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Math Help - moebius, poincare model/complex numbers

  1. #1
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    moebius, poincare model/complex numbers

    imagine a semi circle (x-2)+y^2=2 so centre (2,0) and radius 2 for the upper half plane,
    call the semi circle p.

    I need to give the formula for the reflection across p (hyperbolic reflection)  R_p

    i then have points S=4+3i for example where i need to find  R_p(S) i can make stabby attempts but half the other questions depend on getting the correct formula for  R_p .

    thanks,,v desperate.
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  2. #2
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    Quote Originally Posted by skystar View Post
    imagine a semi circle (x-2)+y^2=2 so centre (2,0) and radius 2 for the upper half plane,
    call the semi circle p.

    I need to give the formula for the reflection across p (hyperbolic reflection)  R_p

    i then have points S=4+3i for example where i need to find  R_p(S) i can make stabby attempts but half the other questions depend on getting the correct formula for  R_p .

    thanks,,v desperate.
    The semi-circle would be (x-2)^2 + y^2 = 4 if it had a radius of 2. What is the definition of the hyperbolic reflection R_p?
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  3. #3
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    A hyperbolic reflection in the Poincare Half-Plane is an inversion. If we choose to think of the upper half plane as the complex plane then inversions are actually Mobius transformations.
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  4. #4
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    sorry excuse my lame example
    a hyperbolic function R as a function of z is  R_m(z)
    and is given by  \frac{k\*\bar{z} + r^2 - k^2}{\bar{z} - k} as a general result.k is constant, r is radius
    if i sub in 2 for both k and r then i can reduce it easily but this is not  R_m
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