History of Topics in Special Relativity/Lorentz transformation (general)

General quadratic form
The general quadratic form q(x) with coefficients of a symmetric matrix A, the associated bilinear form b(x,y), and the linear transformations of q(x) and b(x,y) into q(x′) and b(x′,y′) using the transformation matrix g, can be written as

The case n=1 is the binary quadratic form introduced by Lagrange (1773) and Gauss (1798/1801), n=2 is the ternary quadratic form introduced by Gauss (1798/1801), n=3 is the quaternary quadratic form etc.

Most general Lorentz transformation
The general Lorentz transformation follows from ($$) by setting A=A′=diag(-1,1,...,1) and det g=±1. It forms an indefinite orthogonal group called the Lorentz group O(1,n), while the case det g=+1 forms the restricted Lorentz group SO(1,n). The quadratic form q(x) becomes the Lorentz interval in terms of an indefinite quadratic form of Minkowski space (being a special case of pseudo-Euclidean space), and the associated bilinear form b(x) becomes the Minkowski inner product:

{{NumBlk|:|$$\begin{matrix}\begin{align}-x_{0}^{2}+\cdots+x_{n}^{2} & =-x_{0}^{\prime2}+\dots+x_{n}^{\prime2}\\ -x_{0}y_{0}+\cdots+x_{n}y_{n} & =-x_{0}^{\prime}y_{0}^{\prime}+\cdots+x_{n}^{\prime}y_{n}^{\prime} \end{align} \\ \hline \left.\begin{matrix}\mathbf{x}'=\mathbf{g}\cdot\mathbf{x}\\ \downarrow\\ \begin{align}x_{0}^{\prime} & =x_{0}g_{00}+x_{1}g_{01}+\dots+x_{n}g_{0n}\\ x_{1}^{\prime} & =x_{0}g_{10}+x_{1}g_{11}+\dots+x_{n}g_{1n}\\ & \dots\\ x_{n}^{\prime} & =x_{0}g_{n0}+x_{1}g_{n1}+\dots+x_{n}g_{nn} \end{align} \\ \\ \mathbf{x}=\mathbf{g}^{-1}\cdot\mathbf{x}'\\ \downarrow\\ \begin{align}x_{0} & =x_{0}^{\prime}g_{00}-x_{1}^{\prime}g_{10}-\dots-x_{n}^{\prime}g_{n0}\\ x_{1} & =-x_{0}^{\prime}g_{01}+x_{1}^{\prime}g_{11}+\dots+x_{n}^{\prime}g_{n1}\\ & \dots\\ x_{n} & =-x_{0}^{\prime}g_{0n}+x_{1}^{\prime}g_{1n}+\dots+x_{n}^{\prime}g_{nn} \end{align} \end{matrix}\right|\begin{matrix}\begin{align}\mathbf{A}\cdot\mathbf{g}^{\mathrm{T}}\cdot\mathbf{A} & =\mathbf{g}^{-1}\\ \mathbf{g}^{{\rm T}}\cdot\mathbf{A}\cdot\mathbf{g} & =\mathbf{A}\\ \mathbf{g}\cdot\mathbf{A}\cdot\mathbf{g}^{\mathrm{T}} & =\mathbf{A}\\ \\ \end{align} \\ \begin{align}\sum_{i=1}^{n}g_{ij}g_{ik}-g_{0j}g_{0k} & =\left\{ \begin{align}-1\quad & (j=k=0)\\ 1\quad & (j=k>0)\\ 0\quad & (j\ne k) \end{align} \right.\\ \sum_{j=1}^{n}g_{ij}g_{kj}-g_{i0}g_{k0} & =\left\{ \begin{align}-1\quad & (i=k=0)\\ 1\quad & (i=k>0)\\ 0\quad & (i\ne k) \end{align} \right. \end{align} \end{matrix} \end{matrix}$$|$$}}

The invariance of the Lorentz interval with n=1 between axes and conjugate diameters of hyperbolas was known for a long time since Apollonius (ca. 200 BC). Lorentz transformations ($$) for various dimensions were used by Gauss (1818), Jacobi (1827, 1833), Lebesgue (1837), Bour (1856), Somov (1863), Hill (1882) in order to simplify computations of elliptic functions and integrals. They were also used by Chasles (1829) and Weddle (1847) to describe relations on hyperboloids, as well as by Poincaré (1881), Cox (1881-91), Picard (1882, 1884), Killing (1885, 1893), Gérard (1892), Hausdorff (1899), Woods (1901, 1903), Liebmann (1904/05) to describe hyperbolic motions (i.e. rigid motions in the hyperbolic plane or hyperbolic space), which were expressed in terms of Weierstrass coordinates of the hyperboloid model satisfying the relation $$-x_{0}^{2}+\cdots+x_{n}^{2}=-1$$ or in terms of the Cayley–Klein metric of projective geometry using the "absolute" form $$-x_{0}^{2}+\cdots+x_{n}^{2}=0$$ as discussed by Klein (1871-73). In addition, infinitesimal transformations related to the Lie algebra of the group of hyperbolic motions were given in terms of Weierstrass coordinates $$-x_{0}^{2}+\cdots+x_{n}^{2}=-1$$ by Killing (1888-1897).

Most general Lorentz transformation of velocity
If $$x_{i},\ x_{i}^{\prime}$$ in ($$) are interpreted as homogeneous coordinates, then the corresponding inhomogenous coordinates $$u_{s},\ u_{s}^{\prime}$$ follow by


 * $$\frac{x_{s}}{x_{0}}=u_{s},\ \frac{x_{s}^{\prime}}{x_{0}^{\prime}}=u_{s}^{\prime}\ (s=1,2\dots n)$$

defined by $$u_{1}^{2}+u_{2}^{2}+\dots+u_{n}^{2}\le1$$ so that the Lorentz transformation becomes a homography inside the unit hypersphere, which John Lighton Synge called "the most general formula for the composition of velocities" in terms of special relativity (the transformation matrix g stays the same as in ($$)):

{{NumBlk|:|$$\begin{align}u_{s}^{\prime} & =\frac{g_{s0}+g_{s1}u_{1}+\dots+g_{sn}u_{n}}{g_{00}+g_{01}u_{1}+\dots+g_{0n}u_{n}}\\ \\ u_{s} & =\frac{-g_{0s}+g_{1s}u_{1}^{\prime}+\dots+g_{ns}u_{n}^{\prime}}{g_{00}-g_{10}u_{1}^{\prime}-\dots-g_{n0}u_{n}^{\prime}} \end{align} \left|\begin{align}\sum_{i=1}^{n}g_{ij}g_{ik}-g_{0j}g_{0k} & =\left\{ \begin{align}-1\quad & (j=k=0)\\ 1\quad & (j=k>0)\\ 0\quad & (j\ne k) \end{align} \right.\\ \sum_{j=1}^{n}g_{ij}g_{kj}-g_{i0}g_{k0} & =\left\{ \begin{align}-1\quad & (i=k=0)\\ 1\quad & (i=k>0)\\ 0\quad & (i\ne k) \end{align} \right. \end{align} \right.$$|$$}}

Such Lorentz transformations for various dimensions were used by Gauss (1818), Jacobi (1827–1833), Lebesgue (1837), Bour (1856), Somov (1863), Hill (1882), Callandreau (1885) in order to simplify computations of elliptic functions and integrals, by Picard (1882-1884) in relation to Hermitian quadratic forms, or by Woods (1901, 1903) in terms of the Beltrami–Klein model of hyperbolic geometry. In addition, infinitesimal transformations in terms of the Lie algebra of the group of hyperbolic motions leaving invariant the unit sphere $$-1+u_{1}^{\prime2}+\cdots+u_{n}^{\prime2}=0$$ were given by Lie (1885-1893) and Werner (1889) and Killing (1888-1897).

Equality of difference in squares
Apollonius of Perga (c. 240–190 BC) in his 7th book on conics defined the following well known proposition (the 7th book survived in Arabian translation, and was translated into Latin in 1661 and 1710), as follows:


 * The difference of the squares of the two axes of the hyperbola is equal to the difference of the squares of any two conjugate diameters. (Latin translation 1661 by Giovanni Alfonso Borelli and Abraham Ecchellensis.)


 * In every hyperbola the difference between the squares of the axes is equal to the difference between the squares of any conjugate diameters of the section. (Latin translation 1710 by Edmond Halley.)


 * [..] in every hyperbola the difference of the squares on any two conjugate diameters is equal to the [..] difference [..] of the squares on the axes. (English translation 1896 by w:Thomas Heath.)

Philippe de La Hire (1685) stated this proposition as follows:

I say that the difference of the squares of any two diameters conjugated to each other, AB, DE, is equal to the difference of the squares of any two other diameters conjugated to each other, NM, LK.

and also summarized the related propositions in the 7th book of Apollonius:

In a hyperbola, the difference of the squares of the axes is equal to the difference of the squares of any two conjugate diameters.

Guillaume de l'Hôpital (1707), using the methods of w:analytic geometry, demonstrated the same proposition:

The difference of the squares of any two conjugate diameters "Mm, Ss" is equal to the difference of the squares of the two axes "Aa, Bb." We are to prove that $\overline{CS}^{2}-\overline{CM}^{2}=\overline{CB}^{2}-\overline{CA}^{2}$, or $\overline{CM}^{2}-\overline{CS}^{2}=\overline{CA}^{2}-\overline{CB}^{2}$. (English translation 1723 by Edmund Stone.)

Equality of areas of parallelograms


Apollonius also gave another well known proposition in his 7th book regarding ellipses as well as conjugate sections of hyperbolas (see also Del Centina & Fiocca for further details on the history of this proposition):


 * In the ellipse, and in conjugate sections [the opposite branches of two conjugate hyperbolas] the parallelogram bounded by the axes is equal to the parallelogram bounded by any pair of conjugate diameters, if its angles are equal to the angles the conjugate diameters form at the centre. (English translation by Del Centina & Fiocca based on the Latin translation 1661 by Giovanni Alfonso Borelli and Abraham Ecchellensis. )


 * If two conjugate diameters are taken in an ellipse, or in the opposite conjugate sections; the parallelogram bounded by them is equal to the rectangle bounded by the axes, provided its angles are equal to those formed at the centre by the conjugate diameters. (English translation by Del Centina & Fiocca based on the Latin translation 1710 by Edmond Halley.) )


 * If PP', DD' be two conjugate diameters in an ellipse or in conjugate hyperbolas, and if tangents be drawn at the four extremities forming a parallelogram LL'MM', then the parallelogram LL'MM' = rect. AA'·BB'. (English translation 1896 by w:Thomas Heath.)

Grégoire de Saint-Vincent independently (1647) stated the same proposition:

The parallelograms whose opposite sides are tangent to two conjugate hyperbolas at the extremities of two conjugate diameters are equivalent among them. (English translation by Del Centina & Fiocca. )

Philippe de La Hire (1685), who was aware of both Apollonius 7th book and Saint-Vincent's book, stated this proposition as follows:

If a parallelogram FGHI is circumscribed about conjugate sections NA, DL, BM, KE whose sides are parallel to two conjugate diameters ED, BA drawn through their extremities, and with similar method another parallelogram OPQR is drawn through the extremities of other two conjugate diameters, then the parallelograms FGHI, OPQR are equal. (English translation by Del Centina & Fiocca. )

and also summarized the related propositions in the 7th book of Apollonius:

In conjugate sections and in the ellipse, the parallelogram constructed with the axes, is equal to the parallelogram constructed with any two conjugated diameters, provided the angles are equal to those between the diameters themselves. (English translation by Del Centina & Fiocca. )

Lagrange (1773) – Binary quadratic forms
After the invariance of the sum of squares under linear substitutions was discussed by E:Euler (1771), the general expressions of a binary quadratic form and its transformation was formulated by Joseph-Louis Lagrange (1773/75) as follows


 * $$\begin{matrix}py^{2}+2qyz+rz^{2}=Ps^{2}+2Qsx+Rx^{2}\\

\hline \begin{align}y & =Ms+Nx\\ z & =ms+nx \end{align} \left|\begin{matrix}\begin{align}P & =pM^{2}+2qMm+rm^{2}\\ Q & =pMN+q(Mn+Nm)+rmn\\ R & =pN^{2}+2qNn+rn^{2} \end{align} \\ \downarrow\\ PR-Q^{2}=\left(pr-q^{2}\right)(Mn-Nm)^{2} \end{matrix}\right. \end{matrix}$$

Binary quadratic forms
The theory of binary quadratic forms was considerably expanded by Carl Friedrich Gauss (1798, published 1801) in his Disquisitiones Arithmeticae. He rewrote Lagrange's formalism as follows using integer coefficients α,β,γ,δ:


 * $$\begin{matrix}F=ax^{2}+2bxy+cy^{2}=(a,b,c)\\

F'=a'x^{\prime2}+2b'x'y'+c'y^{\prime2}=(a',b',c')\\ \hline \begin{align}x & =\alpha x'+\beta y'\\ y & =\gamma x'+\delta y'\\ \\ x' & =\delta x-\beta y\\ y' & =-\gamma x+\alpha y \end{align} \left|\begin{matrix}\begin{align}a' & =a\alpha^{2}+2b\alpha\gamma+c\gamma^{2}\\ b' & =a\alpha\beta+b(\alpha\delta+\beta\gamma)+c\gamma\delta\\ c' & =a\beta^{2}+2b\beta\delta+c\delta^{2} \end{align} \\ \downarrow\\ b^{2}-a'c'=\left(b^{2}-ac\right)(\alpha\delta-\beta\gamma)^{2} \end{matrix}\right. \end{matrix}$$

which is equivalent to ($$) (n=1). As pointed out by Gauss, F and F′ are called "proper equivalent" if αδ-βγ=1, so that F is contained in F′ as well as F′ is contained in F. In addition, if another form F″ is contained by the same procedure in F′ it is also contained in F and so forth.

Ternary quadratic forms
Gauss (1798/1801) also discussed ternary quadratic forms with the general expression


 * $$\begin{matrix}f=ax^{2}+a'x^{\prime2}+ax^{\prime\prime2}+2bx'x+2b'xx+2bxx'=\left(\begin{matrix}a, & a', & a''\\

b, & b', & b'' \end{matrix}\right)\\ g=my^{2}+m'y^{\prime2}+my^{\prime\prime2}+2ny'y+2n'yy+2nyy'=\left(\begin{matrix}m, & m', & m''\\ n, & n', & n'' \end{matrix}\right)\\ \hline \begin{align}x & =\alpha y+\beta y'+\gamma y\\ x' & =\alpha'y+\beta'y'+\gamma'y\\ x & =\alphay+\betay'+\gammay'' \end{align} \end{matrix}$$

which is equivalent to ($$) (n=2). Gauss called these forms definite when they have the same sign such as x2+y2+z2, or indefinite in the case of different signs such as x2+y2-z2. While discussing the classification of ternary quadratic forms, Gauss (1801) presented twenty special cases, among them these six variants:


 * $$\left(\begin{matrix}a, & a', & a''\\

b, & b', & b'' \end{matrix}\right)\Rightarrow\begin{matrix}\left(\begin{matrix}1, & -1, & 1\\ 0, & 0, & 0 \end{matrix}\right),\ \left(\begin{matrix}-1, & 1, & 1\\ 0, & 0, & 0 \end{matrix}\right),\ \left(\begin{matrix}1, & 1, & -1\\ 0, & 0, & 0 \end{matrix}\right),\\ \left(\begin{matrix}1, & -1, & -1\\ 0, & 0, & 0 \end{matrix}\right),\ \left(\begin{matrix}-1, & 1, & -1\\ 0, & 0, & 0 \end{matrix}\right),\ \left(\begin{matrix}-1, & -1, & 1\\ 0, & 0, & 0 \end{matrix}\right) \end{matrix}$$

Homogeneous coordinates
Gauss (1818) discussed planetary motions together with formulating elliptic functions. In order to simplify the integration, he transformed the expression


 * $$(AA+BB+CC)tt+aa(t\cos E)^{2}+bb(t\sin E)^{2}-2aAt\cdot t\cos E-2bBt\cdot t\sin E$$

into


 * $$G+G'\cos T^{2}+G''\sin T^{2}$$

in which the eccentric anomaly E is connected to the new variable T by the following transformation including an arbitrary constant k, which Gauss then rewrote by setting k=1:


 * $$\begin{matrix}{\scriptstyle \left(\alpha+\alpha'\cos T+\alpha\sin T\right)^{2}+\left(\beta+\beta'\cos T+\beta\sin T\right)^{2}-\left(\gamma+\gamma'\cos T+\gamma''\sin T\right)^{2}}=0\\

k\left(\cos^{2}T+\sin^{2}T-1\right)=0\\ \hline \begin{align}\cos E & =\frac{\alpha+\alpha'\cos T+\alpha\sin T}{\gamma+\gamma'\cos T+\gamma\sin T}\\ \sin E & =\frac{\beta+\beta'\cos T+\beta\sin T}{\gamma+\gamma'\cos T+\gamma\sin T} \end{align} \left|{\scriptstyle \begin{align}-\alpha\alpha-\beta\beta+\gamma\gamma & =k & \alpha\alpha-\alpha'\alpha'-\alpha\alpha & =-k\\ -\alpha'\alpha'-\beta'\beta'+\gamma'\gamma' & =-k & \beta\beta-\beta'\beta'-\beta\beta & =-k\\ -\alpha\alpha-\beta\beta+\gamma\gamma & =-k & \gamma\gamma-\gamma'\gamma'-\gamma\gamma & =+k\\ -\alpha'\alpha-\beta'\beta+\gamma'\gamma & =0 & \beta\gamma-\beta'\gamma'-\beta\gamma'' & =0\\ -\alpha\alpha-\beta\beta+\gamma\gamma & =0 & \gamma\alpha-\gamma'\alpha'-\gamma\alpha'' & =0\\ -\alpha\alpha'-\beta\beta'+\gamma\gamma' & =0 & \alpha\beta-\alpha'\beta'-\alpha\beta & =0 \end{align} }\right.\\ \hline k=1\\ \begin{align}t\cos E & =\alpha+\alpha'\cos T+\alpha''\sin T\\ t\sin E & =\beta+\beta'\cos T+\beta''\sin T\\ t & =\gamma+\gamma'\cos T+\gamma''\sin T \end{align} \left|{\scriptstyle \begin{align}-\alpha\alpha-\beta\beta+\gamma\gamma & =1\\ -\alpha'\alpha'-\beta'\beta'+\gamma'\gamma' & =-1\\ -\alpha\alpha-\beta\beta+\gamma\gamma & =-1\\ -\alpha'\alpha-\beta'\beta+\gamma'\gamma'' & =0\\ -\alpha\alpha-\beta\beta+\gamma''\gamma & =0\\ -\alpha\alpha'-\beta\beta'+\gamma\gamma' & =0 \end{align} }\right. \end{matrix}$$

Subsequently, he showed that these relations can be reformulated using three variables x,y,z and u,u′,u″, so that


 * $$aaxx+bbyy+(AA+BB+CC)zz-2aAxz-2bByz$$

can be transformed into


 * $$Guu+G'u'u'+Guu''$$,

in which x,y,z and u,u′,u″ are related by the transformation:


 * $$\begin{align}x & =\alpha u+\alpha'u'+\alphau\\

y & =\beta u+\beta'u'+\betau\\ z & =\gamma u+\gamma'u'+\gammau\\ \\ u & =-\alpha x-\beta y+\gamma z\\ u' & =\alpha'x+\beta'y-\gamma'z\\ u & =\alphax+\betay-\gammaz \end{align} \left|{\scriptstyle \begin{align}-\alpha\alpha-\beta\beta+\gamma\gamma & =1\\ -\alpha'\alpha'-\beta'\beta'+\gamma'\gamma' & =-1\\ -\alpha\alpha-\beta\beta+\gamma\gamma & =-1\\ -\alpha'\alpha-\beta'\beta+\gamma'\gamma'' & =0\\ -\alpha\alpha-\beta\beta+\gamma''\gamma & =0\\ -\alpha\alpha'-\beta\beta'+\gamma\gamma' & =0 \end{align} }\right.$$

Jacobi (1827, 1833/34) – Homogeneous coordinates
Following Gauss (1818), Carl Gustav Jacob Jacobi extended Gauss' transformation in 1827:


 * $${\scriptstyle \begin{matrix}\cos P^{2}+\sin P^{2}\cos\vartheta^{2}+\sin P^{2}\sin\vartheta^{2}=1\\

k\left(\cos\psi^{2}+\sin\psi^{2}\cos\varphi^{2}+\sin\psi^{2}\sin\varphi^{2}-1\right)=0\\ \hline {\left.\begin{matrix}\mathbf{(1)}\begin{align}\cos P & =\frac{\alpha+\alpha'\cos\psi+\alpha\sin\psi\cos\varphi+\alpha\sin\psi\sin\varphi}{\delta+\delta'\cos\psi+\delta\sin\psi\cos\varphi+\delta\sin\psi\sin\varphi}\\ \sin P\cos\vartheta & =\frac{\beta+\beta'\cos\psi+\beta\sin\psi\cos\varphi+\beta\sin\psi\sin\varphi}{\delta+\delta'\cos\psi+\delta\sin\psi\cos\varphi+\delta\sin\psi\sin\varphi}\\ \sin P\sin\vartheta & =\frac{\gamma+\beta'\cos\psi+\gamma\sin\psi\cos\varphi+\gamma\sin\psi\sin\varphi}{\delta+\delta'\cos\psi+\delta\sin\psi\cos\varphi+\delta\sin\psi\sin\varphi}\\ \\ \cos\psi & =\frac{-\delta'+\alpha'\cos P+\beta'\sin P\cos\vartheta+\gamma'\sin P\sin\vartheta}{\delta-\alpha\cos P-\beta\sin P\cos\vartheta-\gamma\sin P\sin\vartheta}\\ \sin\psi\cos\varphi & =\frac{-\delta+\alpha\cos P+\beta\sin P\cos\vartheta+\gamma\sin P\sin\vartheta}{\delta-\alpha\cos P-\beta\sin P\cos\vartheta-\gamma\sin P\sin\vartheta}\\ \sin\psi\sin\varphi & =\frac{-\delta+\alpha\cos P+\beta\sin P\cos\vartheta+\gamma\sin P\sin\vartheta}{\delta-\alpha\cos P-\beta\sin P\cos\vartheta-\gamma\sin P\sin\vartheta} \end{align} \\ \\ \hline \mathbf{(2)}\begin{align}\alpha\mu+\beta x+\gamma y+\delta z & =m\\ \alpha'\mu+\beta'x+\gamma'y+\delta'z & =m'\\ \alpha\mu+\betax+\gammay+\deltaz & =m''\\ \alpha\mu+\betax+\gammay+\deltaz & =m'''\\ \\ Am+A'm'+Am+Am & =\mu\\ Bm+B'm'+Bm+Bm & =x\\ Cm+C'm'+Cm+Cm & =y\\ Dm+D'm'+Dm+Dm & =z\\ \\ \end{align} \\ \begin{align}\alpha & =-kA, & \beta & =-kB, & \gamma & =-kC, & \delta & =kD,\\ \alpha' & =kA', & \beta' & =kB', & \gamma' & =kC', & \delta' & =-kD',\\ \alpha & =kA, & \beta & =kB, & \gamma & =kC, & \delta & =-kD,\\ \alpha & =kA, & \beta & =kB, & \gamma & =kC, & \delta & =-kD, \end{align} \end{matrix}\right|\begin{matrix}\begin{align}\alpha\alpha+\beta\beta+\gamma\gamma-\delta\delta & =-k\\ \alpha'\alpha'+\beta'\beta'+\gamma'\gamma'-\delta'\delta' & =k\\ \alpha\alpha+\beta\beta+\gamma\gamma-\delta\delta & =k\\ \alpha\alpha+\beta\beta+\gamma\gamma-\delta\delta & =k\\ \alpha\alpha'+\beta\beta'+\gamma\gamma'-\delta\delta' & =0\\ \alpha\alpha+\beta\beta+\gamma\gamma-\delta\delta & =0\\ \alpha\alpha+\beta\beta+\gamma\gamma-\delta\delta & =0\\ \alpha\alpha+\beta\beta+\gamma\gamma-\delta\delta & =0\\ \alpha\alpha'+\beta\beta'+\gamma\gamma'-\delta\delta' & =0\\ \alpha'\alpha+\beta'\beta+\gamma'\gamma-\delta'\delta & =0\\ \\ -\alpha\alpha+\alpha'\alpha'+\alpha\alpha+\alpha\alpha & =k\\ -\beta\beta+\beta'\beta'+\beta\beta+\beta\beta & =k\\ -\gamma\gamma+\gamma'\gamma'+\gamma\gamma+\gamma\gamma & =k\\ -\delta\delta+\delta'\delta'+\delta\delta+\delta\delta & =-k\\ -\alpha\beta+\alpha'\beta'+\alpha\beta+\alpha\beta & =0\\ -\alpha\gamma+\alpha'\gamma'+\alpha\gamma+\alpha\gamma & =0\\ -\alpha\delta+\alpha'\delta'+\alpha\delta+\alpha\delta & =0\\ -\beta\gamma+\beta'\gamma'+\beta\gamma+\beta\gamma & =0\\ -\gamma\delta+\gamma'\delta'+\gamma\delta+\gamma\delta & =0\\ -\delta\beta+\delta'\beta'+\delta\beta+\delta\beta & =0 \end{align} \end{matrix}} \end{matrix}}$$

Alternatively, in two papers from 1832 Jacobi started with an ordinary orthogonal transformation, and by using an imaginary substitution he arrived at Gauss' transformation (up to a sign change):


 * $${\scriptstyle \begin{matrix}xx+yy+zz=ss+s's'+ss=0\\

\mathbf{(1)}\begin{align}x & =\alpha s+\alpha's'+\alphas\\ y & =\beta s+\beta's'+\betas\\ z & =\gamma s+\gamma's'+\gammas\\ \\ s & =\alpha x+\beta y+\gamma z\\ s' & =\alpha'x+\beta'y+\gamma'z\\ s & =\alphax+\betay+\gammaz \end{align} \left|\begin{align}\alpha\alpha+\beta\beta+\gamma\gamma & =1 & \alpha\alpha+\alpha'\alpha'+\alpha\alpha & =1\\ \alpha'\alpha'+\beta'\beta'+\gamma'\gamma' & =1 & \beta\beta+\beta'\beta'+\beta\beta & =1\\ \alpha\alpha+\beta\beta+\gamma\gamma & =1 & \gamma\gamma+\gamma'\gamma'+\gamma\gamma & =1\\ \alpha'\alpha+\beta'\beta+\gamma'\gamma & =0 & \beta\gamma+\beta'\gamma'+\beta\gamma'' & =0\\ \alpha\alpha+\beta\beta+\gamma\gamma & =0 & \gamma\alpha+\gamma'\alpha'+\gamma\alpha'' & =0\\ \alpha\alpha'+\beta\beta'+\gamma\gamma' & =0 & \alpha\beta+\alpha'\beta'+\alpha\beta & =0 \end{align} \right.\\ \hline \left[\frac{y}{x},\ \frac{z}{x},\ \frac{s'}{s},\ \frac{s''}{s}\right]=\left[-i\cos\varphi,\ -i\sin\varphi,\ i\cos\eta,\ i\sin\eta\right]\\ \left[\alpha',\ \alpha,\ \beta,\ \gamma\right]=\left[i\alpha',\ i\alpha,\ -i\beta,\ -i\gamma\right]\\ \hline \begin{matrix}\mathbf{(2)}\begin{matrix}\left(\alpha-\alpha'\cos\eta-\alpha\sin\eta\right)^{2}=\left(\beta-\beta'\cos\eta-\beta\sin\eta\right)^{2}+\left(\gamma-\gamma'\cos\eta-\gamma''\sin\eta\right)^{2}\\ \left(\alpha-\beta\cos\phi-\gamma\sin\phi\right)^{2}=\left(\alpha'-\beta'\cos\phi-\gamma'\sin\phi\right)^{2}+\left(\alpha-\beta\cos\phi-\gamma''\sin\phi\right)^{2}\\ \hline \begin{align}\cos\phi & =\frac{\beta-\beta'\cos\eta-\beta\sin\eta}{\alpha-\alpha'\cos\eta-\alpha\sin\eta}, & \cos\eta & =\frac{\alpha'-\beta'\cos\phi-\gamma'\sin\phi}{\alpha-\beta\cos\phi-\gamma\sin\phi}\\ \sin\phi & =\frac{\gamma-\gamma'\cos\eta-\gamma\sin\eta}{\alpha-\alpha'\cos\eta-\alpha\sin\eta}, & \sin\eta & =\frac{\alpha-\beta\cos\phi-\gamma''\sin\phi}{\alpha-\beta\cos\phi-\gamma\sin\phi} \end{align} \end{matrix}\\ \hline \\ \mathbf{(3)}\begin{matrix}1-zz-yy=\frac{1-s's'-ss}{\left(\alpha-\alpha's'-\alphas\right)^{2}}\\ \hline \begin{align}y & =\frac{\beta-\beta's'-\betas}{\alpha-\alpha's'-\alphas}, & s' & =\frac{\alpha'-\beta'y-\gamma'z}{\alpha-\beta y-\gamma z},\\ z & =\frac{\gamma-\gamma's'-\gammas}{\alpha-\alpha's'-\alphas'}, & s & =\frac{\alpha-\betay-\gammaz}{\alpha-\beta y-\gamma z}, \end{align} \end{matrix} \end{matrix}\left|\begin{align}\alpha\alpha-\beta\beta-\gamma\gamma & =1\\ \alpha'\alpha'-\beta'\beta'-\gamma'\gamma' & =-1\\ \alpha\alpha-\beta\beta-\gamma\gamma & =-1\\ \alpha'\alpha-\beta'\beta-\gamma'\gamma'' & =0\\ \alpha\alpha-\beta\beta-\gamma''\gamma & =0\\ \alpha\alpha'-\beta\beta'-\gamma\gamma' & =0\\ \\ \alpha\alpha-\alpha'\alpha'-\alpha\alpha & =1\\ \beta\beta-\beta'\beta'-\beta\beta & =-1\\ \gamma\gamma-\gamma'\gamma'-\gamma\gamma & =-1\\ \beta\gamma-\beta'\gamma'-\beta\gamma & =0\\ \gamma\alpha-\gamma'\alpha'-\gamma\alpha & =0\\ \alpha\beta-\alpha'\beta'-\alpha\beta & =0 \end{align} \right. \end{matrix}}$$

Extending his previous result, Jacobi (1833) started with Cauchy's (1829) orthogonal transformation for n dimensions, and by using an imaginary substitution he formulated Gauss' transformation (up to a sign change) in the case of n dimensions:


 * $${\scriptstyle \begin{matrix}x_{1}x_{1}+x_{2}x_{2}+\dots+x_{n}x_{n}=y_{1}y_{1}+y_{2}y_{2}+\dots+y_{n}y_{n}\\

\hline \mathbf{(1)\ }\begin{align}y_{\varkappa} & =\alpha_{1}^{(\varkappa)}x_{1}+\alpha_{2}^{(\varkappa)}x_{2}+\dots+\alpha_{n}^{(\varkappa)}x_{n}\\ x_{\varkappa} & =\alpha_{\varkappa}^{\prime}y_{1}+\alpha_{\varkappa}^{\prime\prime}y_{2}+\dots+\alpha_{\varkappa}^{(n)}y_{n}\\ \\ \frac{y_{\varkappa}}{y_{n}} & =\frac{\alpha_{1}^{(\varkappa)}x_{1}+\alpha_{2}^{(\varkappa)}x_{2}+\dots+\alpha_{n}^{(\varkappa)}x_{n}}{\alpha_{1}^{(n)}x_{1}+\alpha_{2}^{(n)}x_{2}+\dots+\alpha_{n}^{(n)}x_{n}}\\ \frac{x_{\varkappa}}{x_{n}} & =\frac{\alpha_{\varkappa}^{\prime}y_{1}+\alpha_{\varkappa}^{\prime\prime}y_{2}+\dots+\alpha_{\varkappa}^{(n)}y_{n}}{\alpha_{1}^{(n)}x_{1}+\alpha_{2}^{(n)}x_{2}+\dots+\alpha_{n}^{(n)}x_{n}} \end{align} \left|\begin{align}\alpha_{\varkappa}^{\prime}\alpha_{\lambda}^{\prime}+\alpha_{\varkappa}^{\prime\prime}\alpha_{\lambda}^{\prime\prime}+\dots+\alpha_{\varkappa}^{(n)}\alpha_{\lambda}^{(n)} & =0\\ \alpha_{\varkappa}^{\prime}\alpha_{\varkappa}^{\prime}+\alpha_{\varkappa}^{\prime\prime}\alpha_{\varkappa}^{\prime\prime}+\dots+\alpha_{\varkappa}^{(n)}\alpha_{\varkappa}^{(n)} & =1\\ \\ \alpha_{1}^{(\varkappa)}\alpha_{1}^{(\lambda)}+\alpha_{2}^{(\varkappa)}\alpha_{2}^{(\lambda)}+\dots+\alpha_{n}^{(\varkappa)}\alpha_{n}^{(\lambda)} & =0\\ \alpha_{1}^{(\varkappa)}\alpha_{1}^{(\varkappa)}+\alpha_{2}^{(\varkappa)}\alpha_{2}^{(\varkappa)}+\dots+\alpha_{n}^{(\varkappa)}\alpha_{n}^{(\varkappa)} & =1 \end{align} \right.\\ \hline \frac{x_{\varkappa}}{x_{n}}=-i\xi_{\varkappa},\ \frac{y_{\varkappa}}{y_{n}}=i\nu_{\varkappa}\\ 1-\xi_{1}\xi_{1}-\xi_{2}\xi_{2}-\dots-\xi_{n-1}\xi_{n-1}=\frac{y_{n}y_{n}}{x_{n}x_{n}}\left(1-\nu_{1}\nu_{1}-\nu_{2}\nu_{2}-\dots-\nu_{n-1}\nu_{n-1}\right)\\ \alpha_{n}^{(\varkappa)}=i\alpha^{(\varkappa)},\ \alpha_{\varkappa}^{(n)}=-i\alpha_{\varkappa},\ \alpha_{n}^{(n)}=\alpha\\ 1-\xi_{1}\xi_{1}-\xi_{2}\xi_{2}-\dots-\xi_{n-1}\xi_{n-1}=\frac{1-\nu_{1}\nu_{1}-\nu_{2}\nu_{2}-\dots-\nu_{n-1}\nu_{n-1}}{\left[\alpha-\alpha^{\prime}\nu_{1}-\alpha^{\prime\prime}\nu_{2}\dots-\alpha^{(n-1)}\nu_{n-1}\right]^{2}}\\ \hline \mathbf{(2)\ }\begin{align}\nu_{\varkappa} & =\frac{\alpha^{(\varkappa)}-\alpha_{1}^{(\varkappa)}\xi_{1}-\alpha_{2}^{(\varkappa)}\xi_{2}\dots-\alpha_{n-1}^{(\varkappa)}\xi_{n-1}}{\alpha-\alpha_{1}\xi_{1}-\alpha_{2}\xi_{2}\dots-\alpha_{n-1}\xi_{n-1}}\\ \\ \xi_{\varkappa} & =\frac{\alpha_{\varkappa}-\alpha_{\varkappa}^{\prime}\nu_{1}-\alpha_{2}^{\prime\prime}\nu_{2}\dots-\alpha_{\varkappa}^{(n-1)}\nu_{n-1}}{\alpha-\alpha^{\prime}\nu_{1}-\alpha^{\prime\prime}\nu_{2}\dots-\alpha^{(n-1)}\nu_{n-1}} \end{align} \\ \hline \xi_{1}\xi_{1}-\xi_{2}\xi_{2}-\dots-\xi_{n-1}\xi_{n-1}=1\ \Rightarrow\ \nu_{1}\nu_{1}-\nu_{2}\nu_{2}-\dots-\nu_{n-1}\nu_{n-1}=1 \end{matrix}}$$

He also stated the following transformation leaving invariant the Lorentz interval:


 * $$\begin{matrix}uu-u_{1}u_{1}-u_{2}u_{2}-\dots-u_{n-1}u_{n-1}=ww-w_{1}w_{1}-w_{2}w_{2}-\dots-w_{n-1}w_{n-1}\\

\hline {\scriptstyle \begin{align}u & =\alpha w-\alpha'w_{1}-\alpha''w_{2}-\dots-\alpha^{(n-1)}w_{n-1}\\ u_{1} & =\alpha_{1}w-\alpha_{1}^{\prime}w_{1}-\alpha_{1}^{\prime\prime}w_{2}-\dots-\alpha_{1}^{(n-1)}w_{n-1}\\ & \dots\\ u_{n-1} & =\alpha_{n-1}w-\alpha_{n-1}^{\prime}w_{1}-\alpha_{n-1}^{\prime\prime}w_{2}-\dots-\alpha_{n-1}^{(n-1)}w_{n-1}\\ \\ w & =\alpha u-\alpha_{1}u_{1}-\alpha_{2}^{\prime\prime}u_{2}-\dots-\alpha_{n-1}u_{n-1}\\ w_{1} & =\alpha'u-\alpha_{1}^{\prime}u_{1}-\alpha_{2}^{\prime}u_{2}-\dots-\alpha_{n-1}^{\prime}u_{n-1}\\ & \dots\\ w_{n-1} & =\alpha^{(n-1)}u-\alpha_{1}^{(n-1)}u_{1}-\alpha_{2}^{(n-1)}u_{2}-\dots-\alpha_{n-1}^{(n-1)}u_{n-1} \end{align} \left|\begin{align}\alpha\alpha-\alpha'\alpha'-\alpha\alpha\dots-\alpha^{(n-1)}\alpha^{(n-1)} & =+1\\ \alpha_{\varkappa}\alpha_{\varkappa}-\alpha_{\varkappa}^{\prime}\alpha_{\varkappa}^{\prime}-\alpha_{\varkappa}^{\prime\prime}\alpha_{\varkappa}^{\prime\prime}\dots-\alpha_{\varkappa}^{(n-1)}\alpha_{\varkappa}^{(n-1)} & =-1\\ \alpha\alpha_{\varkappa}-\alpha^{\prime}\alpha_{\varkappa}^{\prime}-\alpha^{\prime\prime}\alpha_{\varkappa}^{\prime\prime}\dots-\alpha^{(n-1)}\alpha_{\varkappa}^{(n-1)} & =0\\ \alpha_{\varkappa}\alpha_{\lambda}-\alpha_{\varkappa}^{\prime}\alpha_{\lambda}^{\prime}-\alpha_{\varkappa}^{\prime\prime}\alpha_{\lambda}^{\prime\prime}\dots-\alpha_{\varkappa}^{(n-1)}\alpha_{\lambda}^{(n-1)} & =0\\ \\ \alpha\alpha-\alpha_{1}\alpha_{1}-\alpha_{2}\alpha_{2}\dots-\alpha_{n-1}\alpha_{n-1} & =+1\\ \alpha_{\varkappa}\alpha_{\varkappa}-\alpha_{1}^{\varkappa}\alpha_{1}^{\varkappa}-\alpha_{2}^{\prime\prime}\alpha_{2}^{\prime\prime}\dots-\alpha_{n-1}^{(\varkappa)}\alpha_{n-1}^{(\varkappa)} & =-1\\ \alpha\alpha^{(\varkappa)}-\alpha_{1}\alpha_{1}^{(\varkappa)}-\alpha_{2}\alpha_{2}^{(\varkappa)}\dots-\alpha_{n-1}\alpha_{n-1}^{(\varkappa)} & =0\\ \alpha^{(\varkappa)}\alpha^{(\lambda)}-\alpha_{1}^{(\varkappa)}\alpha_{1}^{\lambda l)}-\alpha_{2}^{(\varkappa)}\alpha_{2}^{(\lambda)}\dots-\alpha_{n-1}^{(\varkappa)}\alpha_{n-1}^{(\lambda)} & =0 \end{align} \text{ }\right.} \end{matrix}$$

Chasles (1829) – Conjugate hyperboloids
Michel Chasles (1829) independently introduced the same equation systems as Gauss (1818) and Jacobi (1827), albeit in the different context of conjugate hyperboloids. He started with two equation systems (a) and (b) from which he derived systems (c), (d) and others:


 * $$\begin{matrix}\left.\begin{align}\alpha^{2}+\beta^{2}-\gamma^{2} & =1\\

\alpha^{\prime2}+\beta^{\prime2}-\gamma^{\prime2} & =1\\ \alpha^{\prime\prime2}+\beta^{\prime\prime2}-\gamma^{\prime\prime2} & =-1 \end{align} \right\} & \dots(a)\\ \\ \left.\begin{align}\alpha\alpha'+\beta\beta'-\gamma\gamma' & =0\\ \alpha\alpha+\beta\beta-\gamma\gamma'' & =0\\ \alpha'\alpha+\beta'\beta-\gamma'\gamma' & =0 \end{align} \right\} & \dots(b)\\ \\ \left.\begin{align}\alpha^{2}+\alpha^{\prime2}-\alpha^{\prime\prime2} & =1\\ \beta^{2}+\beta^{\prime2}-\beta^{\prime\prime2} & =1\\ \gamma^{2}+\gamma^{\prime2}-\gamma^{\prime\prime2} & =-1 \end{align} \right\} & \dots(c)\\ \\ \left.\begin{align}\alpha\beta+\alpha'\beta'-\alpha\beta & =0\\ \alpha\gamma+\alpha'\gamma'-\alpha\gamma & =0\\ \beta\gamma+\beta'\gamma'-\beta\gamma & =0 \end{align} \right\} & \dots(d) \end{matrix}$$

He noted that those quantities become the “frequently employed” formulas of Lagrange [i.e. the coefficients of the Euclidean orthogonal transformation first given by E:Euler (1771)] by setting:


 * $$\begin{matrix}\gamma\quad\Rightarrow\quad-\gamma\sqrt{-1}\\

\gamma'\quad\Rightarrow\quad-\gamma'\sqrt{-1}\\ \alpha\quad\Rightarrow\quad\alpha\sqrt{-1}\\ \beta\quad\Rightarrow\quad\beta\sqrt{-1} \end{matrix}$$

Chasles now showed that equation systems (a,b,c,d) are of importance when discussing the relations between conjugate diameters of hyperboloids. He used the equations of a one-sheet hyperboloid and of a two-sheet hyperboloid having the same principal axes (x,y,z), thus sharing the same conjugate axes, and having the common asymptotic cone $$\tfrac{x^{2}}{a{{}^2}}+\tfrac{y^{2}}{b^{2}}-\tfrac{z^{2}}{c^{2}}=0$$. He then transformed those two hyperboloids to new axes (x',y',z') sharing the property of conjugacy:


 * $$\begin{matrix}\frac{x^{2}}{a{{}^2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}}=1\\

\frac{x^{2}}{a{{}^2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}}=-1\\ \hline \begin{align}x & =lx'+l'y'+l''z'\\ y & =mx'+m'y'+m''z'\\ z & =nx'+n'y'+n''z' \end{align} \\ \left\{ \begin{align}\frac{ll'}{a{{}^2}}+\frac{mm'}{b^{2}}-\frac{nn'}{c^{2}} & =0\\ \frac{ll}{a{{}^2}}+\frac{mm}{b^{2}}-\frac{nn''}{c^{2}} & =0\\ \frac{l'l}{a{{}^2}}+\frac{m'm}{b^{2}}-\frac{n'n''}{c^{2}} & =0 \end{align} \right\} \\ \hline \left(\frac{l^{2}}{a{{}^2}}+\frac{m^{2}}{b^{2}}-\frac{n^{2}}{c^{2}}\right)x^{\prime2}+\left(\frac{l^{\prime2}}{a{{}^2}}+\frac{m^{\prime2}}{b^{2}}-\frac{n^{\prime2}}{c^{2}}\right)y^{\prime2}+\left(\frac{l^{\prime\prime2}}{a{{}^2}}+\frac{m^{\prime\prime2}}{b^{2}}-\frac{n^{\prime\prime2}}{c^{2}}\right)z^{\prime2}=1\\ \left(\frac{l^{2}}{a{{}^2}}+\frac{m^{2}}{b^{2}}-\frac{n^{2}}{c^{2}}\right)x^{\prime2}+\left(\frac{l^{\prime2}}{a{{}^2}}+\frac{m^{\prime2}}{b^{2}}-\frac{n^{\prime2}}{c^{2}}\right)y^{\prime2}+\left(\frac{l^{\prime\prime2}}{a{{}^2}}+\frac{m^{\prime\prime2}}{b^{2}}-\frac{n^{\prime\prime2}}{c^{2}}\right)z^{\prime2}=-1 \end{matrix}$$

He went on to use two semi-diameters of the one-sheet hyperboloid and one semi-diameter of the two-sheet hyperboloid in order to define equation system (A), and went on to suggest that the other equations related to this system can be obtained using the above transformation from oblique coordinates to other oblique ones, but he deemed it more simple to use a geometric argument to obtain system (B), which together with (A) then allowed him to algebraically determine systems (C), (D) and additional ones, leading Chasles to announce that “from these formulas one can very easily conclude the various properties of conjugated diameters of hyperboloids”:


 * $$\begin{matrix}\left.\begin{align}\alpha^{2}+\beta^{2}-\gamma^{2} & =a^{2}\\

\alpha^{\prime2}+\beta^{\prime2}-\gamma^{\prime2} & =b^{2}\\ \alpha^{\prime\prime2}+\beta^{\prime\prime2}-\gamma^{\prime\prime2} & =-c^{2} \end{align} \right\} & \dots(A)\\ \left.\begin{align}\alpha\alpha'+\beta\beta'-\gamma\gamma' & =0\\ \alpha\alpha+\beta\beta-\gamma\gamma'' & =0\\ \alpha'\alpha+\beta'\beta-\gamma'\gamma' & =0 \end{align} \right\} & \dots(B)\\ \left.\begin{align}\alpha^{2}+\alpha^{\prime2}-\alpha^{\prime\prime2} & =a^{2}\\ \beta^{2}+\beta^{\prime2}-\beta^{\prime\prime2} & =b^{2}\\ \gamma^{2}+\gamma^{\prime2}-\gamma^{\prime\prime2} & =-c^{2} \end{align} \right\} & \dots(C)\\ \left.\begin{align}\alpha\beta+\alpha'\beta'-\alpha\beta & =0\\ \alpha\gamma+\alpha'\gamma'-\alpha\gamma & =0\\ \beta\gamma+\beta'\gamma'-\beta\gamma & =0 \end{align} \right\} & \dots(D) \end{matrix}$$

Lebesgue (1837) – Homogeneous coordinates
Victor-Amédée Lebesgue (1837) summarized the previous work of Gauss (1818), Jacobi (1827, 1833), Cauchy (1829). He started with the orthogonal transformation


 * $$\begin{matrix}x_{1}^{2}+x_{2}^{2}+\dots+x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots+y_{n}^{2}\ (9)\\

\hline {\scriptstyle \begin{align}x_{1} & =a_{1,1}y_{1}+a_{1,2}y_{2}+\dots+a_{1,n}y_{n}\\ x_{2} & =a_{2,1}y_{1}+a_{2,2}y_{2}+\dots+a_{2,n}y_{n}\\ \dots\\ x_{n} & =a_{n,1}x_{1}+a_{n,2}x_{2}+\dots+a_{n,n}x_{n}\\ \\ y_{1} & =a_{1,1}x_{1}+a_{2,1}x_{2}+\dots+a_{n,1}x_{n}\\ y_{2} & =a_{1,2}x_{1}+a_{2,2}x_{2}+\dots+a_{n,2}x_{n}\ (12)\ \\ \dots\\ y_{n} & =a_{1,n}x_{1}+a_{2,n}x_{2}+\dots+a_{n,n}x_{n} \end{align} \left|\begin{align}a_{1,\alpha}^{2}+a_{2,\alpha}^{2}+\dots+a_{n,\alpha}^{2} & =1 & (10)\\ a_{1,\alpha}a_{1,\beta}+a_{2,\alpha}a_{2,\beta}+\dots+a_{n,\alpha}a_{n,\beta} & =0 & (11)\\ a_{\alpha,1}^{2}+a_{\alpha,2}^{2}+\dots+a_{\alpha,n}^{2} & =1 & (13)\\ a_{\alpha,1}a_{\beta,1}+a_{\alpha,2}a_{\beta,2}+\dots+a_{\alpha,n}a_{\beta,n} & =0 & (14) \end{align} \right.} \end{matrix}$$

In order to achieve the invariance of the Lorentz interval


 * $$x_{1}^{2}+x_{2}^{2}+\dots+x_{n-1}^{2}-x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots+y_{n-1}^{2}-y_{n}^{2}$$

he gave the following instructions as to how the previous equations shall be modified: In equation (9) change the sign of the last term of each member. In the first n-1 equations of (10) change the sign of the last term of the left-hand side, and in the one which satisfies α=n change the sign of the last term of the left-hand side as well as the sign of the right-hand side. In all equations (11) the last term will change sign. In equations (12) the last terms of the right-hand side will change sign, and so will the left-hand side of the n-th equation. In equations (13) the signs of the last terms of the left-hand side will change, moreover in the n-th equation change the sign of the right-hand side. In equations (14) the last terms will change sign.

He went on to redefine the variables of the Lorentz interval and its transformation:


 * $$\begin{matrix}x_{1}^{2}+x_{2}^{2}+\dots+x_{n-1}^{2}-x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots+y_{n-1}^{2}-y_{n}^{2}\\

\downarrow\\ \begin{align}x_{1} & =x_{n}\cos\theta_{1}, & x_{2} & =x_{n}\cos\theta_{2},\dots & x_{n-1} & =x_{n}\cos\theta_{n-1}\\ y_{1} & =y_{n}\cos\phi_{1}, & y_{2} & =y_{n}\cos\phi_{2},\dots & y_{n-1} & =y_{n}\cos\phi_{n-1} \end{align} \\ \downarrow\\ \cos^{2}\theta_{1}+\cos^{2}\theta_{2}+\dots+\cos^{2}\theta_{n-1}=1\\ \cos^{2}\phi_{1}+\cos^{2}\phi_{2}+\dots+\cos^{2}\phi_{n-1}=1\\ \hline \\ \cos\theta_{i}=\frac{a_{i,1}\cos\phi_{1}+a_{i,2}\cos\phi_{2}+\dots+a_{i,n-1}\cos\phi_{n-1}+a_{i,n}}{a_{n,1}\cos\phi_{1}+a_{n,2}\cos\phi_{2}+\dots+a_{n,n-1}\cos\phi_{n-1}+a_{n,n}}\\ (i=1,2,3\dots n) \end{matrix}$$

Weddle (1847) – Conjugate hyperboloids
Very similar to Chasles (1829), though without reference to him, Thomas Weddle discussed conjugate hyperboloids using the following equation system (&alpha;), from which he derived equations (&beta;) and others:


 * $$\begin{matrix}\left.\begin{align}l_{1}^{2}+m_{1}^{2}-n_{1}^{2} & =1, & l_{1}l_{2}+m_{1}m_{2}-n_{1}n_{2} & =0\\

l_{2}^{2}+m_{2}^{2}-n_{2}^{2} & =1, & l_{1}l_{3}+m_{1}m_{3}-n_{1}n_{3} & =0\\ l_{3}^{2}+m_{3}^{2}-n_{3}^{2} & =-1, & l_{2}l_{3}+m_{2}m_{3}-n_{2}n_{3} & =0 \end{align} \right\} & \dots(\alpha)\\ \\ \left.\begin{align}l_{1}^{2}+l_{2}^{2}-l_{3}^{2} & =1, & l_{1}m_{1}+l_{2}m_{2}-l_{3}m_{3} & =0\\ m_{1}^{2}+m_{2}^{2}-m_{3}^{2} & =1, & l_{1}n_{1}+l_{2}n_{2}-l_{3}n_{3} & =0\\ n_{1}^{2}+n_{2}^{2}-n_{3}^{2} & =-1, & m_{1}n_{1}+m_{2}n_{2}-m_{3}n_{3} & =0 \end{align} \right\} & \dots(\beta) \end{matrix}$$

Using the equations of a one-sheet hyperboloid and of a two-sheet hyperboloid sharing the same conjugate axes, and having the common asymptotic cone $$\tfrac{x^{2}}{a{{}^2}}+\tfrac{y^{2}}{b^{2}}-\tfrac{z^{2}}{c^{2}}=0$$, he defined three conjugate points $$(x_{1}\dots,y_{1}\dots,z_{1}\dots)$$ on those two conjugate hyperboloids, related to each other in the same way as equations (&alpha;, &beta;) stated above:


 * $$\begin{matrix}\frac{x^{2}}{a{{}^2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}}=1\\

\frac{x^{2}}{a{{}^2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}}=-1\\ \hline \begin{align}\frac{x_{1}x_{2}}{a{{}^2}}+\frac{y_{1}y_{2}}{b^{2}}-\frac{z_{1}z_{2}}{c^{2}} & =0\\ \frac{x_{1}x_{3}}{a{{}^2}}+\frac{y_{1}y_{3}}{b^{2}}-\frac{z_{1}z_{3}}{c^{2}} & =0\\ \frac{x_{2}x_{3}}{a{{}^2}}+\frac{y_{2}y_{3}}{b^{2}}-\frac{z_{2}z_{3}}{c^{2}} & =0 \end{align} \quad\begin{align}\frac{x_{1}^{2}}{a{{}^2}}+\frac{y_{1}^{2}}{b^{2}}-\frac{z_{1}^{2}}{c^{2}} & =1\\ \frac{x_{2}^{2}}{a{{}^2}}+\frac{y_{2}^{2}}{b^{2}}-\frac{z_{2}^{2}}{c^{2}} & =1\\ \frac{x_{3}^{2}}{a{{}^2}}+\frac{y_{3}^{2}}{b^{2}}-\frac{z_{3}^{2}}{c^{2}} & =-1 \end{align} \\ \begin{align}x_{1}^{2}+x_{2}^{2}-x_{3}^{2} & =a^{2}\\ y_{1}^{2}+y_{2}^{2}-y_{3}^{2} & =b^{2}\\ z_{1}^{2}+z_{2}^{2}-z_{3}^{2} & =-c^{2} \end{align} \quad\begin{align}x_{1}y_{1}+x_{2}y_{2}-x_{3}y_{3} & =0\\ x_{1}z_{1}+x_{2}z_{2}-x_{3}z_{3} & =0\\ y_{1}z_{1}+y_{2}z_{2}-y_{3}z_{3} & =0 \end{align} \end{matrix}$$

Bour (1856) – Homogeneous coordinates
Following Gauss (1818), Edmond Bour (1856) wrote the transformations:


 * $$\begin{matrix}\cos^{2}E+\sin^{2}E-1=k\left(\cos^{2}T+\sin^{2}T-1\right)\\

\hline \left.\begin{matrix}\mathbf{(1)}\ \begin{align}\cos E & =\frac{\alpha+\alpha'\cos T+\alpha\sin T}{\gamma+\gamma'\cos T+\gamma\sin T}\\ \sin E & =\frac{\beta+\beta'\cos T+\beta\sin T}{\gamma+\gamma'\cos T+\gamma\sin T} \end{align} \\ \hline \\ k=+1\\ t=\gamma+\gamma'\cos T+\gamma''\sin T,\\ 1=u,\ \cos T=u',\ \sin T=u',\\ t=z,\ t\cos E=x,\ t\sin E=y\\ \downarrow\\ \mathbf{(2)}\begin{align}x & =\alpha u+\alpha'u'+\alphau\\ y & =\beta u+\beta'u'+\betau\\ z & =\gamma u+\gamma'u'+\gammau\\ \\ u & =\gamma z-\alpha x-\beta y\\ u' & =\alpha'x+\beta'y'-\gamma'z\\ u & =\alphax+\betay-\gammaz \end{align} \end{matrix}\right|{\scriptstyle \begin{align}-\alpha^{2}-\beta^{2}+\gamma^{2} & =k\\ -\alpha^{\prime2}-\beta^{\prime2}+\gamma^{\prime2} & =-k\\ -\alpha^{\prime\prime2}-\beta^{\prime\prime2}+\gamma^{\prime\prime2} & =-k\\ \alpha\alpha'+\beta\beta'-\gamma\gamma' & =0\\ \alpha\alpha+\beta\beta-\gamma\gamma'' & =0\\ \alpha'\alpha+\beta'\beta-\gamma'\gamma'' & =0\\ \\ \alpha^{2}-\alpha^{\prime2}-\alpha^{\prime\prime2} & =-k\\ \beta^{2}-\beta^{\prime2}-\beta^{\prime\prime2} & =-k\\ \gamma^{2}-\gamma^{\prime2}-\gamma^{\prime\prime2} & =k\\ \beta\gamma-\beta'\gamma'-\beta\gamma & =0\\ \alpha\gamma-\alpha'\gamma'-\alpha\gamma & =0\\ \alpha\beta-\alpha'\beta'-\alpha\beta & =0 \end{align} } \end{matrix}$$

Somov (1863) – Homogeneous coordinates
Following Gauss (1818), Jacobi (1827, 1833), and Bour (1856), Osip Ivanovich Somov (1863) wrote the transformation systems:


 * $$\begin{matrix}\left.\begin{align}\cos\phi & =\frac{m\cos\psi+n\sin\psi+s}{m\cos\psi+n\sin\psi+s''}\\

\sin\phi & =\frac{m'\cos\psi+n'\sin\psi+s'}{m\cos\psi+n\sin\psi+s''} \end{align} \right|\begin{matrix}\cos^{2}\phi+\cos^{2}\phi=1\\ \cos^{2}\psi+\cos^{2}\psi=1 \end{matrix}\\ \hline \mathbf{(1)}\ \begin{align}\cos\phi & =x, & \cos\psi & =x'\\ \sin\phi & =y, & \sin\psi & =y' \end{align} \ \left|\begin{align}x & =\frac{mx'+ny'+s}{mx'+ny'+s''}\\ y & =\frac{m'x'+n'y'+s'}{mx'+ny'+s''} \end{align} \right|\ \begin{matrix}x^{2}+y^{2}=1\\ x^{\prime2}+y^{\prime2}=1 \end{matrix}\\ \hline \begin{align}\cos\phi & =\frac{x}{z}, & \cos\psi & =\frac{x'}{z'}\\ \sin\phi & =\frac{y}{z}, & \sin\psi & =\frac{y'}{z'} \end{align} \ \left|\begin{align}\frac{x}{z} & =\frac{mx'+ny'+sz'}{mx'+ny'+s''z'}\\ \frac{y}{z} & =\frac{m'x'+n'y'+s'z'}{mx'+ny'+s''z'} \end{align} \right|\ \begin{matrix}x^{2}+y^{2}=z^{2}\\ x^{\prime2}+y^{\prime2}=z^{\prime2} \end{matrix}\\ \hline \mathbf{(2)}\ \left.\begin{align}x & =mx'+ny'+sz'\\ y & =m'x'+n'y'+s'z'\\ z & =mx'+ny'+s''z'\\ \\ x' & =mx+m'y-m''z\\ y' & =nx+n'y-n''z\\ z' & =-sx-s'y+s''z\\ \\ dx & =mdx'+ndy'+sdz'\\ dy & =m'dx'+n'dy'+s'dz'\\ dz & =mdx'+ndy'+s''dz' \end{align} \right|{\scriptstyle \begin{align}m^{2}+m^{\prime2}-m^{\prime\prime2} & =1\\ n^{2}+n^{\prime2}-n^{\prime\prime2} & =1\\ -s^{2}-s^{\prime2}+s^{\prime\prime2} & =1\\ ns+n's'-ns & =0\\ sm+s'm'-sm & =0\\ mn+m'n'-mn & =0\\ \\ m^{2}+n^{2}-s^{2} & =1\\ m^{\prime2}+n^{\prime2}-s^{\prime2} & =1\\ -m^{\prime\prime2}-n^{\prime\prime2}+s^{\prime\prime2} & =1\\ -m'm-n'n+s's'' & =0\\ -mm-nn+s''s & =0\\ mm'+nn'-ss' & =0 \end{align} }\\ dx^{2}+dy^{2}-dz^{2}=dx^{\prime2}+dy^{\prime2}-dz^{\prime2} \end{matrix}$$

Klein (1871-73) – Cayley absolute and non-Euclidean geometry
Elaborating on Arthur Cayley's (1859) definition of an "absolute" (Cayley–Klein metric), Felix Klein (1871) defined a "fundamental conic section" in order to discuss motions such as rotation and translation in the non-Euclidean plane. This was elaborated in (1873) when he pointed out that hyperbolic geometry in terms of a surface of constant negative curvature can be related to a quadratic equation, which can be transformed into a sum of squares of which one square has a different sign, and can also be related to the interior of a surface of second degree corresponding to a two-sheet hyperboloid.

Weierstrass coordinates
Wilhelm Killing (1878–1880) described non-Euclidean geometry by using Weierstrass coordinates (named after Karl Weierstrass who described them in lectures in 1872 which Killing attended) obeying the form


 * $$k^{2}t^{2}+u^{2}+v^{2}+w^{2}=k^{2}$$ with $$ds^{2}=k^{2}dt^{2}+du^{2}+dv^{2}+dw^{2}$$

or


 * $$k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=k^{2}$$

where k is the reciprocal measure of curvature, $$k^{2}=\infty$$ denotes Euclidean geometry, $$k^{2}>0$$ elliptic geometry, and $$k^{2}<0$$ hyperbolic geometry. In (1877/78) he pointed out the possibility and some characteristics of a transformation (indicating rigid motions) preserving the above form. In (1879/80) he tried to formulate the corresponding transformations by plugging $$k^{2}$$ into a general rotation matrix:

$$\begin{matrix}k^{2}u^{2}+v^{2}+w^{2}=k^{2}\\ \hline \begin{matrix}\cos\eta\tau+\lambda^{2}\frac{1-\cos\eta\tau}{\eta^{2}}, & \nu\frac{\sin\eta\tau}{\eta}+\lambda\mu\frac{1-\cos\eta\tau}{\eta^{2}}, & -\mu\frac{\sin\eta\tau}{\eta}+\nu\lambda\frac{1-\cos\eta\tau}{\eta^{2}}\\ -k^{2}\nu\frac{\sin\eta\tau}{\eta}+k^{2}\lambda\mu\frac{1-\cos\eta\tau}{\eta^{2}}, & \cos\eta\tau+\mu^{2}\frac{1-\cos\eta\tau}{\eta^{2}}, & \lambda\frac{\sin\eta\tau}{\eta}+k^{2}\mu\nu\frac{1-\cos\eta\tau}{\eta^{2}}\\ k^{2}\mu\frac{\sin\eta\tau}{\eta}+k^{2}\nu\lambda\frac{1-\cos\eta\tau}{\eta^{2}}, & -\lambda\frac{\sin\eta\tau}{\eta}+k^{2}\mu\nu\frac{1-\cos\eta\tau}{\eta^{2}}, & \cos\eta\tau+\nu^{2}\frac{1-\cos\eta\tau}{\eta^{2}} \end{matrix}\\ \left(\lambda^{2}+k^{2}\mu^{2}+k^{2}\nu^{2}=\eta^{2}\right) \end{matrix}$$

In (1885) he wrote the Weierstrass coordinates and their transformation as follows:


 * $$\begin{matrix}k^{2}p^{2}+x^{2}+y^{2}=k^{2}\\

k^{2}p^{2}+x^{2}+y^{2}=k^{2}p^{\prime2}+x^{\prime2}+y^{\prime2}\\ ds^{2}=k^{2}dp^{2}+dx^{2}+dy^{2}\\ \hline \begin{align}k^{2}p' & =k^{2}wp+w'x+w''y\\ x' & =ap+a'x+a''y\\ y' & =bp+b'x+b''y\\ \\ k^{2}p & =k^{2}wp'+ax'+by'\\ x & =w'p'+a'x+b'y'\\ y & =wp'+ax'+b''y' \end{align} \left|{\scriptstyle \begin{align}k^{2}w^{2}+w^{\prime2}+w^{\prime\prime2} & =k^{2}\\ \frac{a^{2}}{k^{2}}+a^{\prime2}+a^{\prime\prime2} & =1\\ \frac{b^{2}}{k^{2}}+b^{\prime2}+b^{\prime\prime2} & =1\\ aw+a'w'+aw & =0\\ bw+b'w'+bw & =0\\ \frac{ab}{k^{2}}+a'b'+ab & =0\\ \\ k^{2}w^{2}+a^{2}+b^{2} & =k^{2}\\ \frac{w^{\prime2}}{k^{2}}+a^{\prime2}+b^{\prime2} & =1\\ \frac{w^{\prime\prime2}}{k^{2}}+a^{\prime\prime2}+b^{\prime\prime2} & =1\\ ww'+aa'+bb' & =0\\ ww+aa+bb'' & =0\\ \frac{w'w}{k^{2}}+a'a+b'b'' & =0 \end{align} }\right. \end{matrix}$$

In (1885) he also gave the transformation for n dimensions:


 * $$\begin{matrix}k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=k^{2}\\

ds^{2}=k^{2}dx_{0}^{2}+dx_{1}^{2}+\dots+dx_{n}^{2}\\ \hline \left.\begin{align}k^{2}\xi_{0} & =k^{2}a_{00}x_{0}+a_{01}x_{1}+\dots+a_{0n}x_{0}\\ \xi_{\varkappa} & =a_{\varkappa0}x_{0}+a_{\varkappa1}x_{1}+\dots+a_{\varkappa n}x_{n}\\ \\ k^{2}x_{0} & =a_{00}k^{2}\xi_{0}+a_{10}\xi_{1}+\dots+a_{n0}\xi_{n}\\ x_{\varkappa} & =a_{0\varkappa}\xi_{0}+a_{1\varkappa}\xi_{1}+\dots+a_{n\varkappa}\xi_{n} \end{align} \right|{\scriptstyle \begin{align}k^{2}a_{00}^{2}+a_{10}^{2}+\dots+a_{n0}^{2} & =k^{2}\\ a_{00}a_{0\varkappa}+a_{10}a_{1\varkappa}+\dots+a_{n0}a_{n\varkappa} & =0\\ \frac{a_{0\iota}a_{0\varkappa}}{k^{2}}+a_{0\iota}a_{1\varkappa}+\dots+a_{n\iota}a_{n\varkappa}=\delta_{\iota\kappa} & =1\ (\iota=\kappa)\ \text{or}\ 0\ (\iota\ne\kappa) \end{align} } \end{matrix}$$

In (1885) he applied his transformations to mechanics and defined four-dimensional vectors of velocity and force. Regarding the geometrical interpretation of his transformations, Killing argued in (1885) that by setting $$k^{2}=-1$$ and using p,x,y as rectangular space coordinates, the hyperbolic plane is mapped on one side of a two-sheet hyperboloid $$p^{2}-x^{2}-y^{2}=1$$ (known as hyperboloid model), by which the previous formulas become equivalent to Lorentz transformations and the geometry becomes that of Minkowski space.

Finally, in (1893) he wrote:


 * $$\begin{matrix}k^{2}t^{2}+u^{2}+v^{2}=k^{2}\\

\hline \begin{align}t' & =at+bu+cv\\ u' & =a't+b'u+c'v\\ v' & =at+bu+c''v \end{align} \left|\begin{align}k^{2}a^{2}+a^{\prime2}+a^{\prime\prime2} & =k^{2}\\ k^{2}b^{2}+b^{\prime2}+b^{\prime\prime2} & =1\\ k^{2}c^{2}+b^{\prime2}+c^{\prime\prime2} & =1\\ k^{2}ab+a'b'+ab & =0\\ k^{2}ac+a'c'+ac & =0\\ k^{2}bc+b'c'+bc & =0 \end{align} \right. \end{matrix}$$

and in n dimensions


 * $$\begin{matrix}k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=k^{2}\\

k^{2}y_{0}y_{0}^{\prime}+y_{1}y_{1}^{\prime}+\cdots+y_{n}y_{n}^{\prime}=k^{2}x_{0}x_{0}^{\prime}+x_{1}x_{1}^{\prime}+\cdots+x_{n}x_{n}^{\prime}\\ ds^{2}=k^{2}dx_{0}^{2}+\dots+dx_{n}^{2}\\ \hline \begin{align}y_{0} & =a_{00}x_{0}+a_{01}x_{1}+\dots+a_{0n}x_{n}\\ y_{1} & =a_{10}x_{0}+a_{11}x_{1}+\dots+a_{1n}x_{n}\\ & \,\,\,\vdots\\ y_{n} & =a_{n0}x_{0}+a_{n1}x_{1}+\dots+a_{nn}x_{n} \end{align} \left|\begin{align}k^{2}a_{00}^{2}+a_{10}^{2}+\dots+a_{n0}^{2} & =k^{2}\\ k^{2}a_{0\varkappa}^{2}+a_{1\varkappa}^{2}+\dots+a_{n\varkappa}^{2} & =1\\ k^{2}a_{00}a_{0\varkappa}+a_{10}a_{1\varkappa}+\dots+a_{n0}a_{n\varkappa} & =0\\ k^{2}a_{0\varkappa}a_{0\lambda}+a_{1\varkappa}a_{1\lambda}+\dots+a_{n\varkappa}a_{n\lambda} & =0\\ (\varkappa,\lambda=1,\dots, n,\ \lambda\lessgtr\varkappa) \end{align} \right. \end{matrix}$$

Infinitesimal transformations and Lie group
After Lie (1885/86) identified the projective group of a general surface of second degree $$\sum f_{ik}x_{i}'x_{k}'=0$$ with the group of non-Euclidean motions, Killing (1887/88) defined the infinitesimal projective transformations (Lie algebra) in relation to the unit hypersphere:


 * $$\begin{matrix}x_{1}^{2}+\dots+x_{m+1}^{2}=1\\

\hline X_{\iota\varkappa}f=x_{i}\frac{\partial f}{\partial x_{\varkappa}}-x_{\varkappa}\frac{\partial f}{\partial x_{\iota}}\\ \text{where}\\ \left(X_{\iota\varkappa},X_{\iota\lambda}\right)=X_{\varkappa\lambda};\ \left(X_{\iota\varkappa},X_{\lambda\mu}\right)=0;\\ \left[\iota\ne\varkappa\ne\lambda\ne\mu\right] \end{matrix}$$

and in (1892) he defined the infinitesimal transformation for non-Euclidean motions in terms of Weierstrass coordinates:


 * $$\begin{matrix}k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=k^{2}\\

\hline X_{\iota\varkappa}=x_{\iota}p_{\varkappa}-x_{\varkappa}p_{\iota},\quad X_{\iota}=x_{0}p_{\iota}-\frac{x_{\iota}p_{0}}{k^{2}}\\ \text{where}\\ \left(X_{\iota}X_{\iota\varkappa}\right)=X_{\varkappa}f;\ \left(X_{\iota}X_{\varkappa\lambda}\right)=0;\ \left(X_{\iota}X_{\varkappa}\right)=-\frac{1}{k^{2}}X_{\iota\varkappa}f; \end{matrix}$$

In (1897/98) he showed the relation between Weierstrass coordinates $$k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=k^{2}$$ and coordinates $$k^{2}+y_{1}^{2}+y_{2}^{2}+\dots+y_{n}^{2}=0$$ used by himself in (1887/88) and by Werner (1889), Lie (1890):


 * $$\begin{matrix}\begin{matrix}k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2} & (a)\\

k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=k^{2} & (b) \end{matrix}\\ \hline V_{\varkappa}=k^{2}x_{0}p_{\varkappa}-x_{\varkappa}p_{0},\quad U_{\iota\varkappa}=p_{\iota}x_{\varkappa}-p_{\varkappa}x_{\iota}\\ \text{where}\\ \left(V_{\iota},V_{\varkappa}\right)=k^{2}U_{\iota\varkappa},\ \left(V_{\iota},U_{\iota\varkappa}\right)=-V_{\varkappa},\ \left(V_{\iota},U_{\varkappa\lambda}\right)=0,\\ \left(U_{\iota\varkappa},U_{\iota\lambda}\right)=U_{\varkappa\lambda},\ \left(U_{\iota\varkappa},U_{\lambda\mu}\right)=0\\ \left[\iota,\varkappa,\lambda,\mu=1,2,\dots n\right]\\ \hline \begin{matrix}y_{1}=\frac{x_{1}}{x_{0}},\ y_{2}=\frac{x_{2}}{x_{0}},\dots y_{n}=\frac{x_{n}}{x_{0}}\\ \downarrow\\ k^{2}+y_{1}^{2}+y_{2}^{2}+\dots+y_{n}^{2}=0\\ \hline q_{\varkappa}+\frac{y_{\varkappa}}{k^{2}}\sum_{\varrho}y_{y}q_{\varrho},\quad q_{\iota}y_{\varkappa}-q_{\varkappa}y_{\iota} \end{matrix} \end{matrix}$$

He pointed out that the corresponding group of non-Euclidean motions in terms of Weierstrass coordinates is intransitive when related to quadratic form (a) and transitive when related to quadratic form (b).

Poincaré (1881) – Weierstrass coordinates
Henri Poincaré (1881) connected the work of E:Hermite (1853) and E:Selling (1873) on indefinite quadratic forms with non-Euclidean geometry (Poincaré already discussed such relations in an unpublished manuscript in 1880). He used two indefinite ternary forms in terms of three squares and then defined them in terms of Weierstrass coordinates (without using that expression) connected by a transformation with integer coefficients:


 * $$\begin{matrix}\begin{align}F & =(ax+by+cz)^{2}+(a'x+b'y+c'z)^{2}-(ax+by+c''z)^{2}\\

& =\xi^{2}+\eta^{2}-\zeta^{2}=-1\\ F & =(ax'+by'+cz')^{2}+(a'x'+b'y'+c'z')^{2}-(ax'+by'+c''z')^{2}\\ & =\xi^{\prime2}+\eta^{\prime2}-\zeta^{\prime2}=-1 \end{align} \\ \hline \begin{align}\xi' & =\alpha\xi+\beta\eta+\gamma\zeta\\ \eta' & =\alpha'\xi+\beta'\eta+\gamma'\zeta\\ \zeta' & =\alpha\xi+\beta\eta+\gamma''\zeta \end{align} \left|\begin{align}\alpha^{2}+\alpha^{\prime2}-\alpha^{\prime\prime2} & =1\\ \beta^{2}+\beta^{\prime2}-\beta^{\prime\prime2} & =1\\ \gamma^{2}+\gamma^{\prime2}-\gamma^{\prime\prime2} & =-1\\ \alpha\beta+\alpha'\beta'-\alpha\beta & =0\\ \alpha\gamma+\alpha'\gamma'-\alpha\gamma & =0\\ \beta\gamma+\beta'\gamma'-\beta\gamma & =0 \end{align} \right. \end{matrix}$$

He went on to describe the properties of "hyperbolic coordinates". Poincaré mentioned the hyperboloid model also in (1887).

Cox (1881–1891) – Weierstrass coordinates
Homersham Cox (1881/82) – referring to similar rectangular coordinates used by Gudermann (1830) and George Salmon (1862) on a sphere, and to Escherich (1874) as reported by Johannes Frischauf (1876) in the hyperbolic plane – defined the Weierstrass coordinates (without using that expression) and their transformation:


 * $$\begin{matrix}z^{2}-x^{2}-y^{2}=1\\

x^{2}-y^{2}-z^{2}=Z^{2}-Y^{2}-X^{2}\\ \hline \begin{align}x & =l_{1}X+l_{2}Y+l_{3}Z\\ y & =m_{1}X+m_{2}Y+m_{3}Z\\ z & =n_{1}X+n_{2}Y+n_{3}Z\\ \\ X & =l_{1}x+m_{1}y-n_{1}z\\ Y & =l_{2}x+m_{2}y-n_{2}z\\ Z & =l_{3}x+m_{3}y-n_{3}z \end{align} \left|{\scriptstyle \begin{align}l_{1}^{2}+m_{1}^{2}-n_{1}^{2} & =1\\ l_{2}^{2}+m_{2}^{2}-n_{2}^{2} & =1\\ l_{3}^{2}+m_{3}^{2}-n_{3}^{2} & =1\\ l_{1}l_{2}+m_{1}m_{2}-n_{1}n_{2} & =0\\ l_{2}l_{3}+m_{2}m_{3}-n_{2}n_{3} & =0\\ l_{3}l_{1}+m_{3}m_{1}-n_{3}n_{1} & =0\\ \\ l_{1}^{2}+l_{2}^{2}-l_{3}^{2} & =1\\ m_{1}^{2}+m_{2}^{2}-m_{3}^{2} & =1\\ n_{1}^{2}+n_{2}^{2}-n_{3}^{2} & =1\\ l_{1}m_{1}+l_{2}m_{2}-l_{3}m_{3} & =0\\ m_{1}n_{1}+m_{2}n_{2}-m_{3}n_{3} & =0\\ n_{1}l_{1}+n_{2}l_{2}-n_{3}l_{3} & =0 \end{align} }\right. \end{matrix}$$

Cox (1881/82) also gave the Weierstrass coordinates and their transformation in hyperbolic space:


 * $$\begin{matrix}w^{2}-x^{2}-y^{2}-z^{2}=1\\

w^{2}-x^{2}-y^{2}-z^{2}=w^{\prime2}-x^{\prime2}-y^{\prime2}-z^{\prime2}\\ \hline \begin{align}x & =l_{1}x'+l_{2}y'+l_{3}z'-l_{4}w'\\ y & =m_{1}x'+m_{2}y'+m_{3}z'-m_{4}w'\\ z & =n_{1}x'+n_{2}y'+n_{3}z'-n_{4}w'\\ w & =r_{1}x'+r_{2}y'+r_{3}z'-r_{4}w'\\ \\ x' & =l_{1}x+m_{1}y+n_{1}z-r_{1}w\\ y' & =l_{2}x+m_{2}y+n_{2}z-r_{2}w\\ z' & =l_{3}x+m_{3}y+n_{3}z-r_{3}w\\ w' & =l_{4}x+m_{4}y+n_{4}z-r_{4}w \end{align} \left|{\scriptstyle \begin{align}l_{1}^{2}+m_{1}^{2}+n_{1}^{2}-r_{1}^{2} & =1\\ l_{2}^{2}+m_{2}^{2}+n_{2}^{2}-r_{2}^{2} & =1\\ l_{3}^{2}+m_{3}^{2}+n_{3}^{2}-r_{3}^{2} & =1\\ l_{4}^{2}+m_{4}^{2}+n_{4}^{2}-r_{4}^{2} & =1\\ l_{2}l_{3}+m_{2}m_{3}+n_{2}n_{3}-r_{2}r_{3} & =0\\ l_{3}l_{1}+m_{3}m_{1}+n_{3}n_{1}-r_{3}r_{1} & =0\\ l_{1}l_{4}+m_{1}m_{4}+n_{1}n_{4}-r_{1}r_{4} & =0\\ l_{2}l_{4}+m_{2}m_{4}+n_{2}n_{4}-r_{2}r_{4} & =0\\ l_{3}l_{4}+m_{3}m_{4}+n_{3}n_{4}-r_{3}r_{4} & =0 \end{align} }\right. \end{matrix}$$

In 1883 he formulated relations between orthogonal circles which he identified with the previously (1881/82) given transformations:


 * $$\begin{matrix}x^{2}+y^{2}+z^{2}-w^{2}=0\\

\hline \begin{align}x & =\lambda_{1}X+\lambda_{2}Y+\lambda_{3}Z+\lambda_{4}W\\ y & =\mu_{1}X+\mu_{2}Y+\mu_{3}Z+\mu_{4}W\\ z & =\nu_{1}X+\nu_{2}Y+\nu_{3}Z+\nu_{4}W\\ -w & =\rho_{1}X+\rho_{2}Y+\rho_{3}Z+\rho_{4}W\\ \\ X & =\lambda_{1}x+\mu_{1}y+\nu_{1}z+\rho_{1}w\\ Y & =\lambda_{2}x+\mu_{2}y+\nu_{2}z+\rho_{2}w\\ Z & =\lambda_{3}x+\mu_{3}y+\nu_{3}z+\rho_{3}w\\ -W & =\lambda_{4}x+\mu_{4}y+\nu_{4}z+\rho_{4}w \end{align} \left|{\scriptstyle \begin{align}\lambda_{1}^{2}+\mu_{1}^{2}+\nu_{1}^{2}-\rho_{1}^{2} & =1\\ \lambda_{2}^{2}+\mu_{2}^{2}+\nu_{2}^{2}-\rho_{2}^{2} & =1\\ \lambda_{3}^{2}+\mu_{3}^{2}+\nu_{3}^{2}-\rho_{3}^{2} & =1\\ \lambda_{4}^{2}+\mu_{4}^{2}+\nu_{4}^{2}-\rho_{4}^{2} & =-1\\ \lambda_{2}\lambda_{3}+\mu_{2}\mu_{3}+\nu_{2}\nu_{3}-\rho_{2}\rho_{3} & =0\\ \lambda_{3}\lambda_{1}+\mu_{3}\mu_{1}+\nu_{3}\nu_{1}-\rho_{3}\rho_{1} & =0\\ \lambda_{1}\lambda_{2}+\mu_{1}\mu_{2}+\nu_{1}\nu_{2}-\rho_{1}\rho_{2} & =0\\ \lambda_{1}\lambda_{4}+\mu_{1}\mu_{4}+\nu_{1}\nu_{4}-\rho_{1}\rho_{4} & =0\\ \lambda_{2}\lambda_{4}+\mu_{2}\mu_{4}+\nu_{2}\nu_{4}-\rho_{2}\rho_{4} & =0\\ \lambda_{3}\lambda_{4}+\mu_{3}\mu_{4}+\nu_{3}\nu_{4}-\rho_{3}\rho_{4} & =0 \end{align} }\right.{\scriptstyle \begin{align}\lambda_{1}^{2}+\lambda_{2}^{2}+\lambda_{3}^{2}-\lambda_{4}^{2} & =1\\ \mu_{1}^{2}+\mu_{2}^{2}+\mu_{3}^{2}-\mu_{4}^{2} & =1\\ \nu_{1}^{2}+\nu_{2}^{2}+\nu_{3}^{2}-\nu_{4}^{2} & =1\\ \rho_{1}^{2}+\rho_{2}^{2}+\rho_{3}^{2}-\rho_{4}^{2} & =-1\\ \lambda_{1}\mu_{1}+\lambda_{2}\mu_{2}+\lambda_{3}\mu_{3}-\lambda_{4}\mu_{4} & =0\\ \lambda_{1}\nu_{1}+\lambda_{2}\nu_{2}+\lambda_{3}\nu_{3}-\lambda_{4}\nu_{4} & =0\\ \lambda_{1}\rho_{1}+\lambda_{2}\rho_{2}+\lambda_{3}\rho_{3}-\lambda_{4}\rho_{4} & =0\\ \mu_{1}\nu_{1}+\mu_{2}\nu_{2}+\mu_{3}\nu_{3}-\mu_{4}\nu_{4} & =0\\ \mu_{1}\rho_{1}+\mu_{2}\rho_{2}+\mu_{3}\rho_{3}-\mu_{4}\rho_{4} & =0\\ \nu_{1}\rho_{1}+\nu_{2}\rho_{2}+\nu_{3}\rho_{3}-\nu_{4}\rho_{4} & =0 \end{align} } \end{matrix}$$

Finally, in a treatise on w:Grassmann's Ausdehnungslehre and circles (1891), he again provided transformations of orthogonal circle systems described by him as being "identical with those for transformation of coordinates in non-Euclidean geometry":


 * $$\begin{matrix}x^{2}+y^{2}+z^{2}=w^{2}\\

\hline \begin{align}x & =\lambda_{1}x'+\lambda_{2}y'+\lambda_{3}z'+\lambda_{4}w' & \text{(4 equations)}\\ x' & =\lambda_{1}x+\mu_{1}y+\nu_{1}z-\rho_{1}w\\ -w' & =\lambda_{4}x+\mu_{4}y+\nu_{4}z-\rho_{4}w \end{align} \\ \hline \begin{align}\lambda_{1}^{2}+\mu_{1}^{2}+\nu_{1}^{2}-\rho_{1}^{2} & =1\\ \lambda_{2}^{2}+\mu_{2}^{2}+\nu_{2}^{2}-\rho_{2}^{2} & =1\\ \lambda_{3}^{2}+\mu_{3}^{2}+\nu_{3}^{2}-\rho_{3}^{2} & =1\\ \lambda_{4}^{2}+\mu_{4}^{2}+\nu_{4}^{2}-\rho_{4}^{2} & =-1\\ \lambda_{1}\lambda_{2}+\mu_{1}\mu_{2}+\nu_{1}\nu_{2}-\rho_{1}\rho_{2} & =0 & \text{(6 equations)}\\ \lambda_{1}^{2}+\lambda_{2}^{2}+\lambda_{3}^{2}-\lambda_{4}^{2} & =1\\ \rho_{1}^{2}+\rho_{2}^{2}+\rho_{3}^{2}-\rho_{4}^{2} & =-1\\ \lambda_{1}\mu_{1}+\lambda_{2}\mu_{2}+\lambda_{3}\mu_{3}-\lambda_{4}\mu_{4} & =0 & \text{(6 equations)} \end{align} \end{matrix}\text{ }$$

Hill (1882) – Homogeneous coordinates
Following Gauss (1818), George William Hill (1882) formulated the equations


 * $$\begin{matrix}k\left(\sin^{2}T+\cos^{2}T-1\right)\\

k\left(\sin^{2}E+\cos^{2}E-1\right)\\ \hline \begin{align} & & \cos E' & =\frac{\alpha+\alpha'\sin T+\alpha\cos T}{\gamma+\gamma'\sin T+\gamma\cos T}\\ & \mathbf{(1)} & \sin E' & =\frac{\beta+\beta'\sin T+\beta\cos T}{\gamma+\gamma'\sin T+\gamma\cos T}\\ \hline \\ & & x & =\alpha u+\alpha'u'+\alphau\\ & & y & =\beta u+\beta'u'+\betau\\ & & z & =\gamma u+\gamma'u'+\gammau\\ & \mathbf{(2)}\\ & & u & =-\alpha x-\beta y+\gamma z\\ & & u' & =\alpha'x+\beta'y'-\gamma'z\\ & & u & =\alphax+\betay-\gammaz \end{align} \left|{\scriptstyle \begin{align}\alpha^{2}+\beta^{2}-\gamma^{2} & =-1\\ \alpha^{\prime2}+\beta^{\prime2}-\gamma^{\prime2} & =1\\ \alpha^{\prime\prime2}+\beta^{\prime\prime2}-\gamma^{\prime\prime2} & =1\\ \alpha\alpha'+\beta\beta'-\gamma\gamma' & =0\\ \alpha\alpha+\beta\beta-\gamma\gamma'' & =0\\ \alpha'\alpha+\beta'\beta-\gamma'\gamma'' & =0\\ \\ (k=-1)\\ \alpha^{2}-\alpha^{\prime2}-\alpha^{\prime\prime2} & =k\\ \beta^{2}-\beta^{\prime2}-\beta^{\prime\prime2} & =k\\ \gamma^{2}-\gamma^{\prime2}-\gamma^{\prime\prime2} & =-k\\ \alpha\beta-\alpha'\beta'-\alpha\beta & =0\\ \alpha\gamma-\alpha'\gamma'-\alpha\gamma & =0\\ \beta\gamma-\beta'\gamma'-\beta\gamma & =0 \end{align} }\right. \end{matrix}$$

Picard (1882-1884) – Quadratic forms
Émile Picard (1882) analyzed the invariance of indefinite ternary Hermitian quadratic forms with integer coefficients and their relation to discontinuous groups, extending Poincaré's Fuchsian functions of one complex variable related to a circle, to "hyperfuchsian" functions of two complex variables related to a hypersphere. He formulated the following special case of an Hermitian form:


 * $$\begin{matrix}\begin{matrix}xx_{0}+yy_{0}-zz_{0}\\

\\ \mathbf{(1)}\ \begin{align}x & =M_{1}X+P_{1}Y+R_{1}Z\\ y & =M_{2}X+P_{2}Y+R_{2}Z\\ z & =M_{3}X+P_{3}Y+R_{3}Z \end{align} \\ \\ \left[\begin{align}[][x,y,z]=\text{complex}\\ \left[x_{0},y_{0},z_{0}\right]=\text{conjugate} \end{align} \right]\\ \\ \hline \\ x^{\prime2}+x^{\prime\prime2}+y^{\prime2}+y^{\prime\prime2}=1\\ x=x'+ix,\quad y=y'+iy\\ \\ \mathbf{(2)}\ \begin{align}X & =\frac{M_{1}x+P_{1}y+R_{1}}{M_{3}x+P_{3}y+R_{3}}\\ Y & =\frac{M_{2}x+P_{2}y+R_{2}}{M_{3}x+P_{3}y+R_{3}} \end{align} \end{matrix}\left|{\scriptstyle \begin{align}M_{1}\mu_{1}+M_{2}\mu_{2}-M_{3}\mu_{3} & =1\\ P_{1}\pi_{1}+P_{2}\pi_{2}-P_{3}\pi_{3} & =1\\ R_{1}\rho_{1}+R_{2}\rho_{2}-R_{3}\rho_{3} & =-1\\ P_{1}\mu_{1}+P_{2}\mu_{2}-P_{3}\mu_{3} & =0\\ M_{1}\rho_{1}+M_{2}\rho_{2}-M_{3}\rho_{3} & =0\\ P_{1}\rho_{1}+P_{2}\rho_{2}-P_{3}\rho_{3} & =0\\ \\ M_{1}\mu_{1}+P_{1}\pi_{1}-R_{1}\rho_{1} & =1\\ M_{2}\mu_{2}+P_{2}\pi_{2}-R_{2}\rho_{2} & =1\\ M_{3}\mu_{3}+P_{3}\pi_{3}-R_{3}\rho_{3} & =-1\\ \mu_{2}M_{1}+\pi_{2}P_{1}-R_{1}\rho_{2} & =0\\ \mu_{2}M_{3}+\pi_{2}P_{3}-R_{3}\rho_{2} & =0\\ \mu_{3}M_{1}+\pi_{3}P_{1}-R_{1}\rho_{3} & =0\\ \\ \left[\begin{align}[][M,P,R\dots]=\text{complex}\\ \left[\mu,\pi,\rho\dots\right]=\text{conjugate} \end{align} \right] \end{align} }\right.\end{matrix}$$

Or in (1884a) in relation to indefinite binary Hermitian quadratic forms:


 * $$\begin{matrix}UU_{0}-VV_{0}=uu_{0}-vv_{0}\\

\hline \begin{align}U & =\mathcal{A}u+\mathcal{B}v\\ V & =\mathcal{C}u+\mathcal{D}v \end{align} \left|\begin{align}\mathcal{A}\mathcal{A}_{0}-\mathcal{C}\mathcal{C}_{0} & =1\\ \mathcal{A}\mathcal{B}_{0}-\mathcal{C}\mathcal{D}_{0} & =0\\ \mathcal{B}\mathcal{B}_{0}-\mathcal{D}\mathcal{D}_{0} & =-1\\ \mathcal{D}\mathcal{D}_{0}-\mathcal{C}\mathcal{C}_{0} & =1 \end{align} \right. \end{matrix}$$

Or in (1884b):


 * $$\begin{matrix}xx_{0}+yy_{0}-1=0\\

\hline \begin{align}X & =\frac{M_{1}x+P_{1}y+R_{1}}{M_{3}x+P_{3}y+R_{3}}\\ Y & =\frac{M_{2}x+P_{2}y+R_{2}}{M_{3}x+P_{3}y+R_{3}} \end{align} \left|{\scriptstyle \begin{align}M_{1}\mu_{1}+M_{2}\mu_{2}-M_{3}\mu_{3}=P_{1}\pi_{1}+P_{2}\pi_{2}-P_{3}\pi_{3} & =1\\ R_{1}\rho_{1}+R_{2}\rho_{2}-R_{3}\rho_{3} & =-1\\ P_{1}\mu_{1}+P_{2}\mu_{2}-P_{3}\mu_{3}=M_{1}\rho_{1}+M_{2}\rho_{2}-M_{3}\rho_{3}=P_{1}\rho_{1}+P_{2}\rho_{2}-P_{3}\rho_{3} & =0\\ M_{1}\rho_{1}+M_{2}\rho_{2}-M_{3}\rho_{3} & =0 \end{align} }\right. \end{matrix}$$

Or in (1884c):


 * $$\begin{matrix}UU_{0}+VV_{0}-WW_{0}=uu_{0}+vv_{0}-ww_{0}\\

\hline \mathbf{(1)}\ \begin{align}U & =Mu+Pv+Rw\\ V & =M'u+P'v+R'w\\ W & =Mu+Pv+R''w\\ \\ u & =M_{0}U+M_{0}^{\prime}V-M_{0}^{\prime\prime}W\\ v & =P_{0}U+P_{0}^{\prime}V-P_{0}^{\prime\prime}W\\ w & =-R_{0}U-R_{0}^{\prime}V+R_{0}^{\prime\prime}W \end{align} \left|{\scriptstyle \begin{align}MM_{0}+M'M_{0}^{\prime}-M''M_{0}^{\prime\prime} & =1\\ PP_{0}+P'P_{0}^{\prime}-P''P_{0}^{\prime\prime} & =1\\ RR_{0}+R'R_{0}^{\prime}-R''R_{0}^{\prime\prime} & =-1\\ MP_{0}+M'P_{0}^{\prime}-M''P_{0}^{\prime\prime} & =0\\ MR_{0}+M'R_{0}^{\prime}-M''R_{0}^{\prime\prime} & =0\\ PR_{0}+P'R_{0}^{\prime}-P''R_{0}^{\prime\prime} & =0\\ \\ MM_{0}+PP_{0}-RR_{0} & =1\\ M'M_{0}^{\prime}+P'P_{0}^{\prime}-R'R_{0}^{\prime} & =1\\ MM_{0}^{\prime\prime}+PP_{0}^{\prime\prime}-R''R_{0}^{\prime\prime} & =-1\\ M_{0}M'+P_{0}P'-R_{0}R' & =0\\ M_{0}M+P_{0}P-R_{0}R'' & =0\\ M_{0}^{\prime}M+P_{0}^{\prime}P-R_{0}^{\prime}R'' & =0 \end{align} }\right.\\ \hline \text{Invariance of unit hypersphere:}\\ \mathbf{(2)}\ \begin{align}\xi' & =\frac{A\xi+A'\eta+A}{C\xi+C'\eta+C}\\ \eta' & =\frac{B\xi+B'\eta+B}{C\xi+C'\eta+C} \end{align} \left|{\scriptstyle \begin{align}AA_{0}+A'A_{0}^{\prime}-A''A_{0}^{\prime\prime} & =1\\ BB_{0}+B'B_{0}^{\prime}-B''B_{0}^{\prime\prime} & =1\\ CC_{0}+C'C_{0}^{\prime}-C''C_{0}^{\prime\prime} & =-1\\ AB_{0}+A'B_{0}^{\prime}-A''B_{0}^{\prime\prime} & =0\\ AC_{0}+A'C_{0}^{\prime}-A''C_{0}^{\prime\prime} & =0\\ BC_{0}+B'C_{0}^{\prime}-B''C_{0}^{\prime\prime} & =0 \end{align} }\right. \end{matrix}$$

Callandreau (1885) – Homography
Following Gauss (1818) and Hill (1882), Octave Callandreau (1885) formulated the equations


 * $$\begin{matrix}k\left(\sin^{2}T+\cos^{2}T-1\right)=\\

{\scriptstyle (\alpha+\alpha'\sin T+\alpha\cos T)^{2}+(\beta+\beta'\sin T+\beta\cos T)^{2}-(\gamma+\gamma'\sin T+\gamma''\cos T)^{2}}\\ \hline \begin{align}\cos\varepsilon' & =\frac{\alpha+\alpha'\sin T+\alpha\cos T}{\gamma+\gamma'\sin T+\gamma\cos T}\\ \sin\varepsilon' & =\frac{\beta+\beta'\sin T+\beta\cos T}{\gamma+\gamma'\sin T+\gamma\cos T} \end{align} \left|{\scriptstyle \begin{align} & \left(k=1\right)\\ \alpha^{2}+\beta^{2}-\gamma^{2} & =-k & \alpha\alpha'+\beta\beta'-\gamma\gamma' & =0\\ \alpha^{\prime2}+\beta^{\prime2}-\gamma^{\prime2} & =+k & \alpha\alpha+\beta\beta-\gamma\gamma'' & =0\\ \alpha^{\prime\prime2}+\beta^{\prime\prime2}-\gamma^{\prime\prime2} & =+k & \alpha'\alpha+\beta'\beta-\gamma'\gamma'' & =0\\ \\ \alpha^{2}-\alpha^{\prime2}-\alpha^{\prime\prime2} & =-1 & \alpha\beta-\alpha'\beta'-\alpha\beta & =0\\ \beta^{2}-\beta^{\prime2}-\beta^{\prime\prime2} & =-1 & \alpha\gamma-\alpha'\gamma'-\alpha\gamma & =0\\ \gamma^{2}-\gamma^{\prime2}-\gamma^{\prime\prime2} & =+1 & \beta\gamma-\beta'\gamma'-\beta\gamma & =0 \end{align} }\right. \end{matrix}$$

Lie (1885-1890) – Lie group, hyperbolic motions, and infinitesimal transformations
In (1885/86), Sophus Lie identified the projective group of a general surface of second degree $$\sum f_{ik}x_{i}'x_{k}'=0$$ with the group of non-Euclidean motions. In a thesis guided by Lie, Hermann Werner (1889) discussed this projective group by using the equation of a unit hypersphere as the surface of second degree (which was already given before by Killing (1887)), and also gave the corresponding infinitesimal projective transformations (Lie algebra):


 * $$\begin{matrix}x_{1}^{2}+x_{2}^{2}+\dots+x_{n}^{2}=1\\

\hline x_{i}p_{\varkappa}-x_{\varkappa}p_{i},\quad p_{i}-x_{i}\sum_{1}^{n}{\scriptstyle j}\ x_{j}p_{j}\quad(i,\varkappa=1,\dots, n)\\ \text{where}\\ \left(Q_{i},Q_{\varkappa}\right)=R_{i,\varkappa};\ \left(Q_{i},Q_{j,\varkappa}\right)=\varepsilon_{i,j}Q_{\varkappa}-\varepsilon_{i,\varkappa}Q_{j};\\ \left(R_{i,\varkappa},R_{\mu,\nu}\right)=\varepsilon_{\varkappa,\mu}R_{i,\nu}-\varepsilon_{\varkappa,\nu}R_{i,\mu}-\varepsilon_{,\mu}R_{\varkappa,\nu}+\varepsilon_{i,\nu}R_{\varkappa,\mu}\\ \left[\varepsilon_{i,\varkappa}\equiv0\ \text{for}\ i\ne\varkappa;\ \varepsilon_{i,i}=1\right] \end{matrix}$$

More generally, Lie (1890) defined non-Euclidean motions in terms of two forms $$x_{1}^{2}+x_{2}^{2}+x_{3}^{2}\pm1=0$$ in which the imaginary form with $$+1$$ denotes the group of elliptic motions (in Klein's terminology), the real form with −1 the group of hyperbolic motions, with the latter having the same form as Werner's transformation:


 * $$\begin{matrix}x_{1}^{2}+\dots+x_{n}^{2}-1=0\\

\hline p_{k}-x_{k}\sum j_{1}^{0}x_{j}p_{j},\quad x_{i}p_{k}-x_{k}p_{i}\quad(i,k=1\dots n) \end{matrix}$$

Summarizing, Lie (1893) discussed the real continuous groups of the conic sections representing non-Euclidean motions, which in the case of hyperbolic motions have the form:


 * $$x^{2}+y^{2}-1=0$$ or $$x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-1=0$$ or $$x_{1}^{2}+\dots+x_{n}^{2}-1=0$$.

Gérard (1892) – Weierstrass coordinates
Louis Gérard (1892) – in a thesis examined by Poincaré – discussed Weierstrass coordinates (without using that name) in the plane using the following invariant and its Lorentz transformation equivalent to ($$) (n=2):


 * $$\begin{matrix}X^{2}+Y^{2}-Z^{2}=1\\

X^{2}+Y^{2}-Z^{2}=X^{\prime2}+Y^{\prime2}-Z^{\prime2}\\ \hline \begin{align}X & =aX'+a'Y'+a''Z'\\ Y & =bX'+b'Y'+b''Z'\\ Z & =cX'+c'Y'+c''Z'\\ \\ X' & =aX+bY-cZ\\ Y' & =a'X+b'Y-c'Z\\ Z' & =-aX-bY+c''Z \end{align} \left|\begin{align}a^{2}+b^{2}-c^{2} & =1\\ a^{\prime2}+b^{\prime2}-c^{\prime2} & =1\\ a^{\prime\prime2}+b^{\prime\prime2}-c^{\prime\prime2} & =-1\\ aa'+bb'-cc' & =0\\ a'a+b'b-c'c'' & =0\\ aa+bb-c''c & =0 \end{align} \right. \end{matrix}$$

He gave the case of translation as follows:


 * $$\begin{align}X & =Z_{0}X'+X_{0}Z'\\

Y & =Y'\\ Z & =X_{0}X'+Z_{0}Z' \end{align} \ \text{with}\ \begin{align}X_{0} & =\operatorname{sh}OO'\\ Z_{0} & =\operatorname{ch}OO' \end{align} $$

Hausdorff (1899) – Weierstrass coordinates
Felix Hausdorff (1899) – citing Killing (1885) – discussed Weierstrass coordinates in the plane using the following invariant and its transformation:


 * $$\begin{matrix}p^{2}-x^{2}-y^{2}=1\\

\hline \begin{align}x & =a_{1}x'+a_{2}y'+x_{0}p'\\ y & =b_{1}x'+b{}_{2}y'+y_{0}p'\\ p & =e_{1}x'+e_{2}y'+p_{0}p'\\ \\ x' & =a_{1}x+b_{1}y-e_{1}p\\ y' & =a_{2}x+b_{2}y-e_{2}p\\ -p' & =x_{0}x+y_{0}y-p_{0}p \end{align} \left|{\scriptstyle \begin{align}a_{1}^{2}+b_{1}^{2}-e_{1}^{2} & =1\\ a_{2}^{2}+b_{2}^{2}-e_{2}^{2} & =1\\ -x_{0}^{2}-y_{0}^{2}+p_{0}^{2} & =1\\ a_{2}x_{0}+b_{2}y_{0}-e_{2}p_{0} & =0\\ a_{1}x_{0}+b_{1}y_{0}-e_{1}p_{0} & =0\\ a_{1}a_{2}+b_{1}b_{2}-e_{1}e_{2} & =0\\ \\ a_{1}^{2}+a_{2}^{2}-x_{0}^{2} & =1\\ b_{1}^{2}+b_{2}^{2}-y_{0}^{2} & =1\\ -e_{1}^{2}-e_{2}^{2}+p_{0}^{2} & =1\\ b_{1}e_{1}+b_{2}e_{2}-y_{0}p_{0} & =0\\ a_{1}e_{1}+a_{2}e_{2}-x_{0}p_{0} & =0\\ a_{1}b_{1}+a_{2}b_{2}-x_{0}y_{0} & =0 \end{align} }\right. \end{matrix}$$

Woods (1901-05) – Beltrami and Weierstrass coordinates
In (1901/02) Frederick S. Woods defined the following invariant quadratic form and its projective transformation in terms of Beltrami coordinates (he pointed out that this can be connected to hyperbolic geometry by setting $$k=\sqrt{-1}R$$ with R as real quantity):


 * $$\begin{matrix}k^{2}\left(u^{2}+v^{2}+w^{2}\right)+1=0\\

\hline \begin{align}u' & =\frac{\alpha_{1}u+\alpha_{2}v+\alpha_{3}w+\alpha_{4}}{\delta_{1}u+\delta_{2}v+\delta_{3}w+\delta_{4}}\\ v' & =\frac{\beta_{1}u+\beta_{2}v+\beta_{3}w+\beta_{4}}{\delta_{1}u+\delta_{2}v+\delta_{3}w+\delta_{4}}\\ w' & =\frac{\gamma_{1}u+\gamma_{2}v+\gamma_{3}w+\gamma_{4}}{\delta_{1}u+\delta_{2}v+\delta_{3}w+\delta_{4}} \end{align} \left|\begin{align}k^{2}\left(\alpha_{i}^{2}+\beta_{i}^{2}+\gamma_{i}^{2}\right)+\delta_{i}^{2} & =k^{2}\\ (i=1,2,3)\\ k^{2}\left(\alpha_{4}^{2}+\beta_{4}^{2}+\gamma_{4}^{2}\right)+\delta_{4}^{2} & =1\\ \alpha_{i}\alpha_{h}+\beta_{i}\beta_{h}+\gamma_{i}\gamma_{h}+\delta_{i}\delta_{h} & =0\\ (i,h=1,2,3,4;\ i\ne h) \end{align} \right. \end{matrix}$$

Alternatively, Woods (1903, published 1905) – citing Killing (1885) – used the invariant quadratic form in terms of Weierstrass coordinates and its transformation (with $$k=\sqrt{-1}k$$ for hyperbolic space):


 * $$\begin{matrix}x_{0}^{2}+k^{2}\left(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}\right)=1\\

ds^{2}=\frac{1}{k^{2}}dx_{0}^{2}+dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}\\ \hline \begin{align}x_{1}^{\prime} & =\alpha_{1}x_{1}+\alpha_{2}x_{2}+\alpha_{3}x_{3}+\alpha_{0}x_{0}\\ x_{2}^{\prime} & =\beta_{1}x_{1}+\beta_{2}x_{2}+\beta_{3}x_{3}+\beta_{0}x_{0}\\ x_{3}^{\prime} & =\gamma_{1}x_{1}+\gamma_{2}x_{2}+\gamma_{3}x_{3}+\gamma_{0}x_{0}\\ x_{0}^{\prime} & =\delta_{1}x_{1}+\delta_{2}x_{2}+\delta_{3}x_{3}+\delta_{0}x_{0} \end{align} \left|\begin{align}\delta_{0}^{2}+k^{2}\left(\alpha_{0}^{2}+\beta_{0}^{2}+\gamma_{0}^{2}\right) & =1\\ \delta_{i}^{2}+k^{2}\left(\alpha_{i}^{2}+\beta_{i}^{2}+\gamma_{i}^{2}\right) & =k^{2}\\ (i=1,2,3)\\ \delta_{i}\delta_{h}+k^{2}\left(\alpha_{i}\alpha_{h}+\beta_{i}\beta_{h}+\gamma_{i}\gamma_{h}\right) & =0\\ (i,h=0,1,2,3;\ i\ne h) \end{align} \right. \end{matrix}$$

Liebmann (1904–05) – Weierstrass coordinates
Heinrich Liebmann (1904/05) – citing Killing (1885), Gérard (1892), Hausdorff (1899) – used the invariant quadratic form and its Lorentz transformation equivalent to ($$) (n=2)


 * $$\begin{matrix}p^{\prime2}-x^{\prime2}-y^{\prime2}=1\\

\hline \begin{align}x_{1} & =\alpha_{11}x+\alpha_{12}y+\alpha_{13}p\\ y_{1} & =\alpha_{21}x+\alpha_{22}y+\alpha_{23}p\\ x_{1} & =\alpha_{31}x+\alpha_{32}y+\alpha_{33}p\\ \\ x & =\alpha_{11}x_{1}+\alpha_{21}y_{1}-\alpha_{31}p_{1}\\ y & =\alpha_{12}x_{1}+\alpha_{22}y_{1}-\alpha_{32}p_{1}\\ p & =-\alpha_{13}x_{1}-\alpha_{23}y_{1}+\alpha_{33}p_{1} \end{align} \left|\begin{align}\alpha_{33}^{2}-\alpha_{13}^{2}-\alpha_{23}^{2} & =1\\ -\alpha_{31}^{2}+\alpha_{11}^{2}+\alpha_{21}^{2} & =1\\ -\alpha_{32}^{2}+\alpha_{12}^{2}+\alpha_{22}^{2} & =1\\ \alpha_{31}\alpha_{32}-\alpha_{11}\alpha_{12}-\alpha_{21}\alpha_{22} & =0\\ \alpha_{32}\alpha_{33}-\alpha_{12}\alpha_{13}-\alpha_{22}\alpha_{23} & =0\\ \alpha_{33}\alpha_{31}-\alpha_{23}\alpha_{11}-\alpha_{23}\alpha_{21} & =0 \end{align} \right. \end{matrix}$$