For a

surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is t ...

in three dimension the focal surface, surface of centers or evolute is formed by taking the centers of the curvature spheres, which are the

tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...

ial

sphere A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...

s 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. Elliptical umbilic focal surface

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 A ridge or a mountain ridge is a geographical feature consisting of a chain of mountains or hills that form a continuous elevated crest for an extended distance. The sides of the ridge slope away from the narrow top on either side. The line ...

the focal surface has a cuspidal edge, three such edges pass through an elliptical umbilic and only one through a hyperbolic umbilic. At points where the

Gaussian curvature In differential geometry, the Gaussian curvature or Gauss curvature of a surface at a point is the product of the principal curvatures, and , at the given point: K = \kappa_1 \kappa_2. The Gaussian radius of curvature is the reciprocal of . F ...

is zero, one sheet of the focal surface will have a point at infinity corresponding to the zero principal curvature. If

\vec p

is a point of the given surface,

\vec n

the unit normal and

k_1,k_2

the principal curvatures at

\vec p

, then :

\vec b_1(\vec p)=\vec p+\frac \quad

and

\quad \vec b_2(\vec p)=\vec p+\frac\

are the corresponding two points of the focal surface.

Special cases

#The focal surface of a

consists of a single point, its center. #One part of the focal surface of a

surface of revolution A surface of revolution is a surface in Euclidean space created by rotating a curve (the generatrix) around an axis of rotation. Examples of surfaces of revolution generated by a straight line are cylindrical and conical surfaces depending o ...

consists of the axis of rotation. #The focal surface of a

Torus In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle. If the axis of revolution does not ...

consists of the directrix circle and the axis of rotation. #The focal surface of a

Dupin cyclide In mathematics, a Dupin cyclide or cyclide of Dupin is any geometric inversion of a standard torus, cylinder or double cone. In particular, these latter are themselves examples of Dupin cyclides. They were discovered by (and named after) Charles ...

consists of a pair of focal conics. The Dupin cyclides are the only surfaces, whose focal surfaces degenerate into two curves. #One part of the focal surface of a channel surface degenerates to its directrix. #Two confocal quadrics (for example an ellipsoid and a hyperboloid of one sheet) can be considered as focal surfaces of a surface.Hilbert Cohn-Vossen p. 197.

Notes

References

* . {{DEFAULTSORT:Focal Surface Surfaces

Special cases

See also

Notes

References