Origin of Laguerre geometry?

Question

Laguerre geometry is described as either the geometry of oriented lines and circles in the Euclidean plane, equipped with a certain unusual symmetry group (see https://en.wikipedia.org/wiki/Laguerre_transformations) or as the geometry of vertical parabolas and non-vertical lines (see https://en.wikipedia.org/wiki/Laguerre_plane).

The group of symmetries of the Laguerre plane is the same as the group $PGL(2,\mathbb D)$ where $\mathbb D$ is the dual numbers. It makes me wonder: Did Laguerre start with the symmetry group $PGL(2,\mathbb D)$ and then work out the geometry, or did he formulate a geometry somehow and people later discovered that its symmetry group was $PGL(2,\mathbb D)$?

(Don't know what tags to use.)

Answer 1

The formulation of Laguerre geometry in terms of dual numbers is a decidedly 'synthetic' one, meant to exhibit how this set of transformations can be regarded as a different 'real form' of the well-known linear fractional transformations of the complex plane.

If you want to know more about the history of how Laguerre geometry (itself a special case of Lie sphere geometry) was developed (long before the introduction of 'dual numbers'), you might want to look at a few other articles, including the Wikipedia article on spherical wave transformations.