Integral geometry

Integral geometry is a branch of geometry that deals with dimensions that are invariant among groups of transformations of space. It has its roots in geometric probability theory ( Buffon's needle problem , Crofton's intersection formula ). Another early classic result is Cauchy's surface formula , which expresses the surface area of ​​a convex body as the mean over the surfaces of the parallel projections of the convex body in all spatial directions. The name “integral geometry” comes from Wilhelm Blaschke , who wanted to detach the area from the geometric probability theory and was inspired by a lecture by Gustav Herglotz .

Blaschke applied integral geometry - in addition to affine subspaces - above all to convex bodies in Euclidean space. The body can be moved in space and in integral geometry integrals (mean values) are formed over the movement group of the body (rotations and translations in Euclidean space). The dimension invariant under the movement group is called the kinematic density. Crofton used kinematic densities in simple cases and then Henri Poincaré for the case of the intersection of a curve with a moving second curve.

Luis Santaló and SS Chern extended the integral geometry to smooth (non-convex) surfaces and non-Euclidean spaces, Hugo Hadwiger to convex rings (finite union of convex sets).

The reconstruction of functions from their integrals over affine subspaces ( Radon transformation ) is a sub-area that is used in computed tomography . Another application is stochastic geometry , which emerged from the 1970s .

The straight line defined by intersects twice, i.e. H. . ${\displaystyle \varphi ,p}$ ${\displaystyle \gamma }$ ${\displaystyle n_{\gamma }(\varphi ,p)=2}$

Crofton formula

For solid , only the straight lines in the blue area intersect the segment , i.e. H. it has to be. ${\displaystyle \varphi }$ ${\displaystyle [-{\tfrac {L}{2}},+{\tfrac {L}{2}}]}$ ${\displaystyle 0\leq p\leq |{\cos(\varphi )}|{\tfrac {L}{2}}}$

The Crofton formula was already known to Augustin Louis Cauchy and is sometimes named after both. It expresses the arc length of a flat curve by an integral over the number of points of intersection with a straight line; whose distance from the origin is (length of the perpendicular from the origin to the straight line) and the angle of the perpendicular to the x-axis (see Hessian normal form of the straight line equation). Then there is a kinematically invariant measure (invariant under rotations and translations of the Euclidean plane). be the number of points of intersection of the straight line parameterized by the curve. Crofton's formula then reads: ${\displaystyle s(\gamma )}$ ${\displaystyle \gamma }$ ${\displaystyle n_{\gamma }}$ ${\displaystyle p}$ ${\displaystyle \varphi }$ ${\displaystyle dpd\varphi }$ ${\displaystyle n_{\gamma }}$ ${\displaystyle p,\varphi }$

${\displaystyle s(\gamma )={\frac {1}{2}}\int \int n_{\gamma }(\varphi ,p)\;d\varphi \;dp}$

The formula can be made plausible if one looks at an example of a line of length on the x-axis, with the center at the origin. Crofton's formula then gives: ${\displaystyle \gamma }$ ${\displaystyle L}$

${\displaystyle s(\gamma )={\frac {1}{2}}\int _{0}^{2\pi }\int _{0}^{|{\cos(\varphi )}|{\frac {L}{2}}}n_{\gamma }(\varphi ,p)\;dp\;d\varphi \ ={\frac {L}{4}}\int _{0}^{2\pi }|{\cos(\varphi )}|d\varphi ={\frac {L}{4}}4=L}$.

This can be transferred to any curve by approximation using straight lines.

Another simple example is the unit circle . For each , the straight line intersects the circle exactly for and twice for . thats why ${\displaystyle \gamma }$ ${\displaystyle \varphi \in [0,2\pi ]}$ ${\displaystyle p}$ ${\displaystyle 0\leq p\leq 1}$ ${\displaystyle 0\leq p<1}$

${\displaystyle s(\gamma )={\frac {1}{2}}\int _{0}^{2\pi }\int _{0}^{1}n_{\gamma }(\varphi ,p)\;dp\;d\varphi \ ={\frac {1}{2}}\int _{0}^{2\pi }\int _{0}^{1}2\;dp\;d\varphi \ ={\frac {1}{2}}\cdot 2\cdot 2\pi =2\pi }$,

which, as expected, is the known circumference.

Main kinematic formula

One result of Blaschke is his main kinematic formula.

The special case of the plane and areas and , which are delimited by piecewise smooth curves, is considered. is moved, whereby the movement group here consists of two translations and one rotation. The kinematic density is . The curvature of sei (with the Euler-Poincaré characteristic ), the area and the perimeter (analogous to the unmoved area ). The curvature of the intersection of and is . Then the main kinematic formula is: ${\displaystyle D_{0}}$ ${\displaystyle D_{1}}$ ${\displaystyle D_{1}}$ ${\displaystyle dK_{1}}$ ${\displaystyle D_{1}}$ ${\displaystyle c_{1}=2\pi \chi (D_{1})}$ ${\displaystyle \chi }$ ${\displaystyle F_{1}}$ ${\displaystyle L_{1}}$ ${\displaystyle D_{0}}$ ${\displaystyle D_{0}}$ ${\displaystyle D_{1}}$ ${\displaystyle c_{01}}$

${\displaystyle \int _{D_{0}\cap D_{1}\neq \emptyset }\,c_{01}dK_{1}=2\pi (F_{0}c_{1}+F_{1}c_{0}+L_{0}L_{1})}$

For convex regions are and one has: ${\displaystyle c_{1}=c_{0}=c_{01}=1}$

${\displaystyle \int _{D_{0}\cap D_{1}\neq \emptyset }dK_{1}=2\pi (F_{0}+F_{1})+L_{0}L_{1}}$

There is also an n-dimensional version.

literature

• Wolfgang Blaschke: Lectures on integral geometry , 2 volumes 1935, 1937, 3rd edition VEB Deutscher Verlag der Wissenschaften 1955
• Luis Santaló: Introduction to Integral Geometry , Hermann, Paris 1953
• Hugo Hadwiger: Lectures on content, surface and isoperimetry, Springer 1957
• MI Stoka: Géométrie Intégrale , Gauthier-Villars 1968
• Luis Santaló: Integral geometry and geometric probability , Addison-Wesley 1976, Cambridge UP 2004
• Rolf Schneider, Wolfgang Weil: Integralgeometrie , Teubner 1992
• Ren De-lin: Topics in integral geometry , World Scientific 1994

Web links

• SF Shushurin, Integral geometry , Encyclopedia of mathematics, Springer

Individual evidence

1. ↑ For example Tsukerman, Veomett, A simple proof of Cauchy's surface area formula, Arxiv 2016
2. ↑ Blaschke, lectures on integral geometry, Hamburger Mathematische Einzelschriften, 1935/37, Chelsea 1949
3. ↑ Sigurdur Helgason : Integral geometry and Radon tranforms , Springer 2011
4. ^ Israel Gelfand , MI Graev , Semjon Grigorjewitsch Gindikin : Selected topics in integral geometry , American Mathematical Society, 2003
5. ^ Crofton On the Theory of Local Probability . Transactions of the Royal Society, Vol. 158, 1868, p. 181
6. ↑ Adam Weyhaupt, Cauchy Crofton`s formula , Indiana University
7. ^ Ren De-lin, Topics in integral geometry, World Scientific 1994, p. 44