Evolute (surface)
focal surface
The set of cuspidal edges of the developable surfaces formed by the normals to a given surface along a family of curvature lines of
. The evolute consists of two sheets
and
, each of which is the set of centres of normal curvature of the corresponding family
or
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 and
of
and
, respectively, are
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where and
are the radii of normal curvature of the curvature lines of
and
,
is the radius vector of the surface
, and
is the unit normal to
.
The envelope of the planes that are parallel to the tangent planes to a given surface 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)
. Its radius vector is
, where
and
are the mean and Gaussian curvatures of
, respectively; consequently,
and
are parallel surfaces. In addition, if
is the Laplace operator corresponding to the third fundamental form of
and
, then
. If
, that is, if the mean evolute degenerates to a plane, then
is called a Bonnet surface; if
, then
is homothetic to
and is called a Goursat surface. In particular, for
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 at
the value
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) |
Evolute (surface). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Evolute_(surface)&oldid=16367