First, let the axis a be the line at infinity.
where a^{-1} is defined by the equation aa^{-1}=1.
All the groups described above fix the line at infinity. Unfortunately, sometimes you need to know central-axial collineations moving the line at infinity. We will compute one such group, using [0] as axis and (0,0) as center.
Let a be an element of K, distinct from 0. Define a' by a+a'=0. For each t in K, there exists exactly one element
The map is a bijection with Moreover, if x is an element of K distinct from a', there exists exactly one
and the map
is a bijection. Using these notations, we define maps defined on the sets of points and lines of P(K,T), respectively.
This map will be called Finally, define
We get