Is there any way of explaining the Cayley–Klein metric to undergrads? How to explain the Cayley-Klein or sometimes called Beltrami–Klein metric concept to find the distance between two points in a hyperbolic space to an audience with no higher education than maybe a second-year undergrad? What are a, b, C and what are p, q on quadric Q.
${\displaystyle d(a,b)=C\log {\frac {\left|bp\right|\left|qa\right|}{\left|ap\right|\left|qb\right|}}}$
Furthermore, is there any exercise in finding distance using the above concept?
 A: Explaining what to do would be easy. Explaining why it's done that way is a lot harder.
In general $a$ and $b$ would be two points whose distance you want to compute. So $d(a,b)$ would be that distance. In the case of hyperbolic geometry, the points would be hyperbolic points in the Beltrami-Klein model, i.e. points inside the unit circle.
Connect these two points with a line. Intersect that line with the unit circle. You get two points of intersection. These are $p$ and $q$. Your choice which one is which will affect the sign of the result, so either ignore that sign or pick the points in a deliberate way.
$C$ is just some constant. In the case of hyperbolic geometry I seem to recall that one picks $C=\frac12$ to get a constant Gaussian curvature of $-1$, which is what most people use. Using a different positive real number will essentially just scale your metric, resulting in a different curvature.
The things between the vertical bars would be distances. So measure four distances. Compute two products and one fraction, which together form what's called a cross ratio. Take the natural logarithm of that. Multiply by that factor $C$ and your have the distance.
You can also measure angles. There is a duality in projective geometry that swaps the roles of lines and points. So take two lines $a$ and $b$. Intersect them (which is dual to connecting points). Construct the tangents from the point of intersection to the unit circle (dual to intersecting the line with the circle) and call these $p$ and $q$. The distances in the formula are a bit more tricky now, but if you intersect all four lines with some other line (not through the point of intersection of $a$ and $b$) then you can use distances of the intersection points on that line.
One big caveat here is that if the two original lines intersect within the circle, then your don't really see how a tangent through that point of intersection could work. The formulas for computing them suddenly involve square roots of negative numbers. So you need complex numbers. You get complex tangents, complex distances, and in the end a purely imaginary logarithm. Too turn that into an angle, one would use a purely imaginary constant such as $C=\frac1{2i}$. So the constant for distance measurement and the one for angle measurement is independent.
The whole thing works for different kinds of geometries. In the above I've used the unit circle as the fundamental quadric, which is the typical model of hyperbolic geometry. So that was my $Q$. But any other ellipse would have worked the same. With different fundamental quadric you get different geometries. But your question only asked about hyperbolic geometry, so maybe this is going too far.
If you want an exercise, pick two points in the unit circle and compute their distance. Some geometry software like e.g. Cinderella will be able to do hyperbolic measurements and confirm your computation.
