Focal surface
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For a surface in three dimension the focal surface, surface of centers or evolute is formed by taking the centers of the two circles whose radii correspond to the principal curvatures. For each point on the surface there will generally be two points, both on the normal to the surface and the set of all these points form the focal surface, which consist of two sheets. At umbilical points the two sheets will come together. At points where the Gaussian curvature is zero one sheet of the focal surface will have a point at infinity corresponding the the zero principal curvature.
[edit] Special cases
The spheres is the only surfaces where both sheets of the focal surface degenerate to a single point.
Both sheets of the focal surface of Cyclides form degenerate circles. For the torus one of these are is the straight line along the axis of symmetry.
One sheet of the focal surface of a canal surface will forms a degenerate curve. This family includes all surfaces of revolution.
[edit] See also
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