# moebius, poincare model/complex numbers

• May 1st 2008, 10:20 AM
skystar
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) $\displaystyle R_p$

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

thanks,,v desperate.
• May 1st 2008, 10:24 AM
icemanfan
Quote:

Originally Posted by skystar
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) $\displaystyle R_p$

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

thanks,,v desperate.

The semi-circle would be $\displaystyle (x-2)^2 + y^2 = 4$ if it had a radius of 2. What is the definition of the hyperbolic reflection $\displaystyle R_p$?
• May 1st 2008, 11:47 AM
ThePerfectHacker
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.
• May 2nd 2008, 06:52 AM
skystar
sorry excuse my lame example
a hyperbolic function R as a function of z is $\displaystyle R_m(z)$
and is given by $\displaystyle \frac{k\*\bar{z} + r^2 - k^2}{\bar{z} - k}$ 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 $\displaystyle R_m$