FOCAL OF A SURFACE

 Other name: caustic, evolute.

The notion of focal is the analogue for surfaces of the notion of evolute for curves.

For all points of the surface, there exist two principal curvature centers; the two focals of the surface are the two surfaces loci of the centers of curvature, and the complete focal is the union of the two focals.

Like for the evolutes of curves, the complete focal is the envelope of the normals to the surface. More precisely, along a curvature line, the normal at the surface stays tangent to a curve which is the cuspidal edge of the corresponding normal surface. The focals, associated to each of the two families of curvature lines, are the two surfaces unions of these cuspidal edges.

Examples:
- a surface is developable iff one of its focals is a curve at infinity;
- a surface is the envelope of spheres iff one of the focals degenerate into a curve; when the two focals degenerate into curves, we get a Dupin cyclide.
- in the theory of boat hulls, the focals are the loci of the metacenters.
- the focal of the ellipsoid is called the Cayley astroid.