Laguerre plane

Definition

Let $(\mathcal{P}, \mathcal{Z}, \in)$ be an incidence structure 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:

any three pairwise non-parallel points lie on exactly one circle;
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 $z \cap z' = \set{P}$ ;
for every circle $z$ and every point $P$ , there is exactly one point $P' \in z$ with $P \parallel P'$ ;
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.

Tag	`337Z2AXRK2`
URI	`m2://337Z2AXRK2`
Kind	definition
Writing	W0 stub
MSC	51B15
Defines	Laguerre plane
Uses	Incidence structure

Laguerre plane

Definition

Remarks

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