Minkowski plane

Definition

A Minkowski plane is an incidence structure $(\mathcal{P}, \mathcal{Z}, \in)$ of points and circles, where circles are subsets of $\mathcal{P}$ . It comes with two equivalence relations $\parallel_+$ and $\parallel_-$ on $\mathcal{P}$ . Two points are parallel if they are $(+)$ -parallel or $(-)$ -parallel. The following conditions hold:

for any points $P, Q$ there is exactly one point $R$ with $P \parallel_+ R$ and $R \parallel_- Q$ ;
for every circle $z$ , every point $P$ and each sign $\pm$ , exactly one point of $z$ is $(\pm)$ -parallel to $P$ ;
any three pairwise non-parallel points lie on exactly one circle;
for every circle $z$ , every point $P \in z$ and every point $Q \notin z$ not parallel to $P$ , there is exactly one circle $z'$ with $P, Q \in z'$ and $z \cap z' = \set{P}$ ;
every circle has at least three points, and there is a circle $z$ and a point not on $z$ .

Remarks

Example: let $L = \R \cup \set{\infty}$ be the real projective line. The classical real Minkowski plane has points $L \times L$ , with $(x_1, y_1) \parallel_+ (x_2, y_2)$ when $x_1 = x_2$ and $(x_1, y_1) \parallel_- (x_2, y_2)$ when $y_1 = y_2$ . Its circles are the graphs of the maps $x \mapsto (ax + b)/(cx + d)$ of $L$ with $ad - bc \ne 0$ . Clause 1 holds with $R = (x_1, y_2)$ , and clause 2 holds because each circle is the graph of a bijection of $L$ . Clause 3 holds because for distinct $x_1, x_2, x_3$ and distinct $y_1, y_2, y_3$ there is exactly one such map with $x_i \mapsto y_i$ .

Non-example: the same points and parallelisms with the graphs of all bijections of $L$ as circles fail clause 3, since many bijections take three given values at three given points.

Equivalently, the classical plane comes from a hyperboloid of one sheet. Its circles are the plane sections that contain no line of the hyperboloid, and its two families of lines give the parallel classes.

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Incidence structure concept
Minkowski plane definition

Tag	`HJFPH6NVVH`
URI	`m2://HJFPH6NVVH`
Kind	definition
Writing	W0 stub
MSC	51B20
Defines	Minkowski plane
Uses	Incidence structure

Minkowski plane

Definition

Remarks

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