Talk:Cayley–Klein metric

Section "Cross ratio and distance" is mathematically incorrect and incomprehensible
It starts by saying that "hyperbolic distance" and "Elliptic distance" between two points $$a$$ and $$b$$ differ by a scalar multiple $$C$$. But surely, then they are the same distance measure. It gets worse because it then tries to give alternative forms for the distance measure d using quantities which it calls $$\{\Omega_{xx}, \Omega_{xy}, \Omega_{yy}\}$$, but it defines two of those quantities $$\Omega_{xx}$$ and $$\Omega_{yy}$$ to equal $$0$$, so it ends up saying that $$d = \arccos(\Omega_{xy}/0) $$. --Svennik (talk) 11:12, 23 December 2021 (UTC)

Suggestions for how to greatly improve the article
This article would benefit from an explicit construction of models of elliptic geometry, Euclidean geometry and hyperbolic geometry. In each case, a quadric should be defined, and it should be shown how to go from that to the corresponding non-Euclidean geometry. I can see how this might work for hyperbolic geometry, but I'm at a loss as to how it can produce a model of Elliptic geometry. As it stands, the article doesn't show this.

The article would also benefit from less history and more focus on definitions in the first section following the introduction. History should go at the end. The historical background features specialised jargon like "algebra of throws". As it stands, this doesn't seem like the "Foundations" of the subject but merely historical background. --Svennik (talk) 11:34, 23 December 2021 (UTC)

A draft of a new article
I've produced a draft of a new article on this topic. Check it out here: https://en.wikipedia.org/wiki/User:Svennik/sandbox --Svennik (talk) 18:36, 29 December 2021 (UTC)

Technical distraction removed
The following details do not serve the general reader.

Normal forms of the absolute
Any quadric (or surface of second order) with real coefficients of the form $\Omega = \sum \omega_{\alpha\beta} x_{\alpha} x_{\beta} = 0$ can be transformed into normal or canonical forms in terms of sums of squares, while the difference in the number of positive and negative signs doesn't change under a real homogeneous transformation of determinant ≠ 0 by Sylvester's law of inertia, with the following classification ("zero-part" means real equation of the quadric, but no real points):   Proper surfaces of second order.   $$x_1^2 + x_2^2 + x_3^2 + x_4^2=0$$. Zero-part surface.  $$x_1^2 + x_2^2 + x_3^2 - x_4^2=0$$. Oval surface.  Ellipsoid  Elliptic paraboloid  Two-sheet hyperboloid    $$x_1^2 + x_2^2 - x_3^2 - x_4^2=0$$. Ring surface.  One-sheet hyperboloid </li> <li> Hyperbolic paraboloid </li> </ol> </li> </ol> </li> <li> Conic surfaces of second order. <ol> <li> $$x_1^2 + x_2^2 + x_3^2=0$$. Zero-part cone. <li> Zero-part cone </li> <li> Zero-part cylinder </li> </ol> </li> <li> $$x_1^2 + x_2^2 - x_3^2=0$$. Ordinary cone. <li> Cone </li> <li> Elliptic cylinder </li> <li> Parabolic cylinder </li> <li> Hyperbolic cylinder </li> </ol> </li> </ol> </li> <li> Plane pairs. <ol> <li> $$x_1^2 + x_2^2=0$$. Conjugate imaginary plane pairs. <li> Mutually intersecting imaginary planes.</li> <li> Parallel imaginary planes.</li> </ol> </li> <li> $$x_1^2 - x_2^2=0$$. Real plane pairs. <li> Mutually intersecting planes.</li> <li> Parallel planes.</li> <li> One plane is finite, the other one infinitely distant, thus not existent from the affine point of view.</li> </ol> </li> </ol> </li> <li> Double counting planes. <ol> <li> $$x_1^2 = 0$$. <li> Double counting finite plane. </li> <li> Double counting infinitely distant plane, not existent in affine geometry. </li> </ol> </li> </ol> </li> </ol> The collineations leaving invariant these forms can be related to linear fractional transformations or Möbius transformations. Such forms and their transformations can now be applied to several kinds of spaces, which can be unified by using a parameter ε (where ε=0 for Euclidean geometry, ε=1 for elliptic geometry, ε=−1 for hyperbolic geometry), so that the equation in the plane becomes $$x_1^2 + x_2^2+\tfrac1{\varepsilon}x_3^2=0$$ and in space $$x_1^2 + x_2^2 + x_3^2+\tfrac1{\varepsilon}x_4^2=0$$. For instance, the absolute for the Euclidean plane can now be represented by $$x_1^2 + x_2^2=0,\ x_3=0$$.

The elliptic plane or space is related to zero-part surfaces in homogeneous coordinates: $$\begin{array}{c|c} \Omega=x_1^2 + x_2^2 + x_3^2=0 & \Omega=x_1^2 + x_2^2 + x_3^2 + x_4^2 = 0 \\ \hline d = \arccos\frac{x_1 y_1 + x_2 y_2 + x_3 y_3}{\sqrt{x_1^2 + x_2^2 + x_3^2}\sqrt{y_1^2 + y_2^2 + y_3^2}} & d = \arccos\frac{x_1 y_1 + x_2 y_2 + x_3 y_3 + x_4 y_4}{\sqrt{x_1^2 + x_2^2 + x_3^2 + x_4^2}\sqrt{y_1^2 + y_2^2 + y_3^2 + x_4^2}} \end{array}$$ or using inhomogeneous coordinates $\left[\mathfrak{x}, \mathfrak{y}, \dots, 1\right] = \left[\tfrac{x_1}{x_n}, \tfrac{x_2}{x_n}, \dots, \tfrac{x_n}{x_n}\right]$ by which the absolute becomes the imaginary unit circle or unit sphere:

$$\begin{array}{c|c} \Omega=\mathfrak{x}^2+\mathfrak{y}^2 + 1=0 & \Omega=\mathfrak{x}^2+\mathfrak{y}^2+\mathfrak{z}^2 + 1=0\\ \hline d=\arccos\frac{\mathfrak{x}_1\mathfrak{x}_2+\mathfrak{y}_1\mathfrak{y}_2 + 1}{\sqrt{\mathfrak{x}_1^2+\mathfrak{y}_1^2 + 1}\sqrt{\mathfrak{x}_2^2+\mathfrak{y}_2^2 + 1}} & d=\arccos\frac{\mathfrak{x}_1\mathfrak{x}_2+\mathfrak{y}_1\mathfrak{y}_2+\mathfrak{z}_1\mathfrak{z}_2 + 1}{\sqrt{\mathfrak{x}_1^2+\mathfrak{y}_1^2+\mathfrak{z}_1^2 + 1}\sqrt{\mathfrak{x}_2^2+\mathfrak{y}_2^2+\mathfrak{y}_1^2 + 1}} \end{array}$$

or expressing the homogeneous coordinates in terms of the condition $x_1^2 + \dots + x_n^2 = y_1^2 + \dots + y_n^2 = 1$ (Weierstrass coordinates) the distance simplifies to:

$$\begin{array}{c|c} d = \arccos\left(x_1 y_1 + x_2 y_2 + x_3 y_3\right) & d = \arccos\left(x_1 y_1 + x_2 y_2 + x_3 y_3 + x_4 y_4\right)\end{array}$$

The hyperbolic plane or space is related to the oval surface in homogeneous coordinates:

$$\begin{array}{c|c} \Omega=x_1^2 + x_2^2 - x_3^2=0 & \Omega=x_1^2 + x_2^2 + x_3^2 - x_4^2=0\\ \hline d=\operatorname{arcosh}\frac{x_1 y_1 + x_2 y_2 - x_3 y_3}{\sqrt{x_1^2 + x_2^2 - x_3^2}\sqrt{y_1^2 + y_2^2 - y_3^2}} & d=\operatorname{arcosh}\frac{x_1 y_1 + x_2 y_2 + x_3 y_3 - x_4 y_4}{\sqrt{x_1^2 + x_2^2 + x_3^2 - x_4^2}\sqrt{y_1^2 + y_2^2 + y_3^2 - x_4^2}} \end{array}$$

or using inhomogeneous coordinates $$\left[\mathfrak{x},\mathfrak{y},\dots,1\right]=\left[\tfrac{x_1}{x_n},\tfrac{x_2}{x_{n}},\dots,\tfrac{x_{n}}{x_{n}}\right]$$ by which the absolute becomes the unit circle or unit sphere:

$$\begin{array}{c|c} \Omega=\mathfrak{x}^2+\mathfrak{y}^2 - 1=0 & \Omega=\mathfrak{x}^2+\mathfrak{y}^2+\mathfrak{z}^2 - 1=0\\ \hline d=\operatorname{arcosh}\frac{\mathfrak{x}_1\mathfrak{x}_2+\mathfrak{y}_1\mathfrak{y}_2 - 1}{\sqrt{\mathfrak{x}_1^2+\mathfrak{y}_1^2 - 1}\sqrt{\mathfrak{x}_2^2+\mathfrak{y}_2^2 - 1}} & d=\operatorname{arcosh}\frac{\mathfrak{x}_1\mathfrak{x}_2+\mathfrak{y}_1\mathfrak{y}_2+\mathfrak{z}_1\mathfrak{z}_2 - 1}{\sqrt{\mathfrak{x}_1^2+\mathfrak{y}_1^2+\mathfrak{z}_1^2 - 1}\sqrt{\mathfrak{x}_2^2+\mathfrak{y}_2^2+\mathfrak{y}_1^2 - 1}} \end{array}$$

or expressing the homogeneous coordinates in terms of the condition $$x_1^2 + x_2^2+\dots - x_{n}^2=y_1^2 + y_2^2+\dots - y_{n}^2=-1$$ (Weierstrass coordinates of the hyperboloid model) the distance simplifies to:

$$\begin{array}{c|c} d=\operatorname{arcosh}\left(x_1 y_1 + x_2 y_2 - x_3 y_3\right) & d=\operatorname{arcosh}\left(x_1 y_1 + x_2 y_2 + x_3 y_3 - x_4 y_4\right)\end{array}$$

Per WP:BOLD — Rgdboer (talk) 01:53, 15 January 2023 (UTC)