Hilbert metric for the Euclidean plane, a reference? 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

Thanks!
[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.]
 A: Clemens, A Scrapbook of Complex Curve Theory, p. 17 gives the formula for the sphere and hyperbolic space, with a parameter $K$ which you can set to $k=0$ (bottom of page 19).
A: Some formulas for Cayley-Klein metrics in terms of cross-ratios are given in sections 6 and 7 of my paper http://arxiv.org/abs/0708.0929 They are based on the Klein's memoir "On the so-called Non-Euclidean geometry". This memoir is reviewed in the historical context in http://arxiv.org/abs/1406.7309 (On Klein's So-called Non-Euclidean geometry, by Norbert A'Campo and  Athanase Papadopoulos). As the authors of this review wrote, Klein's approach was received by the mathematical community in diverse manners. For example, Darboux wrote: 

Mr. Felix Klein is the one who removed these very serious objections
  [concerning non-Euclidean geometry] by showing in a beautiful memoir
  that a geometry invented by the famous Cayley and in which a conic
  called the absolute provides the elements of all measures and enables,
  in particular, to deﬁne the distance between two points, gives the most
  perfect and adequate representation of non-Euclidean geometry.

while Genocchi was not so optimistic:

From the geometric point of view, the spirit may be shocked by certain
  deﬁnitions adopted by Mr. Klein: the notions of distance and angle,
  which are so simple, are replaced by complicated deﬁnitions... The statements are extravagant.

A: See 
1) for technical aspects : around page 96 of "Geometry" (V. V. Prasolov and V. M. Tikhomirov) TMM, Vol. 200, American Mathematical Society, 2001. Very rich book.
2) for a general perspective: around page 94 of "The Princeton Companion to Mathematics" (T. Gowers, editor), 2008.
A: For 3) you should look at https://people.kth.se/~akarl/hilbertfk.pdf . They also prove that a Hilbert Geometry is never a Hilbert Space.
