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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.

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© Robert FERRÉOL 2017