---
tag: B4BN9PM331
kind: definition
name: Integral geometry
msc: [52A22, 53C65]
defines: TAN5G5EQ96
uses: [0XPR1E85W0, 0H3YY8CPKH, PR11BZQQPG, 0MP9BCF6DP]
sources:
  - url: https://en.wikipedia.org/wiki/Integral_geometry
---

## Definition

Let $G$ be a group acting on a space $X$.
*Integral geometry* is the study of $G$-invariant measures on spaces of geometric objects in $X$.
It also studies the integrals of geometric quantities against these measures.
In the classical case, $X$ is the [[Euclidean space|0XPR1E85W0]] $\R^n$ and $G$ is the [[Euclidean group|0H3YY8CPKH]] of rigid motions.
The objects are then the affine subspaces of a fixed dimension.

The basic example is the measure on [[lines|0MP9BCF6DP]] in $\R^2$.
For $\theta \in [0, \pi)$ and $p \in \R$, let $L(\theta, p) = \set{x \in \R^2 : x_1 \cos \theta + x_2 \sin \theta = p}$; every line is $L(\theta, p)$ for exactly one pair $(\theta, p)$.
Transport the [[measure|PR11BZQQPG]] $dp \, d\theta$ from $[0, \pi) \times \R$ to the set of lines by this parametrization.
The result is invariant under all rigid motions of $\R^2$, and up to a constant factor it is the only such measure.

## Remarks

Example: the lines that meet the closed disc of radius $r$ centred at the origin are the $L(\theta, p)$ with $\abs{p} \le r$, a set of measure $2 r \pi$.
A translation by a vector $a$ maps $L(\theta, p)$ to $L(\theta, p + a_1 \cos \theta + a_2 \sin \theta)$ and a rotation by $\alpha$ shifts $\theta$ by $\alpha$, so both preserve $dp \, d\theta$.
Non-example: writing non-vertical lines as $y = m x + c$, the measure $dm \, dc$ is not invariant under rotations.
In these coordinates the invariant measure is $(1 + m^2)^{-3/2} \, dm \, dc$.
The same ideas give invariant measures on affine $k$-planes of $\R^n$, and lead to the Crofton and kinematic formulas.