J.W. Bruce
Dupin indicatrices and families of curve congruences
Bruce, J.W.; Tari, F.
Authors
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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