Evolute (surface)

focal surface

The set of cuspidal edges of the developable surfaces formed by the normals to a given surface $F$ along a family of curvature lines of $F$. The evolute consists of two sheets $F_u$ and $F_v$, each of which is the set of centres of normal curvature of the corresponding family $u$ or $v$ of curvature lines. The surface itself is called the evolvent (the evolvent surface) of its evolute. For example, the evolute of a torus is its axis of revolution and the circle described by the centre of its rotating circle.

The radius vectors $R_u$ and $R_v$ of $F_u$ and $F_v$, respectively, are

$$R_u=r+\rho_un,\quad R_v=r+\rho_vn,$$

where $\rho_u$ and $\rho_v$ are the radii of normal curvature of the curvature lines of $u$ and $v$, $r$ is the radius vector of the surface $F$, and $n$ is the unit normal to $F$.

The envelope of the planes that are parallel to the tangent planes to a given surface $F$ and pass through the middle of the segment between the centres of normal curvature of the curvature lines is called the mean evolute (mean enveloping surface) $\Phi$. Its radius vector is $R_m=r+(H/K)n$, where $H$ and $K$ are the mean and Gaussian curvatures of $F$, respectively; consequently, $F$ and $\Phi$ are parallel surfaces. In addition, if $\Delta'$ is the Laplace operator corresponding to the third fundamental form of $F$ and $w=(rn)$, then $\Delta'w=-2(w+H/K)$. If $w=-H/K$, that is, if the mean evolute degenerates to a plane, then $F$ is called a Bonnet surface; if $w+H/K=cw$, then $F$ is homothetic to $\Phi$ and is called a Goursat surface. In particular, for $c=1$ one obtains a minimal surface.

Comments

Both leaves of the evolute of a surface can be obtained as the set of critical values of the end-point mapping, defined on the normal bundle of the surface and assigning to the normal vector $v$ at $r$ the value $r+v$ in space. This can be easily generalized to define focal sets or evolute submanifolds in higher dimensions and codimensions.

References

[a1]	G. Darboux, "Leçons sur la théorie générale des surfaces et ses applications géométriques du calcul infinitésimal" , 1 , Gauthier-Villars (1887) pp. 1–18
[a2]	M. Berger, B. Gostiaux, "Differential geometry: manifolds, curves, and surfaces" , Springer (1988) (Translated from French)