Dupin theorem

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)