Computable Distance in a Projective Space?

Question

So this is absolutely not my area of expertise. Still, I wondered if someone could point me in the direction of a computational distance measure between points in a projective space? (ie. an algorithm I could implement to find distance between two points in projective space).

I am trying to find a distance measure for objects which should be scale invariant and can be embedded as points in $\mathbb{R}^n$, naturally, for scale invariance I thought of 1d subspaces and projective spaces. I know the Grassmanian over a real vector space and thus real projective spaces are smooth manifolds and not necessarily endowed with a metric, so this may bear no fruit. But I have seen some mention of metrics on Grassmannians while trying to understand the literature myself.

Any links to papers (preferable readable by a non-specialist) are much appreciated.

Answer 1

Well, the representation of points in a projective space is not unique. But you can always find a canonical representative of a point, namely $(0,\ldots,0,1,p_1,\ldots,p_k)$, where $p_1,\ldots,p_k$ are arbitrary. From here you could use the Euclidean distance.

Answer 2

A detour by Affine Geometry, using barycentric coordinates $(p,q,r)$ (cousin of projective coordinates : see p. 13 here) with respect to a fixed affine basis characterized by triangle $ABC$.

One can imbed metric properties into Affine Geometry with barycentric coordinates. Let us take the example of the equation of the circumcircle (which is a metric concept, not invariant by a general affine transform, as are usually the properties in this geometry) with classical notations $a=BC,b=CA,c=AB$ is :

$$a^2qr+b^2rp+c^2pq=0$$

In fact, the presence of lengths $a,b,c$ provide the "metrical information"...

Here is the general formula (2) which can be placed at the basis of this metric.

Let us define the

$$\text{Barycentrical coordinates of} \ \vec{MM'}=M'-M : (p',q',r')-(p,q,r)\tag{1}$$

(where $(p,q,r)$ and $(p',q',r')$ are the barycentrical coordinates of $M,M'$ resp.)

The fundamental formula is the way (ordinary Euclidean) dot product between vectors can be expressed using their barycentrical coordinates:

$$\vec{u} · \vec{v} = 2 {\frak{A}} (\alpha p_1p_2 + \beta q_1q_2 + \gamma r_1r_2)\tag{2}$$

where $\frak{A}$ is the area of triangle $ABC$ with $\alpha=\operatorname{cotan}(\hat{A}), \ \beta=\operatorname{cotan}(\hat{B}), \ \gamma=\operatorname{cotan}(\hat{C}).$

Here is a reference in croatian which is rather understandable (using Google translate...), where formula (2) is established.

I am going looking for an equivalent reference in English.

See as well here.