Spherical wave transformation
Spherical wave transformations leave the form of spherical waves as well as the laws of optics and electrodynamics invariant in all inertial frames. They were defined between 1908 and 1909 by Harry Bateman and Ebenezer Cunningham, with Bateman giving the transformation its name. They correspond to the conformal group of "transformations by reciprocal radii" in the framework of Lie sphere geometry, which were already known in the 19th century. Time is used as fourth dimension as in Minkowski space, so spherical wave transformations are similar to the Lorentz transformation of special relativity. It turns out that the conformal group includes the Lorentz group and the Poincaré group as subgroups, but only the latter represent symmetries of all laws of nature including mechanics, whereas the conformal group is only related to certain areas such as electrodynamics.
A special case of Lie sphere geometry is the "transformation by reciprocal directions" or Laguerre inversion, being a generator of the group of Laguerre transformations (Laguerre group). It transforms not only spheres into spheres but also planes into planes. If time is used as fourth dimension, a close analogy to the Lorentz transformation and the Lorentz group was pointed out by several authors such as Bateman, Cartan or Poincaré.
Inversions preserving angles between circles were first discussed by Durrande (1820), with Quetelet (1827) and Plücker (1828) writing down the corresponding transformation formula, being the radius of inversion:
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