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 . ${\displaystyle 1/4}$

The theorem was proved by Cauchy in 1841 and 1850 for and in the general case by T. Kubota , Hermann Minkowski and Tommy Bonnesen . ${\displaystyle n=2,3}$

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 . ${\displaystyle r\,}$ ${\displaystyle \pi r^{2}\,}$ ${\displaystyle 4\pi r^{2}\,}$

The following two examples are only intended to clarify the situation (the values ​​in the right-hand column vary around the value ): ${\displaystyle 1/4}$

• The image of a cube with edge length differs depending on the direction of projection: ${\displaystyle a}$

Direction of projection	image	Area of ​​the image	Relation to the cube surface ${\displaystyle 6a^{2}\,}$
parallel to an edge	a square with side length ${\displaystyle a\,}$	${\displaystyle a^{2}\,}$	${\displaystyle 1:6\,}$
parallel to a surface diagonal	a rectangle with side lengths and ${\displaystyle a\,}$ ${\displaystyle a{\sqrt {2}}}$	${\displaystyle a^{2}{\sqrt {2}}}$	${\displaystyle 1:3{\sqrt {2}}\approx 1:4{,}2}$
parallel to a room diagonal	a regular hexagon with side length ${\displaystyle {\frac {a}{3}}{\sqrt {6}}}$	${\displaystyle a^{2}{\sqrt {3}}}$	${\displaystyle 1:2{\sqrt {3}}\approx 1:3{,}5}$
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: ${\displaystyle a}$

Direction of projection	image	Area of ​​the image	Relationship to the tetrahedron surface ${\displaystyle {a^{2}}{\sqrt {3}}}$
along an edge	an isosceles triangle with base and height ${\displaystyle a\,}$ ${\displaystyle {\frac {a}{2}}{\sqrt {2}}}$	${\displaystyle {\frac {a^{2}}{4}}{\sqrt {2}}}$	${\displaystyle 1:2{\sqrt {6}}\approx 1:4{,}90}$
parallel to a height	an equilateral triangle with side length ${\displaystyle a\,}$	${\displaystyle {\frac {a^{2}}{4}}{\sqrt {3}}}$	${\displaystyle 1:4\,}$
normal to two crooked edges	a square with side length ${\displaystyle {\frac {a}{2}}{\sqrt {2}}}$	${\displaystyle {\frac {a^{2}}{2}}}$	${\displaystyle 1:2{\sqrt {3}}\approx 1:3{,}46}$
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. ${\displaystyle n}$ ${\displaystyle {\frac {nv_{n}}{v_{n-1}}}}$ ${\displaystyle v_{n}}$ ${\displaystyle n}$

See also

• Dispersity analysis

Individual evidence

1. ^ 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
2. ^ Cauchy, Mémoire sur la rectification des courbes et la quadrature des surfaces courbes, Mém. Acad. Sci., Volume 22 (3), 1850
3. ↑ Kubota, On Convex-Closed Manifolds in N-Dimensional Space, Sci. Rep. Tohoku University, Vol. 14, 1925, pp. 85-99
4. ↑ Minkowski, Theory of Convex Bodies, in particular the justification of their concept of surface, Collected Treatments, Volume 2, pp. 131–229
5. ^ Bonnesen, Lesproblemèmes des isopérimètres et isoepiphanes, 1929
6. ^ Evidence can be found, for example, in Gian-Carlo Rota , Daniel Klain: Introduction to geometric probability, Cambridge UP 1997
7. ↑ Tsukerman, Veomett, A simple proof of Cauchy's surface area formula, Arxiv 2016