In projective geometry the dual of the cross ratio dual is an angle measurement?

Question

I am trying to get my head around angles in projective geometry.

I understand (more or less) the cross ratio and that it can be seen as an distance measurement. (for example in the Beltrami Cayley Klein model of hyperbolic geometry)

But then there is its dual where the cross ratio is some kind of angle measurement of lines?

I just draw a blank here

which lines are mend here? can anybody give some light?

Answer 1

If you're willing to look at two points of a (real) projective plane and draw a line through them and consider the cross-ratio as a "kind of distance," then the dual plane (another projective plane) would have lines corresponding to those points, and a point corresponding to that line, and you can think of the cross-ratio as a "kind of angle."

Alternatively, given any set of four distinct lines in a projective plane, you can cut them with a fifth line that isn't concurrent with the four, and the points of intersection will be four points you can compute the cross-ratio for. It turns out that no matter what fifth line you pick, you always get the same answer for the cross-ratio. This indicates that this cross ratio is an invariant of the original set of four lines.

I think the next exploration might be (I think GeoGebra could handle this) would be to make a software construction like the last paragraph and play with it. Plot two movable points and two collinear (maybe immovable) points. Have the computer keep track of the cross ratio. Observe what happens to the ratio as you move the mobile points toward each other. Pick a point off the line and send a line through each mobile point from that new point. What happens to the angle between them as you move the points?

Answer 2

You can define angles in terms of the cross ratio (of lines) and the circular points at infinity.

If you have two lines that intersect at the point P, you can draw two more lines from P to the circular points at infinity. Taking the natural logarithm of this will give you the Euclidean angle between the two original lines times i (up to multiples of 2πi depending on branch).

The circular points at infinity add just enough structure to projective geometry to let you tell circles apart from other conics, this turns out to give enough structure to define angles.

The same construction can be used to define hyperbolic angles, a.k.a. rapidity. In that case, instead of drawing lines to the circular points at infinity, you pick a pair of real points that define asymptotes for a unit hyperboloid in Minkowski space. This gives the relation between the distances using cross ratios in the Beltrami-Klein model and hyperbolic angles in the hyperboloid model.