---
tag: TAN5G5EQ96
kind: concept
name: Integral geometry
msc: [52A22, 53C65]
sources:
  - url: https://en.wikipedia.org/wiki/Integral_geometry
same_as:
  wikidata: Q2725319
  wikipedia: Integral geometry
---

## Summary

Integral geometry studies measures on families of geometric objects, such as the lines or the planes of Euclidean space.
The measures are chosen so that they do not change under rigid motions.
Its main results express a geometric size, such as the length of a curve, as an average over such a family.
For example, the length of a curve is an average of the number of times a random line meets it.
The subject grew out of geometric probability, starting with [[Buffon's needle problem|06X5PF6YFP]], and is closely tied to convex geometry and to stochastic geometry.

## Explanations

### L1 human

Throw a long straight stick at random onto a drawing and count how often it crosses the curves of the drawing.
On average the count is proportional to the total length of the curves, whatever their shape.
Integral geometry makes sense of "a random line" with a measure on the set of all lines.
This measure does not depend on where the origin is or how the axes are turned.
With this measure, lengths, areas and volumes become averages of counts and sections.