---
tag: GA0NCJMZHB
kind: concept
name: Laguerre geometry
msc: [51B15, 51B25]
sources:
  - url: https://en.wikipedia.org/wiki/Laguerre_transformations
---

## Summary

Laguerre geometry is the geometry of oriented lines and oriented circles of the plane.
Its transformations preserve oriented contact and map oriented lines to oriented lines.
Points count as circles of radius $0$, and these transformations need not map points to points.
A typical example moves every oriented line parallel to itself by the same distance and adds that distance to every signed radius.
It is a subgeometry of [[Lie sphere geometry|TFQSF9RA1G]], and its abstract incidence version is the [[Laguerre plane|F0JQ1TGKB0]]; it is named after Edmond Laguerre.

## Explanations

### L1 human

Think of lines with arrows on them and circles with a direction of travel.
A directed line and a directed circle touch properly when they are tangent and run the same way at the touching point.
Laguerre geometry keeps only this touching relation and the fact of being a line.
So it may grow every circle by one centimetre and push every line one centimetre sideways at once.
All touching pairs still touch, even though a point has become a small circle.