Focal surface

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.[1][2]

A surface with an elliptical umbilic, and its focal surface.

A surface with a hyperbolic umbilic and its focal surface.

Focal surfaces (blue, pink) of a hyperbolic paraboloid(white) ${\displaystyle z=x^{2}-y^{2},\;0\leq x,y\leq 0.5}$

Focal surfaces (green and red) of a monkey saddle (blue). At the center point of the monkey saddle the Gauss curvature is 0, otherwise negative.

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. 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.[3] 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.

If ${\displaystyle {\vec {p}}}$ is a point of the given surface, ${\displaystyle {\vec {n}}}$ the unit normal and ${\displaystyle k_{1},k_{2}}$ the principal curvatures at ${\displaystyle {\vec {p}}}$, then

${\displaystyle {\vec {b}}_{1}({\vec {p}})={\vec {p}}+{\frac {\vec {n}}{k_{1}}}\quad }$ and ${\displaystyle \quad {\vec {b}}_{2}({\vec {p}})={\vec {p}}+{\frac {\vec {n}}{k_{2}}}\ }$

are the corresponding two points of the focal surface.