Projective characterization of isometries

Question

An homography on a projective plane is a similarity if and only if it commutes with the absolute involution on the line at infinity. There is a similar characterization for isometries?

Answer 1

I'm not used to the term “absolute involution”, but to me similarities are characterized by preserving (the absolute values of) all angles, and angles in a projective setup can be measured using Laguerre's formula, which essentially is some cross ratio involving the ideal circle points $[1:i:0]$ and $[1:-i:0]$. So your whole similarity geometry is essentially defined by two designated points. Your absolute involution presumably exchanges their roles, thus negating all angles.

For isometries, the situation is different. Knowing the line at infinity allows you to compare lengths on the same line. Knowing the designated ideal circle points allows comparing lengths anywhere in the plane, and essentially describing circles (as conics through these points). But that only establishes relative length measurements. To obtain absolute lengths, you need to designate some reference length as your unit of length. You can't do that using additional information on the line at infinity, as points there have no finite distance from one another. You can do this by fixing two finite points, but that is too restrictive: there are only very few (i.e. finitely many) isometries that fix a given pair of points. So I can't think of a formulation which would resemble the one you gave, due to this different nature.

The problem you describe is inherently Euclidean. For other geometries, say hyperbolic or elliptic, there exists an absolute measure of length. Knowing the fundamental conic defining the geometry, lengths can be computed, and any transformation fixing that conic will be an isometry of that geometry. Using the machinery used for such absolute lengths, and applying it to Euclidean geometry, the immediate result indicates that all Euclidean lengths are zero anyway. In this sense, any similarity would preserve “lengths”, but that is obviously not the usual interpretation of the term isometry.