Hyperbolic Geometry: Parameterization of Curves for Hyperbolic Distance

[Phoenixtears]

Homework Statement

Consider the points P = (1/2, √3/2) and Q = (1,1). They lie on the half circle of radius one centered at (1,0).

a) Use the deifnition and properites of the hyperbolic distance (and length) to compute d_H(P,Q).

b) Compute the coordinates of the images of Pa nd Q through the standard inversion and use that to evaluate again d_H(P,Q).

Homework Equations

Our professor gave us a few trig identites to use, but other than that we need the equation for Hyperbolic distance:

∫ $\sqrt{}x'(t)²+y'(t)²$ $\frac{}{}y(t)$
From a to b where a < t < b

The Attempt at a Solution

Ignoring the question almost entirely, the part that I can't figure out is how to parameterize this correctly. My professor gave us a solution that gave one option for a parameterization, but I'm not sure how he got there.

His solution:
x(t) = cost +1
y(t)= sint

I believe that more than one parameterization would work, but how do you come up with the one that makes things simplest?

Thank you so much!

Phoenix

 

[Phoenixtears]

Okay, I feel a little silly for not understanding his parameterization, but I do understand it now. My question then moves to if there is a quick trick to parameterizing potentially more difficult curves in the hyperbolic plane?