Finding an angle of a triangle in the upper half plane model given three points

Question

I've been given three points in the upper half plane $(i, 3i, 1 + 2i$), and one of the homework questions that asked is to find the angles of the given triangle. A previous problem asks to find the length of the sides of the triangle, and then I'm also asked to find the measure of the angles of the triangle. Would they relate? if they do not, how would I find the angles?

Answer 1

Recall that the geodesics in the upperhalf plane model of hyperbolic geometry, $H$, are semicircles centered on the real line and vertical straight lines with constant real value. Try to find the circular/vertical arcs/line segments which connect your three points.

Another point you might like to recall is that the upperhalf plane model of hyperbolic geometry actually preserves angles from the Euclidean geometry of the complex plane $\mathbb{C}$. This means that if you can find the angles of the triangle at the point of intersection in the euclidean case, with the same curved edges *, then the angles will be the same in the hyperbolic case, and so you're done. Note that in hyperbolic geometry, there exist triangles with zero angles (although any such vertices with zero angle lie on the boundary of $H$, so don't apply to this problem).

My biggest tip I can give you for solving this problem is to draw pictures!

* if a curve intersects another curve, their angle of intersection is the difference between the angles of straight lines tangent to the curves at their point of intersection