[M 1] They correspond to the conformal group of "transformations by reciprocal radii" in relation to the framework of Lie sphere geometry, which were already known in the 19th century.
[M 2][8][M 3][9][10][11][12][13] Inversions preserving angles between circles were first discussed by Durrande (1820), with Quetelet (1827) and Plücker (1828) writing down the corresponding transformation formula,
gives Thomson's transformation by reciprocal radii (inversions):[M 5] Subsequently, Liouville's theorem was extended to
[21] This method was occasionally and implicitly employed by Lie (1871)[M 6] himself and explicitly introduced by Laguerre (1880).
This method was known for some time in circle geometry (though without using the concept of orientation) and can be further differentiated depending on whether the additional coordinate is treated as imaginary or real:
was used by Cousinery (1826), Druckenmüller (1842), and in the "cyclography" of Fiedler (1882), therefore the latter method was also called "cyclographic projection" – see E. Müller (1910) for a summary.
Bateman (1909) also noticed the equivalence to the previously mentioned Lie sphere transformations in R3, because the radius
It consists of elementary transformations that represent a generalized inversion into a four-dimensional hypersphere:[30] which become real spherical wave transformations in terms of Lie sphere geometry if the real radius
[M 1] Felix Klein (1921) pointed out the similarity of these relations to Lie's and his own researches of 1871, adding that the conformal group doesn't have the same meaning as the Lorentz group, because the former applies to electrodynamics whereas the latter is a symmetry of all laws of nature including mechanics.
For instance, Fricke and Klein (1897) started by defining an "absolute" Cayley metric in terms of a one-part curvilinear surface of second degree, which can be represented by a sphere whose interior represents hyperbolic space with the equation[34] where
, the above relations assume the form in terms of the unit sphere in R3:[36] which is identical to the stereographic projection of the
The relation between the Lorentz group and the Cayley metric in hyperbolic space was also pointed out by Klein (1910)[M 23] as well as Pauli (1921).
[M 10][5][6] According to Darboux[M 24] and Bateman,[M 2] similar relations were discussed before by Albert Ribaucour (1870)[M 25] and by Lie himself (1871).
[M 27] Lines, circles, planes, or spheres with radii of certain orientation are called by Laguerre half-lines, half-circles (cycles), half-planes, half-spheres, etc.
coordinates as well:[M 30] with consequently he obtained the relation As mentioned above, oriented spheres in R3 can be represented by points of four-dimensional space R4 using minimal (isotropy) projection, which became particularly important in Laguerre's geometry.
[5] For instance, E. Müller (1898) based his discussion of oriented spheres on the fact that they can be mapped upon the points of a plane manifold of four dimensions (which he likened to Fiedler's "cyclography" from 1882).
He systematically compared the transformations by reciprocal radii (calling it "inversion at a sphere") with the transformations by reciprocal directions (calling it "inversion at a plane sphere complex").
[M 31] Following Müller's paper, Smith (1900) discussed Laguerre's transformation and the related "group of the geometry of reciprocal directions".
[M 32] Smith obtained the same transformation as Laguerre and Darboux in different notation, calling it "inversion into a spherical complex":[M 33] with the relations In 1905 both Poincaré and Einstein pointed out that the Lorentz transformation of special relativity (setting
[42] Poincaré showed that the Lorentz transformation can be seen as a rotation in four-dimensional space with time as fourth coordinate, with Minkowski deepening this insight much further (see History of special relativity).
[M 34] Bateman (1910) also sketched geometric representations of relativistic light spheres using such spherical systems.
[M 35][43] However, Kubota (1925) responded to Bateman by arguing that the Laguerre inversion is involutory whereas the Lorentz transformation is not.
He concluded that in order to make them equivalent, the Laguerre inversion has to be combined with a reversal of direction of the cycles.
: According to Müller, the Lorentz transformation can be seen as the product of an even number of such Laguerre inversions that change the sign.
Timerding (1911)[M 38] used Laguerre's concept of oriented spheres in order to represent and derive the Lorentz transformation.
as the distance between its center and the central plane, he obtained the relations to a corresponding sphere resulting in the transformation By setting
After the first comparison of the Lorentz transformation and the Laguerre inversion by Bateman (1910) as mentioned above, the equivalence of both groups was pointed out by Cartan in 1912[M 45] and 1914,[M 46] and he expanded upon it in 1915 (published 1955) in the French version of Klein's encyclopedia.
These two groups are isomorphic, so that mathematically these two theories, one physical, the other one geometric, show no essential difference.
L-transforms are more easily understood if we use the so-called cyclographic model of Laguerre geometry.
In the cyclographic model, an L-transform is seen as a special affine map (Lorentz transformation),...