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Dupin theorem

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Given three families of surfaces forming a triorthogonal system of surfaces, then the line of intersection of any two surfaces of different families will be a curvature line for each of these surfaces. For instance, co-focal central surfaces of the second order intersect along curvature lines. The theorem is named after Ch. Dupin, who gave the first proof of it [1].

References

[1] Ch. Dupin, "Développements de géométrie" , Paris (1813)
[2] V.F. Kagan, "Foundations of the theory of surfaces in a tensor setting" , 1 , Moscow-Leningrad (1947) (In Russian)


Comments

For the history see [a2], p.398 or [a3], p.361.

References

[a1] C.C. Hsiung, "A first course in differential geometry" , Wiley (1981) pp. Chapt. 3, Sect. 4
[a2] M. Berger, B. Gostiaux, "Differential geometry: manifolds, curves, and surfaces" , Springer (1988) (Translated from French)
[a3] H.S.M. Coxeter, "Introduction to geometry" , Wiley (1961) pp. 11; 258
[a4] D. Laugwitz, "Differentialgeometrie" , Teubner (1960)
[a5] W. Blaschke, K. Leichtweiss, "Elementare Differentialgeometrie" , Springer (1973)
How to Cite This Entry:
Dupin theorem. E.V. Shikin (originator), Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dupin_theorem&oldid=15665
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098