, as a robust feature of the surface. These ridges come together at umbilics, where the contact between the surface and the unique sphere of curvature is of type D. A dual notion, that of a subparabolic curve is also of some interest. In this paper we describe the way in which the ridge and subparabolic curves can evolve in a generic 1-parameter family of surfaces.

Type
Research Article
Copyright
Cambridge Philosophical Society 1999

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