The Euclidean plane can be seen as an affine chart of the projective plane, together with a scalar product on the line at infinity, so that the absolute consists of two conjugate complex points at infinity, I and J, the cyclic (or circular) points at infinity.
The Laguerre formula gives the Euclidean angle between two lines in terms of a logarithm of a cross ratio involving I and J (=what I call a Hilbert metric, but that is probably sometimes called a projective metric or a Cayley-Klein metric).
I am sure to have seen a formula for the Euclidean distance between two points in terms of the Hilbert metric - up to fix a unit vector. I spent a lot of time trying to find it again without success...
Question 1: What are references containing an expression of the Euclidean metric on the plane written in terms of a logarithm of a cross-ratio?
There is a related formula in the book of Richter-Gebert "Perspectives on projective geometry", but this is not the one I am thinking of.
Question 2: Question 1 for any dimension
Question 3: Question 2 for the (Lorentzian) Minkowski space
[Edit: I am not asking for references about Hyperbolic or spherical metric as Hilbert metrics, nor for Euclidean geometry seen as a limit of those ones.]