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Dupin indicatrices and families of curve congruences

Bruce, J.W.; Tari, F.

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Authors

J.W. Bruce

F. Tari



Abstract

We study a number of natural families of binary differential equations (BDE's) on a smooth surface M in R-3. One, introduced by G. J. Fletcher in 1996, interpolates between the asymptotic and principal BDE's, another between the characteristic and principal BDE's. The locus of singular points of the members of these families determine curves on the surface. In these two cases they are the tangency points of the discriminant sets ( given by a fixed ratio of principle curvatures) with the characteristic (resp. asymptotic) BDE. More generally, we consider a natural class of BDE's on such a surface M, and show how the pencil of BDE's joining certain pairs are related to a third BDE of the given class, the so-called polar BDE. This explains, in particular, why the principal, asymptotic and characteristic BDE's are intimately related.

Citation

Bruce, J., & Tari, F. (2005). Dupin indicatrices and families of curve congruences. Transactions of the American Mathematical Society, 357(1), 267-285

Journal Article Type Article
Publication Date 2005
Deposit Date Aug 27, 2008
Publicly Available Date Aug 27, 2008
Journal Transactions of the American Mathematical Society
Print ISSN 0002-9947
Electronic ISSN 1088-6850
Publisher American Mathematical Society
Peer Reviewed Peer Reviewed
Volume 357
Issue 1
Pages 267-285
Keywords Implicit differential equations, Differential geometry.
Publisher URL http://www.ams.org/tran/2005-357-01/S0002-9947-04-03497-X/home.html

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Copyright Statement
First published in Transactions of the American Mathematical Society 357(1) 2005, published by the American Mathematical Society




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