---
tag: J03P12X1X4
kind: definition
name: Laguerre geometry
msc: [51B15, 51B25]
defines: GA0NCJMZHB
uses: [0MP9BCF6DP, 0S52TNNTH9]
sources:
  - url: https://en.wikipedia.org/wiki/Laguerre_transformations
---

## Definition

Work in the Euclidean plane $\R^2$ with the [[dot product|0S52TNNTH9]].

1. A *cycle* is a pair $(c, r)$ with $c \in \R^2$ and $r \in \R$. For $r \ne 0$ it is the circle with centre $c$ and radius $\abs{r}$, oriented by the sign of $r$. For $r = 0$ it is the point $c$.
2. A *spear* is an oriented [[line|0MP9BCF6DP]], given by a unit vector $N \in \R^2$ and $h \in \R$ as the line $\set{x : x \cdot N = h}$ with the orientation determined by $N$.
3. A cycle $(c, r)$ and a spear $(N, h)$ are in *oriented contact* when $c \cdot N - h = r$.

A *Laguerre transformation* is a pair of bijections, one of the set of spears and one of the set of cycles.
It is required that a cycle and a spear are in oriented contact if and only if their images are.
*Laguerre geometry* is the study of the properties of spears and cycles that are invariant under all Laguerre transformations.

## Remarks

Two cycles $(c_1, r_1)$ and $(c_2, r_2)$ touch with matching orientations when $\abs{c_1 - c_2} = \abs{r_1 - r_2}$.
In the cyclographic model, the cycle $(c, r)$ is the point $(c, r) \in \R^3$, and this condition says that the difference of the two points is a null vector of the form $x^2 + y^2 - z^2$.
Example: for $t \in \R$, the map sending the cycle $(c, r)$ to $(c, r + t)$ and the spear $(N, h)$ to $(N, h - t)$ is a Laguerre transformation.
Check: $c \cdot N - (h - t) = r + t$ holds exactly when $c \cdot N - h = r$, and both maps are bijections.
It sends each point $(c, 0)$ to a circle of radius $\abs{t}$, so it is not a map of points.
Non-example: inversion in the unit circle is not a Laguerre transformation in this sense.
It sends the lines not through the origin to circles, so it does not map spears to spears.
Euclidean isometries and dilations $x \mapsto \lambda x$, $\lambda > 0$, acting on centres and radii, are also Laguerre transformations.