Focal surface

From Wikipedia, the free encyclopedia

For a surface in three dimension the focal surface, surface of centers or evolute is formed by taking the centers of the curvature spheres, which are the tangential spheres whose radii are the reciprocals of one of the principal curvatures at the point of tangency. Equivalently it is the surface formed by the centers of the circles which osculate the curvature lines.

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 points, these two points of the focal surface are distinct; at umbilical points the two sheets come together. 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.

[edit] Special cases

The sphere is the only surface where both sheets of the focal surface degenerate to a single point.

Both sheets of the focal surface of Dupin 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 channel surface will form a degenerate curve. Such surfaces includes all surfaces of revolution, where the degenerate curve is the axis of revolution.