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

Lorentz transformation via sphere transformation
If one only requires the invariance of the light cone represented by the differential equation $$-dx_{0}^{2}+\dots+dx_{n}^{2}=0$$, which is the same as asking for the most general transformation that changes spheres into spheres, the Lorentz group can be extended by adding dilations represented by the factor λ. The result is the group Con(1,p) of spacetime conformal transformations in terms of special conformal transformations and inversions producing the relation


 * $$-dx_{0}^{2}+\dots+dx_{n}^{2}=\lambda\left(-dx_{0}^{\prime2}+\dots+dx_{n}^{\prime2}\right)$$.

One can switch between two representations of this group by using an imaginary sphere radius coordinate x0=iR with the interval $$dx_{0}^{2}+\dots+dx_{n}^{2}$$ related to conformal transformations, or by using a real radius coordinate x0=R with the interval $$-dx_{0}^{2}+\dots+dx_{n}^{2}$$ related to Lie's (1871) sphere transformation (or spherical wave transformations) in terms of contact transformations preserving circles and spheres. It was shown by Bateman & Cunningham (1909–1910), that the group Con(1,3) is the most general one leaving invariant the equations of Maxwell's electrodynamics.

It turns out that Con(1,3) is isomorphic to the special orthogonal group SO(2,4), and contains the Lorentz group SO(1,3) as a subgroup by setting λ=1. More generally, Con(q,p) is isomorphic to SO(q+1,p+1) and contains SO(q,p) as subgroup. This implies that Con(0,p) is isomorphic to the Lorentz group of arbitrary dimensions SO(1,p+1). Consequently, the conformal group in the plane Con(0,2) – known as the group of Möbius transformations – is isomorphic to the Lorentz group SO(1,3). This can be seen using tetracyclical coordinates satisfying the form $$-x_{0}^{2}+x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=0$$, which were discussed by Pockels (1891), Klein (1893), Bôcher (1894). The relation between Con(1,3) and the Lorentz group was noted by Bateman & Cunningham (1909–1910) and others. (For a different take on the Möbius group, see also E:Lorentz transformation via Cayley–Klein parameters, Möbius and spin transformations).

A subgroup of Lie's group of sphere transformations is the Laguerre group (or group of transformations by reciprocal directions) dealing with oriented spheres, planes and lines, which was implicit in the work of Ribaucour (1870), Lie (1871), Darboux (1873), while Stephanos (1883) argued that Lie's geometry of oriented spheres in terms of contact transformations, as well as the special case of the transformations of oriented planes into each other (such as by Laguerre), provides a geometrical interpretation of Hamilton's biquaternions. The Laguerre group is generated by Laguerre inversions introduced by Laguerre (1882) and discussed by Darboux (1887) and Smith (1900) leaving invariant $$X^{2}+Y^{2}+Z^{2}-R^{2}$$ with R as radius, thus the Laguerre group is isomorphic to the Lorentz group as pointed out by Bateman (1910), Cartan (1912, 1915/55), Poincaré (1912/21) and others. The Laguerre inversions were written as follows:

which correspond to antichronous Lorentz transformations, and become orthochronous by changing the sign of $$R'$$. A special case of formulas (5a) with $$a=\sqrt{-1}$$ was given by Bonnet (1856), while its complete form was provided by Laguerre (1882), Darboux (1887), Smith (1900). Laguerre transformations in trigonometric form were given by Scheffers (1899). The axis of transformation used in Laguerre inversions is identical to the radical axis introduced by Gaultier (1812/13), consisting of all centers of circles intersecting the given circles orthogonally (orthogonal circles). The role of general Lorentz transformations in relation to orthogonal circles was shown by Cox (1883-91).

Furthermore, setting $$v=\tfrac{2a}{1+a^{2}}$$ together with $$\cos\alpha'=\tfrac{X'}{-R'}$$ and $$\cos\alpha=\tfrac{X}{R}$$ in formulas (5a) gives:

This was used by Darboux (1873) as a sphere transformation, and in 1881 he showed that it can also be used to perform Laguerre transformations of planes. In special relativity, it turns out that formula (5b) describes the aberration of light, see E:velocity addition and aberration.

Gaultier (1812/13) - Radical axis
Louis Gaultier (written 1812, published 1813) showed that if a, o be the centers of circles A, O, and RA, RO the respective radii, then by the Pythagorean theorem it follows in case a is outside of O and o is outside of A:


 * $$\overline{ao}^{2}=\overline{RA}^{2}+\overline{RO}^{2}$$

or both a, o are inside O:


 * $$\overline{ao}^{2}=\overline{RA}^{2}-\overline{RO}^{2}$$

Furthermore, let circle O (center o, radius RO) satisfy the previous relations with respect to two circles A (center a, radius RA) and B (center b, radius RB):


 * $$\overline{oa}^{2}=\overline{RA}^{2}\pm\overline{RO}^{2}$$ and $$\overline{ob}^{2}=\overline{RB}^{2}\pm\overline{RO}^{2}$$

producing
 * $$\overline{oa}^{2}-\overline{ob}^{2}=\overline{RA}^{2}-\overline{RB}^{2}$$

and perpendicular to line ab let the line oh be drawn from o to point h, then the two right triangles oha and ohb satisfy:


 * $$\overline{ah}^{2}-\overline{bh}^{2}=\overline{oa}^{2}-\overline{ob}^{2}=\overline{RA}^{2}-\overline{RB}^{2}$$

He concluded:


 * We will denote the radical axis of AB as the line ho raised to the determined point h perpendicular to line ab, which contains the centers of all the radical circles common to A and B.

Gaultier's concept of the radical axis (L'axe radical) was also discussed by Jean-Victor Poncelet (1822: Corde idèale), Jacob Steiner (1826: Potenzlinie), Julius Plücker (1828: Chordale) and became well known.

Bonnet (1856)
Pierre Ossian Bonnet (1856) defined a reciprocal transformation preserving lines of curvatures. He noted that his transformation implies the following relation between curvature radii $$\rho,\rho_{1}$$ and ordinates $$\zeta,\zeta_{1}$$ of the respective curvature centers:


 * $$\rho_{1}=i\zeta,\quad\rho=-i\zeta_{1}$$ where $$i=\sqrt{-1}$$

Ribaucour (1870)
Albert Ribaucour (1870), defined what was later called "Ribaucour transformations" preserving lines of curvature:


 * p. 330: If circles are normal to three surfaces, they are normal to a family of surfaces belonging to a triply orthogonal system. This results in a class of orthogonal triple systems which I will propose to call cyclic systems, intimately linked to the deformation of surfaces. Given a surface (A), we can propose to seek all the cyclic systems which derive from it; the $$ds^{2}$$ of this surface being put in the form $$ds^{2}=\lambda^{2}.dx\ dy$$ [...]


 * p. 332: If spheres have their contact chords normal to surfaces, the circles passing through the centers of these spheres and their points of contact with their enveloping surfaces are normal to an infinity of surfaces forming part of a cyclic system. [...] If surfaces are part of an orthogonal system, the osculating circles of their orthogonal trajectories corresponding to all the points of one of these surfaces are normal to a family of surfaces belonging to a cyclic system. [...] I will point out the simple case where (A) is a plane, a case which leads to a general transformation of the surfaces with correspondence of the lines of curvature [...].

Lie (1871) - Lie sphere transformation
In several papers between 1847 and 1850 it was shown by Joseph Liouville that the relation λ(δx2+δy2+δz2) is invariant under the group of conformal transformations generated by w:inversions transforming spheres into spheres, which can be related special conformal transformations or Möbius transformations. (The conformal nature of the linear fractional transformation $$\tfrac{a+bz}{c+dz}$$ of a complex variable $$z$$ was already discussed by Euler (1777)).

Liouville's theorem was extended to all dimensions by Sophus Lie (1871a). In addition, Lie described a manifold whose elements can be represented by spheres, where the last coordinate yn+1 can be related to an imaginary radius by iyn+1:


 * $$\begin{matrix}\sum_{i=1}^{i=n} (x_i-y_i)^2+y_{n+1}^2=0 \\ \downarrow\\ \sum_{i=1}^{i=n+1} (y_i^{\prime}-y_i^{\prime\prime})^2=0 \end{matrix}$$

If the second equation is satisfied, two spheres y′ and y″ are in contact. Lie then defined the correspondence between w:contact transformations in Rn and conformal point transformations in Rn+1: The sphere of space Rn consists of n+1 parameter (coordinates plus imaginary radius), so if this sphere is taken as the element of space Rn, it follows that Rn now corresponds to Rn+1. Therefore, any transformation (to which he counted E:orthogonal transformations and inversions) leaving invariant the condition of contact between spheres in Rn, corresponds to the conformal transformation of points in Rn+1. He pointed out that conformal point transformations consist of motions (such as rigid transformations and orthogonal transformations), similarity transformations, and inversions.

In the same paper, Lie also mentioned the "well known fact" that "parallel transformations" (dilatations having the property of transforming planes to parallel planes) preserve lines of curvature, and he alluded to Bonnet's (1856) transformation as an example. Generally, all of the discussed transformations that preserve lines of curvature are either inversions or parallel transformations. In a footnote he specifically remarked that line transformations under which "(const=0)" remains unchanged, give all transformations of R by which surfaces of common spherical image pass into other such surfaces, and that the new spherical image emerges from the former by a conformal point transformation of the image-sphere, and that Bonnet's (1856) transformation belongs here.

Klein, Pockels, Bôcher (1871-91) - Polyspherical coordinates
In relation to line geometry, Felix Klein (1871/72) used coordinates satisfying the condition $$s_{1}^{2}+s_{2}^{2}+s_{2}^{2}+s_{2}^{2}+s_{5}^{2}=0$$. They were introduced in 1868 (belatedly published in 1873) by Gaston Darboux as a system of five coordinates in R3 (later called "pentaspherical" coordinates) in which the last coordinate is imaginary. Sophus Lie (1871) more generally used n+2 coordinates in Rn (later called "polyspherical" coordinates) satisfying $$\scriptstyle \sum_{i=1}^{i=n+2}x_{i}^{2}=0$$ in which the last coordinate is imaginary, as a means to discuss conformal transformations generated by inversions. These simultaneous publications can be explained by the fact that Darboux, Lie, and Klein corresponded with each other by letter.

When the last coordinate is defined as real, the corresponding polyspherical coordinates satisfy the form of a sphere. Initiated by lectures of Klein between 1889–1890, his student Friedrich Carl Alwin Pockels (1891) used such real coordinates, emphasizing that all of these coordinate systems remain invariant under conformal transformations generated by inversions:


 * $$x_1^2+x_2^2+\cdots+x_{n+1}^2-x_{n+2}^2=0 \text{ or } \sum_1^{n+1} x_h^2-x_{n+2}^2=0$$

Special cases were described by Klein (1893):


 * $$y_1^2+y_2^2+y_3^2+y_4^2-y_5^2=0$$ (pentaspherical).


 * $$x_1^2+x_2^2+x_3^2-x_4^2=0$$ (tetracyclical).

Both systems were also described by Maxime Bôcher (1894) in an expanded version of a thesis supervised by Klein.

Darboux (1873-87) - Laguerre inversion
In 1873, Gaston Darboux stated the following proposition:


 * Given a surface $$\left(\Sigma\right)$$, we add a fixed sphere $$\left({\rm S}\right)$$ to it, and we construct all spheres tangent to the surface and intersecting $$\left({\rm S}\right)$$ at a constant angle $$\alpha$$. Through the intersection of each of these spheres and $$\left({\rm S}\right)$$ new spheres pass intersecting $$\left({\rm S}\right)$$ at a constant angle $$\beta$$. These new spheres envelop a surface $$\left(\Sigma_{1}\right)$$, corresponding point by point to $$\left(\Sigma\right)$$ with conservation of lines of curvature. The corresponding points on the two surfaces are on circles normal both to the two surfaces and to the sphere $$\left({\rm S}\right)$$.

which he generalized by making a second proposition:


 * Consider a surface $$\left(\Sigma\right)$$, envelope of a series of variable spheres $$\left(\rm U\right)$$ intersecting under any angles the sphere $$\left(\rm S\right)$$. At each of the spheres $$\left(\rm U\right)$$ intersecting $$\left(\rm S\right)$$ at an angle I call $$\varphi$$ we match a sphere $$\left(\rm U_1\right)$$ passing through the intersection of $$\left(\rm S\right)$$ and from $$\left(\rm U \right)$$, and intersecting $$\left(\rm S\right)$$ at an angle $$\varphi_1$$ determined by equation
 * $$\frac{\cos\varphi-\cos\varphi_{1}}{1-\cos\varphi\cos\varphi_{1}}=h$$
 * Then the new spheres $$\left(\rm U_1\right)$$ envelop a surface $$\left(\Sigma_1\right)$$ which corresponds point by point at $$\left(\Sigma\right)$$ with curvature lines preserved. If we subject the spheres $$\left(\rm U\right)$$ tangent to $$\left(\Sigma \right)$$ to cut $$\left(\rm S\right)$$ under a constant angle, $$\varphi$$ will be constant; it will be the same for $$\varphi_1$$, by virtue of the previous equation, and we find the theorem given above. »

In 1881 he quoted his above propositions, gave priority to the first one to Ribaucour (1870), and then showed that Laguerre's transformation of reciprocal directions is included as well:


 * This proposal gave a new means of realizing a mode of transformation of surfaces with preservation of the lines of curvature, to which Ribaucour had devoted a few lines in the communication Sur la deformation des surfaces made to the Academy in 1870.
 * [..] Suppose, in particular, that the sphere $$\left(\rm S\right)$$ reduces to a plane $$\left(\pi\right)$$. Then to any plane $$\left(\rm P\right)$$ will correspond a plane $$\left(\rm P'\right)$$ passing through the intersection of $$\left(\pi\right)$$ and $$\left(\rm P\right)$$, and the angles $$\varphi,\varphi'$$ that the planes $$\left(\rm P\right)$$, $$\left(\rm P'\right)$$ make with $$\left(\pi\right)$$ will be linked by relation (1). It is not difficult to recognize, in this transformation from one plane to another, that which has recently been studied by Laguerre under the name of transformation by reciprocal directions. We see that it is included in the transformation of spheres which is defined by our second proposition. I have recalled these results only to arrive at the proposition which is the main object of this Communication. I will show, in accordance with a general theorem of Lie, that the transformation first proposed by Ribaucour boils down to dilatations (transition from a surface to the parallel surface) and to transformations by reciprocal vector rays.

He went on to rewrite his 1873 equation as:


 * $$\mathrm{tang}\frac{\varphi}{2}=\mathrm{tang}\frac{\varphi_{1}}{2}\sqrt{\frac{1-h}{1 +h}}$$

In 1887, Darboux gave a much more detailed account. For instance, he re-derived and extended the transformation of oriented half-lines given by Laguerre (1882) using coordinates x,y,z,R:


 * $$\begin{matrix}x^{\prime2}+y^{\prime2}+z^{\prime2}-R^{\prime2}=x^{2}+y^{2}+z^{2}-R^{2}\\ \hline \begin{align}x' & =x, & z' & =\frac{1+k^{2}}{1-k^{2}}z-\frac{2kR}{1-k^{2}},\\ y' & =y, & R' & =\frac{2kz}{1-k^{2}}-\frac{1+k^{2}}{1-k^{2}}R, \end{align} \end{matrix}$$ or $$\begin{align}z'+R' & =\frac{1+k}{1-k}(z-R)\\ z'-R' & =\frac{1-k}{1+k}(z+R) \end{align}$$

He went on to derive expressions and theorems similar to those given by him in 1873, and added that Bonnet's (1856) transformation is a special case.

Regarding the history of such transformations (before Laguerre's research) he wrote:
 * In the memoir already quoted, inserted in volume V of Mathematische Annalen, Lie has made known all the contact transformations which preserve the lines of curvature; he even pointed out (p. 186) the particular case of transformation by reciprocal directions; but this transformation had already been given in different works by Ribaucour. See, in particular, Ribaucour's note sur la deformation des surfaces (Comptes rendus, t. LXX, p. 332, 1870). In a different form, it was the subject of the author's studies published in Notes V and IX of Mémoire sur une classe remarquable de courbes et de surfaces algébriques, 1873.

Laguerre (1880-82) - Laguerre inversion
A systematic formulation of a geometry of orientation was given by Edmond Laguerre (1880), including geometric transformations of oriented planes into oriented planes and oriented spheres into oriented spheres, which he called "w:transformation by reciprocal directions". Besides the focus on the transformation of planes, a distinguishing feature to previous authors was the employment of the concept of orientation (i.e. attributing a certain sign to lines and radii) which became an indispensable tool in Lie sphere geometry and Laguerre geometry.

In 1882 he developed the "transformation of oriented half-lines" which was later called "Laguerre inversion", using two cycles (=oriented circles) K and K' whose radical axis is the axis of transformation and whose common tangents are parallel to the directions of the half-lines which transform into themselves. This led him to the following algebraic formulation ($$R,R'$$ being the radii and $$D,D'$$ the distances of their centers to the axis):


 * $$\left.\begin{align}D' & =\frac{D\left(1+\alpha^{2}\right)-2\alpha R}{1-\alpha^{2}}\\ R' & =\frac{2\alpha D-R\left(1+\alpha^{2}\right)}{1-\alpha^{2}} \end{align} \right|\begin{align}D^{2}-D^{\prime2} & =R^{2}-R^{\prime2}\\ D-D' & =\alpha(R-R')\\ D+D' & =\frac{1}{\alpha}(R+R') \end{align} $$

Stephanos (1883) - Lie sphere transformation
Cyparissos Stephanos (1883) showed that Hamilton's biquaternion a0+a1ι1+a2ι2+a3ι3 can be interpreted as an oriented sphere in terms of Lie's sphere geometry (1871), having the vector a1ι1+a2ι2+a3ι3 as its center and the scalar $$a_{0}\sqrt{-1}$$ as its radius. Its norm $$a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+a_{4}^{2}$$ is thus equal to the power of a point of the corresponding sphere. In particular, the norm of two quaternions N(Q1-Q2) (the corresponding spheres are in contact with N(Q1-Q2)=0) is equal to the tangential distance between two spheres. The general contact transformation between two spheres then can be given by a homography using 4 arbitrary quaternions A,B,C,D and two variable quaternions X,Y:


 * $$XAY+XB+CY+D=0$$ (or $$X=-\frac{CY+D}{AY+B}$$).

Stephanos pointed out that the special case A=0 denotes transformations of oriented planes (see Laguerre (1882)).

Cox (1883-91) – Orthogonal circles
In 1883, Homersham Cox formulated transformations between coordinates and systems of orthogonal circles, which he identified with the transformations of homogeneous coordinates in imaginary (=hyperbolic) geometry:


 * $$\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}$$

In 1891, he again discussed coordinate transformations between orthogonal circles, which he identified with the transformations of coordinates in non-Euclidean (hyperbolic) 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{ }$$

Scheffers (1899) - Laguerre transformations
Georg Scheffers (1899) synthetically determined all finite contact transformations preserving circles in the plane, consisting of dilatations, inversions, and the following one preserving circles and lines (compare with Laguerre inversion by Laguerre (1882) and Darboux (1887)):


 * $$\begin{matrix}\sigma^{\prime2}-\rho^{\prime2}=\sigma^{2}-\rho^{2}\\ \hline \rho'=\frac{\rho}{\cos\omega}+\sigma\tan\omega,\quad\sigma'=\rho\tan\omega+\frac{\sigma}{\cos\omega} \end{matrix}$$

Smith (1900) - Laguerre inversion
Percey F. Smith (1900) followed Laguerre (1882) and Darboux (1887) and defined the Laguerre inversion as follows:


 * $$\begin{matrix}p^{\prime2}-p^{2}=R^{\prime2}-R^{2}\\

\hline \kappa=\frac{R'-R}{p'-p}\\ p'=\frac{\kappa^{2}+1}{\kappa^{2}-1}p-\frac{2\kappa}{\kappa^{2}-1}R,\quad R'=\frac{2\kappa}{\kappa^{2}-1}p-\frac{\kappa^{2}+1}{\kappa^{2}-1}R \end{matrix}$$

He added that Bonnet's (1856) transformation is a special case with $$\kappa^{2}=-1$$, and he also gave credit to Lie (1871) for defining the corresponding "group of the geometry of reciprocal directions".

Bateman and Cunningham (1909–1910) - Spherical wave transformations
In line with Lie's (1871) research on the relation between sphere transformations with an imaginary radius coordinate and 4D conformal transformations, it was pointed out by Harry Bateman and Ebenezer Cunningham (1909–1910), that by setting u=ict as the imaginary fourth coordinates one can produce spacetime conformal transformations. Not only the quadratic form $$\lambda\left(dx^{2}+dy^{2}+dz^{2}+du^{2}\right)$$, but also Maxwells equations are covariant with respect to these transformations, irrespective of the choice of λ. These variants of conformal or Lie sphere transformations were called spherical wave transformations by Bateman. However, this covariance is restricted to certain areas such as electrodynamics, whereas the totality of natural laws in inertial frames is covariant under the Lorentz group. In particular, by setting λ=1 the Lorentz group SO(1,3) can be seen as a 10-parameter subgroup of the 15-parameter spacetime conformal group Con(1,3).

Bateman (1910/12) also alluded to the identity between the Laguerre inversion and the Lorentz transformations. In general, the isomorphism between the Laguerre group and the Lorentz group was pointed out by Élie Cartan (1912, 1915/55), Henri Poincaré (1912/21) and others.