12.5: Axiom II

Note that once the following claim is proved, Axiom II follows from Corollary 10.5.2.

Claim $\PageIndex{1}$

A subset of the h-plane is an h-line if and only if it forms a line for the h-distance in the sense of Definition 1.5.1.

Proof

Let $\ell$ be an h-line. Applying the main observation (Theorem 12.3.1) we can assume that $\ell$ contains the center of the absolute. In this case, $\ell$ is an intersection of a diameter of the absolute and the h-plane. Let $A$ and $B$ be the endpoints of the diameter.

Consider the map $\iota : \ell \to \mathbb{R}$ defined as

Note that $\iota :\ell\to \mathbb{R}$ is a bijection.

Further, if $X,Y\in \ell$ and the points $A$, $X$, $Y$, and $B$ appear on $[AB]$ in the same order, then

$\iota(Y)-\iota(X)=\ln \dfrac{AY}{YB}-\ln \dfrac{AX}{XB}=\ln \dfrac{AY\cdot BX}{YB\cdot XB}=XY_h.$

We proved that any h-line is a line for h-distance. The converse follows from Claim 12.4.3.