Cauchy's theorem (geometry)
The Cauchy (also Cauchy theorem , Cauchy`s surface formula) is a result of integral geometry , which the French mathematician Augustin-Louis Cauchy back and says that for every convex body the average area of its parallel projections into the plane always a quarter its surface is.
In other words: the expected value with a randomly selected projection direction for the ratio between the surface area of the projection and the content of the surface of the original body is .
The theorem was proved by Cauchy in 1841 and 1850 for and in the general case by T. Kubota , Hermann Minkowski and Tommy Bonnesen .
Examples
The validity of a sphere has to be shown trivially: the image of a sphere from the radius with parallel projection into the plane is always a circle of the same radius. This means that the area of each image is exactly a quarter of the surface of the sphere .
The following two examples are only intended to clarify the situation (the values in the right-hand column vary around the value ):
- The image of a cube with edge length differs depending on the direction of projection:
Direction of projection | image | Area of the image | Relation to the cube surface |
---|---|---|---|
parallel to an edge | a square with side length | ||
parallel to a surface diagonal | a rectangle with side lengths and | ||
parallel to a room diagonal | a regular hexagon with side length | ||
other directions | irregular (but point-symmetrical) hexagons | differently | differently |
- Likewise, the image of a regular tetrahedron with edge length differs depending on the direction of projection:
Direction of projection | image | Area of the image | Relationship to the tetrahedron surface |
---|---|---|---|
along an edge | an isosceles triangle with base and height | ||
parallel to a height | an equilateral triangle with side length | ||
normal to two crooked edges | a square with side length | ||
other directions | irregular triangles or squares | differently | differently |
generalization
In the case of a convex body in -dimensional Euclidean space, the factor 4 has to be replaced by if the volume of the -dimensional unit sphere denotes.
See also
Individual evidence
- ^ Cauchy, Note sur divers théorèmes à la rectification des courbes et à la quadrature des surfaces, Compte Rendu Acad. Sci., Vol. 13, 1841, pp. 1060-1065
- ^ Cauchy, Mémoire sur la rectification des courbes et la quadrature des surfaces courbes, Mém. Acad. Sci., Volume 22 (3), 1850
- ↑ Kubota, On Convex-Closed Manifolds in N-Dimensional Space, Sci. Rep. Tohoku University, Vol. 14, 1925, pp. 85-99
- ↑ Minkowski, Theory of Convex Bodies, in particular the justification of their concept of surface, Collected Treatments, Volume 2, pp. 131–229
- ^ Bonnesen, Lesproblemèmes des isopérimètres et isoepiphanes, 1929
- ^ Evidence can be found, for example, in Gian-Carlo Rota , Daniel Klain: Introduction to geometric probability, Cambridge UP 1997
- ↑ Tsukerman, Veomett, A simple proof of Cauchy's surface area formula, Arxiv 2016