Triorthogonal system of surfaces

A set of surfaces in three-dimensional space that splits into three one-parameter families in such a way that any two surfaces of different families form a right angle at each point of their intersection. It is assumed that the surfaces of the triorthogonal system are regular, and in this case the curves along which the surfaces of the system intersect are lines of curvature of the surfaces (Dupin's theorem).

Triorthogonal systems of surfaces are given by the systems of coordinate surfaces in an orthogonal curvilinear coordinate system of space. Thus, in the spherical coordinate system a triorthogonal system of surfaces is made up by: one family of spheres with common centre at the origin, a second family of cones of revolution with vertex at the origin and axis through which the planes of the third family of coordinate surfaces pass. With each triorthogonal system of surfaces some orthogonal curvilinear coordinate system of space is associated. The line element of space in orthogonal coordinates $u,v,w$ has the form

$$ds^2=H_1^2du^2+H_2^2dv^2+H_3^2dw^2,$$

where $H_i(u,v,w)$, $i=1,2,3$, are the so-called Lamé functions, for which the Riemann tensor of this space form vanishes identically. A triorthogonal system of surfaces is determined by these functions up to a translation (or reflection). A triorthogonal system of surfaces can be associated with any regular surface, and the latter occurs as a combination of it. If a one-parameter family of regular surfaces, occurring in the combination of a triorthogonal system of surfaces, is given, and if this family contains at least one surface that is neither a plane nor a sphere, then the entire triorthogonal family of surfaces of this family is completely determined.

The confocal surfaces of a second-order surface in Euclidean space form a triorthogonal system of surfaces; the equation of the system of these surfaces in a Cartesian coordinate system has the form

$$\frac{x^2}{a+\lambda}+\frac{y^2}{b+\lambda}+\frac{z^2}{c+\lambda}=1,$$

where $a,b,c$ are fixed quantities, $0<c<b<a$ and $\lambda$ is a parameter. For $\lambda<c$ this equation defines a family of ellipsoids, for $c<\lambda<b$ a family of one-sheet hyperboloids, and for $b<\lambda<a$ a family of two-sheet hyperboloids. Three surfaces of this system pass through each point of space: a one-sheet hyperboloid, a two-sheet hyperboloid and an ellipsoid. Spherical transformations are automorphisms of triorthogonal systems of surfaces in Euclidean space.

References

[1]	G. Darboux, "Leçons sur les systèmes orthogonaux et les coordonnées curvilignes" , Gauthier-Villars , Paris (1910)
[2]	V.F. Kagan, "Foundations of the theory of surfaces in a tensor setting" , 1–2 , Moscow-Leningrad (1947–1948) (In Russian)

References

[a1]	M. Berger, B. Gostiaux, "Differential geometry: manifolds, curves, and surfaces" , Springer (1988) (Translated from French)
[a2]	D. Hilbert, S.E. Cohn-Vossen, "Geometry and the imagination" , Chelsea (1952) (Translated from German)