Laguerre geometry

Remarks

Two cycles $(c_1, r_1)$ and $(c_2, r_2)$ touch with matching orientations when $\abs{c_1 - c_2} = \abs{r_1 - r_2}$. In the cyclographic model, the cycle $(c, r)$ is the point $(c, r) \in \R^3$, and this condition says that the difference of the two points is a null vector of the form $x^2 + y^2 - z^2$. Example: for $t \in \R$, the map sending the cycle $(c, r)$ to $(c, r + t)$ and the spear $(N, h)$ to $(N, h - t)$ is a Laguerre transformation. Check: $c \cdot N - (h - t) = r + t$ holds exactly when $c \cdot N - h = r$, and both maps are bijections. It sends each point $(c, 0)$ to a circle of radius $\abs{t}$, so it is not a map of points. Non-example: inversion in the unit circle is not a Laguerre transformation in this sense. It sends the lines not through the origin to circles, so it does not map spears to spears. Euclidean isometries and dilations $x \mapsto \lambda x$, $\lambda > 0$, acting on centres and radii, are also Laguerre transformations.