Focal surface (geometry)

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

In geometry , a focal surface is a surface associated with a given surface. she consists

• from the totality of the centers of the circles oscillating the lines of curvature .

Since there are usually two sets of lines of curvature on a surface, the focal surface is usually divided into two parts. The concept of the focal surface is a transfer of the concept of the evolute of a plane curve to surfaces. Similar to an Evolute, a focal surface is touched by the surface normals.

Burning surfaces (green and red) of a monkey saddle (blue). In the central point the Gaussian curvature is 0, otherwise negative.

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

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

the corresponding points of the two focal surfaces. If there is a main curvature , the corresponding center point of the circle lies in and the focal surface has a pole (see second picture: monkey saddle). If the Gaussian curvature is negative, the focal surfaces lie on different sides of the surface, as in the example of the hyperbolic paraboloid (see picture). ${\displaystyle 0}$ ${\displaystyle \infty }$

There are important special cases:

1. If the surface is a sphere , the focal surface degenerates into the center of the sphere.
2. If the surface is a surface of revolution , the cured an internal surface of the rotation axis.
3. With the torus , the focal surface consists of the guide circle and the torus axis.
4. In a Dupin's cyclid the focal surface consists of two focal cones . The Dupin's cyclides are the only surfaces whose focal surfaces degenerate into two curves.
5. In the case of a channel surface, a focal surface degenerates into a curve (guide curve).
6. Two confocal, dissimilar quadrics (e.g. an ellipsoid and a single-shell hyperboloid) can be understood as the focal surfaces of a surface.

Individual evidence

1. ↑ David Hilbert, Stephan Cohn-Vossen: Illustrative Geometry , Springer-Verlag, 2011, ISBN 3642199488 , p. 197.
2. ^ Morris Kline: Mathematical Thought From Ancient to Modern Times , Volume 2, Oxford University Press, 1990, ISBN 0199840423
3. ↑ Georg Glaeser, Hellmuth Stachel, Boris Odehnal: The Universe of Conics , Springer, 2016, ISBN 3662454505 , p. 147.
4. ↑ Hilbert Cohn-Vossen p. 197.