Möbius plane

Remarks

Clause 2 is the touching axiom: $z'$ touches $z$ at $P$.

Example: the classical real Möbius plane has as points $\R^2 \cup \set{\infty}$, the Riemann sphere. Its circles are the Euclidean circles and the sets $g \cup \set{\infty}$ for lines $g$. Three distinct points lie on exactly one such circle: on a line with $\infty$ if they are collinear or one is $\infty$, and on the circumcircle otherwise. Given $z$, $P \in z$ and $Q \notin z$, the circles through $P$ tangent to $z$ at $P$ form a pencil, and exactly one of them passes through $Q$.

Non-example: the Euclidean plane with only its circles as blocks fails clause 1, since three collinear points lie on no circle.

The plane sections of an ovoid in a three-dimensional projective space form a Möbius plane, called ovoidal; the classical plane comes from a sphere. A finite Möbius plane has an order $n$: every circle has $n + 1$ points and there are $n^2 + 1$ points.