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Frame and direction mappings for surfaces in ℝ3

Part of: Theory of singularities and catastrophe theory General theory in ordinary differential equations Classical differential geometry

Published online by Cambridge University Press:  26 January 2019

J. W. Bruce  and
F. Tari
J. W. Bruce
Affiliation:
Department of Mathematical Sciences, University of Liverpool, Mathematical Sciences Building, Liverpool L69 7ZL, UK ([email protected])
F. Tari
Affiliation:
Instituto de Ciências Matemáticas e de Computacão - USP, Avenida Trabalhador são-carlense, 400 - Centro, CEP: 13566-590, São Carlos, SP, Brazil ([email protected])
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Abstract

We study frames in ℝ3 and mapping from a surface M in ℝ3 to the space of frames. We consider in detail mapping frames determined by a unit tangent principal or asymptotic direction field U and the normal field N. We obtain their generic local singularities as well as the generic singularities of the direction field itself. We show, for instance, that the cross-cap singularities of the principal frame map occur precisely at the intersection points of the parabolic and subparabilic curves of different colours. We study the images of the asymptotic and principal foliations on the unit sphere by their associated unit direction fields. We show that these curves are solutions of certain first order differential equations and point out a duality in the unit sphere between some of their configurations.

Keywords

Framesprincipal directions and curvesasymptotic directions and curvessingularitiesfirst order differential equationsdivergent dagrams

MSC classification

Primary: 53A05: Surfaces in Euclidean space
Secondary: 58K05: Critical points of functions and mappings 34A09: Implicit equations, differential-algebraic equations
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 149 , Issue 3 , June 2019 , pp. 795 - 830
Copyright
Copyright © Royal Society of Edinburgh 2019 

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