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Hyperbolic Geometry: Parameterization of Curves for Hyperbolic Distance

  1. Oct 11, 2012 #1
    1. The problem statement, all variables and given/known data

    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 dH(P,Q).

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

    2. Relevant equations

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

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

    3. 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!

  2. jcsd
  3. Oct 11, 2012 #2
    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?
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