For a surface
in three dimension the focal surface, surface of centers or evolute is formed by taking the centers of the curvature sphere
s, which are the tangent
s whose radii are the reciprocals
of one of the principal curvature
s at the point of tangency. Equivalently it is the surface formed by the centers of the circles which osculate
the curvature line
As the principal curvatures are the eigenvalues of the second fundamental form, there are two at each point, and these give rise to two points of the focal surface on each normal direction
to the surface. Away from umbilical point
s, these two points of the focal surface are distinct; at umbilical points the two sheets come together. When the surface has a ridge
the focal surface has a cuspidal edge
, three such edges pass through an elliptical umbilic and only one through a hyperbolic umbilic.
At points where the Gaussian curvature
is zero, one sheet of the focal surface will have a point at infinity corresponding to the zero principal curvature.
is a point of the given surface,
the unit normal and
the principal curvature
are the corresponding two points of the focal surface.
#The focal surface of a sphere
consists of a single point, its center.
#One part of the focal surface of a surface of revolution
consists of the axis of rotation.
#The focal surface of a Torus
consists of the directrix circle and the axis of rotation.
#The focal surface of a Dupin cyclide
consists of a pair of focal conics
. The Dupin cyclides are the only surfaces, whose focal surfaces degenerate into two curves.
#One part of the focal surface of a channel surface
degenerates to its directrix.
#Two confocal quadrics
(for example an ellipsoid and a hyperboloid of one sheet) can be considered as focal surfaces of a surface.
[Hilbert Cohn-Vossen p. 197.]