---
tag: 337Z2AXRK2
kind: definition
name: Laguerre plane
msc: [51B15]
defines: F0JQ1TGKB0
uses: [FGKNAFNHA3]
sources:
  - url: https://en.wikipedia.org/wiki/Laguerre_plane
---

## Definition

Let $(\mathcal{P}, \mathcal{Z}, \in)$ be an [[incidence structure|FGKNAFNHA3]] of points and circles, where circles are subsets of $\mathcal{P}$.
Let $\parallel$ be an equivalence relation on $\mathcal{P}$, called parallelism.
The structure is a *Laguerre plane* if:

1. any three pairwise non-parallel points lie on exactly one circle;
2. for every circle $z$, every point $P \in z$ and every point $Q \notin z$ with $P \not\parallel Q$, there is exactly one circle $z'$ with $P, Q \in z'$ and $z \cap z' = \set{P}$;
3. for every circle $z$ and every point $P$, there is exactly one point $P' \in z$ with $P \parallel P'$;
4. every circle has at least three points, and there is a circle $z$ and a point not on $z$.

## Remarks

Example: the real parabola model has points $\R^2$, parallel classes the vertical lines $x = \text{const}$, and circles the graphs $y = ax^2 + bx + c$ with $a, b, c \in \R$.
Clause 3 holds because each circle is the graph of a function of $x$.
Clause 1 holds because through three points with distinct $x$-coordinates there is exactly one polynomial of degree at most $2$.
For clause 2, two such polynomials whose graphs touch at $P = (x_0, y_0)$ differ by $\lambda (x - x_0)^2$, and exactly one value of $\lambda$ makes the graph pass through $Q$.

Non-example: keep the same circles but let parallelism be equality of points.
Then three points on a vertical line are pairwise non-parallel and lie on no circle, so clause 1 fails.

The classical model is the set of plane sections of a circular cylinder not parallel to its axis, with the generators of the cylinder as parallel classes.
More generally, an oval cone in three-dimensional projective space with its vertex removed gives an ovoidal Laguerre plane.