Skip to main content
\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)
Mathematics LibreTexts

12.5: Axiom II

  • Page ID
    23658
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)

    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

    \(\iota(X)=\ln \dfrac{AX}{XB}.\) 

    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.