The following is from Hermann Weyl's Space-Time-Matter.
Although the structure was thus erected, it was by no means definitely decided whether, in absolute geometry, the axiom of parallels would not after all be shown to be a dependent theorem. The strict proof that non-Euclidean geometry is absolutely consistent in itself had yet to follow. This resulted almost of itself in the further development of non-Euclidean geometry. As often happens, the simplest way of proving this was not discovered at once. It was discovered by Klein as late as 1870 and depends on the construction of a Euclidean model for non-Euclidean geometry. Let us confine our attention to the plane! In a Euclidean plane with rectangular co-ordinates $x$ and $y$ we shall draw a circle $U$ of radius unity with the origin as centre. Introducing homogeneous co-ordinates $$ x=\frac{x_{1}}{x_{3}}\text{, }y=\frac{x_{2}}{x_{3}} $$ (so that the position of a point is defined by the ratio of three numbers, i.e. $x_{1}:x_{2}:x_{3}$), the equation to the circle becomes $$ -x_{1}^{2}-x_{2}^{2}+x_{3}^{2}=0. $$ Let us denote the quadratic form on the left by $\Omega\left(x\right)$ and the corresponding symmetrical bilinear form of two systems of value, $x_{i},x_{i}^{\prime}$ by $\Omega\left(x,x^{\prime}\right)$. A transformation which assigns to every point$x$ a transformed point$x^{\prime}$ according to the linear formulae $$ x_{i}^{\prime}=\sum_{k=1}^{3}\alpha_{ik}x_{k}\qquad\left(|\alpha_{ik}|\neq0\right) $$ is called, as we know, a collineation (affine transformations are a special class of collineations). It transforms every straight line, point for point, into another straight line and leaves the cross-ratio of four points on a straight line unaltered. We shall now set up a little dictionary by which we translate the conceptions of Euclidean geometry into a new language, that of non-Euclidean geometry; we use inverted commas to distinguish its words. The vocabulary of this dictionary is composed of only three words.
The word ''point'' is applied to any point on the inside of $U$ (Fig 4).
A ''straight line'' signifies the portion of a straight line lying wholly in $U$. The collineations which transform the circle $U$ into itself are of two kinds; the first leaves the sense in which $U$ is
described unaltered, whereas the second reverses it. The former are called ''congruent'' transformations; two figures composed of points are called ''congruent'' if they can be transformed into one another by such a transformation. All the axioms of Euclid except the postulate of parallels hold for these ''points,'' ''straight lines,'' and the conception ''congruence''. A whole sheaf of ''straight lines'' passing through the ''point'' $P$ which do not cut the one ''straight line'' $g$ is shown in Fig 4. This suffices to prove the consistency of non-Euclidean geometry, for things and relations are shown for which all the theorems of Euclidean geometry are valid provided that the appropriate nomenclature be adopted. It is evident, without further explanation, that Klein's model is also applicable to spatial geometry.
We now determine the non-Euclidean distance between two ''points'' in this model, viz. between $$ A=\left(x_{1}:x_{2}:x_{3}\right)\text{ and }A^{\prime}=\left(x_{1}^{\prime}:x_{2}^{\prime}:x_{3}^{\prime}\right). $$ Let the straight line $AA^{\prime}$ cut the circle $U$ in the two points, $B_{1}$, $B_{2}$. The homogeneous co-ordinates $y_{i}$ of these two points are of the form $$ y_{i}=\lambda x_{i}+\lambda^{\prime}x_{i}^{\prime} $$ and the corresponding ratio of the parameters, $\lambda:\lambda^{\prime}$, is given by the equation $\Omega\left(y\right)=0$, viz. $$ \frac{\lambda}{\lambda^{\prime}}=\frac{-\Omega\left(x,x^{\prime}\right)\pm\sqrt{\Omega^{2}\left(x,x^{\prime}\right)-\Omega\left(x\right)\Omega\left(x^{\prime}\right)}}{\Omega\left(x\right)}. $$ Hence the cross-ratio of the four points, $A,A^{\prime},B_{1},B_{2}$ is $$ \left[AA^{\prime}\right]=\frac{\Omega\left(x,x^{\prime}\right)+\sqrt{\Omega^{2}\left(x,x^{\prime}\right)-\Omega\left(x\right)\Omega\left(x^{\prime}\right)}}{\Omega\left(x,x^{\prime}\right)-\sqrt{\Omega^{2}\left(x,x^{\prime}\right)-\Omega\left(x\right)\Omega\left(x^{\prime}\right)}}. $$ This quantity which depends on the two arbitrary ''points,'' $A,A^{\prime}$, is not altered by a ''congruent'' transformation. If $A,A^{\prime},A^{\prime\prime}$ are any three ''points'' lying on a ''straight line'' in the order written, then $$ \left[AA^{\prime\prime}\right]=\left[AA^{\prime}\right]\cdot\left[A^{\prime}A^{\prime\prime}\right]. $$ The quantity $$ \tfrac{1}{2}\log\left[AA^{\prime}\right]=\overline{AA^{\prime}}=r $$ has thus the functional property $$ \overline{AA^{\prime}}+\overline{A^{\prime}A^{\prime\prime}}=\overline{AA^{\prime\prime}}. $$ As it has the same value for ''congruent'' distances $AA^{\prime}$ too, we must regard it as the non-Euclidean distance between the two points, $A,A'$. Assuming the logs to be taken to the base $e$, we get an absolute determination for the unit of measure, as was recognised by Lambert. The definition may be written in the shorter form: \begin{gather*} \cosh r=\frac{\Omega^{2}\left(x,x^{\prime}\right)}{\sqrt{\Omega\left(x\right)\cdot\Omega\left(x^{\prime}\right)}}\\ \text{(cosh denotes the hyperbolic cosine).} \end{gather*} This measure-determination had already been enunciated before Klein by Cayley who referred it to an arbitrary real or imaginary conic section $\Omega(x)=0$: he called it the ''projective measure-determination''. But it was reserved for Klein to recognise that in the case of a real conic it leads to non-Euclidean geometry.
By the cross ratio of the points $A,A^{\prime},B_{1},B_{2}$ I believe Weyl means something of the form
$$\frac{AB_{1}\cdot A^{\prime}B_{2}}{AB_{2}\cdot A^{\prime}B_{1}}.$$
But it's not clear to me what the terms $AB_{1}$, etc., should be algebraically. Will someone please explain this to me? Also, are all the values of $\lambda$ and $\lambda^{\prime}$ different (in general) for $B_1$ and $B_2?$
The following sources provide clear justification for including the "mathematical physics" tag in this question:
Hyperbolic Geometry is Projective Relativistic Geometry, Norman Wildberger (full lecture video)
MINKOWSKI SPACE-TIME AND HYPERBOLIC GEOMETRY, J F Barrett (pdf)
This is my illustration of a future light-cone enclosing the upper hyperboloid sheet determined by the coincidence set of velocity vectors (4-vectors) at the vertex. The Klein disk is the central projection onto the Euclidean plane tangent to the velocity hyperboloid at the tip of the rest-frame velocity vector. The red curve is a surface geodesic of the hyperboloid, and its projection onto the Klein disk is the blue line $\overline{B_{1}B_{2}}$.
I believe without proof that the depicted geodesic is a hyperbola branch in the $\mathbb{E}^3$ embedding space, and the arc between the tips of the two red arrows has the arc-length given by Weyl's formula. Thus the "sum of distances" is identical with the composition of Lorenz boosts.
I have never seen this connection between the Klein disk and special relativity given more than a cursory treatment in the scientific literature; despite Weyl's discussion of the topic over a century ago.
I don't have the book so I might be missing context, but here is how I read that text.
Yes, they differ in the sign in front of the square root. Observing that this leads to two different solutions for the numerator but equivalent denominators I'd write $B_1=(\lambda_1:\lambda')$ and $B_2=(\lambda_2:\lambda')$ as the homogeneous coordinates of the points of intersection with respect to the coordinate system defined by the 3d vectors for $A$ and $A'$.
\begin{align*} \lambda_1&=-\Omega\left(x,x^{\prime}\right)+\sqrt{\Omega^{2}\left(x,x^{\prime}\right)-\Omega\left(x\right)\Omega\left(x^{\prime}\right)}\\ \lambda_2&=-\Omega\left(x,x^{\prime}\right)-\sqrt{\Omega^{2}\left(x,x^{\prime}\right)-\Omega\left(x\right)\Omega\left(x^{\prime}\right)}\\ \lambda'&=\Omega\left(x\right) \end{align*}
When you have homogeneous coordinates on a projective line, it's easiest to define the cross ratio in terms of $2\times2$ determinants. For that your should note $A=(1:0)$ and $A'=(0:1)$ are the homogeneous line coordinates of the two defining points, as you can see by describing these points as linear combinations of themselves. So you'd have
$$ \frac{AB_{1}\cdot A^{\prime}B_{2}}{AB_{2}\cdot A^{\prime}B_{1}}= \frac{\begin{vmatrix}1&\lambda_1\\0&\lambda'\end{vmatrix} \cdot\begin{vmatrix}0&\lambda_2\\1&\lambda'\end{vmatrix}} {\begin{vmatrix}1&\lambda_2\\0&\lambda'\end{vmatrix} \cdot\begin{vmatrix}0&\lambda_1\\1&\lambda'\end{vmatrix}}= \frac{\lambda'\cdot(-\lambda_2)}{\lambda'\cdot(-\lambda_1)}= \frac{-\lambda_2}{-\lambda_1} $$