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.]

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).

• Thank you, this reference contains a smart argument to see that Hilbert distance with a complex absolute is indeed the spherical metric! But I don't want to see Euclidean metric as a limit of spherical and/or hyperbolic metric, I want to define it directly from the imaginary absolute at infinity. Mar 16 '16 at 8:31

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.

• Thank you also for those reference, but I am looking for a precise formula for the Euclidean distance, not for the hyperbolic one. To precise my post, a formula similar to what I am looking at is Theorem 18.10 in Richter-Gebert "Perspectives in Projective Geometry" (but his formula is not explicitely a Hilbert-like one) Mar 16 '16 at 8:19

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.

• Thank you, but seemly we are not speaking about the same thing. May question is about Minkowski space, in the sense of Lorentzian geometry: $\mathbb{R}^n$ endowed with a non-degenerate symmetric bilinear form with exactly one negative direction. As Euclidean space, this space can be defined with the help of degenerated quadric on the projective space. Hence it is natural to ask if its "peudo distance" can be recovered with the help of the Hilbert distance. Mar 16 '16 at 7:57

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.

• Thank you also for these references. As you say in your interesting paper, the Hilbert distance on a parabolic line is zero. So Hilbert distance cannot be used directly to recover the Euclidean distance. But I am pretty sure to have seen once a formula for the Euclidean distance with: the logarithm+the cross ratio+cyclic points I and J, and, necessarily, other termes. Mar 16 '16 at 8:30
• Busemann and Kelly write in their book "Projective Geometry and Projective Metrics" (p.231): "There remains the question whether Euclidean distance and angle can be expressed directly as the logarithms of cross ratios instead as the limit of such ratios. This is not possible for distance, but the following heuristic argument shows how it can be done for angles". The heuristic argument leads to the Laguerre formula. Mar 17 '16 at 8:30
• Yes, Euclidean metric is not a Hilbert metric in the strict sense as the distance is then always zero. BUT I saw once a formula for Euclidean distance with: Logarith+cross ratio+ I and J plus other terms (a unit have to be fixed). I am looking for this reference that I can't find again. For a formula for Euclidean metric with I,J,and a unit fixed, see Richter-Gerbert "Perspctives on projective geometry" Theorem 18.10 Mar 17 '16 at 9:53