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   Author  Topic: point of symmetry  (Read 7929 times)
trusure
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point of symmetry  
« on: Apr 20th, 2009, 5:19pm »
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while I was trying to find a Linear Fractional Transformation from the upper half plane to itself, with 0 mapped to 1 and i mapped to 2i, I used that the point of symmetric of i is -i mapped to -2i which is the point of symmetric for 2i,(using the symmetric principle), so now I have 3 pints mapped to 3 points, and  using the cross ratio I found f(z)=2.5 *z +1 !  which is not correct ??
 so, where is my mistake ??  
 
can anyone help me ?
 
Thank you
« Last Edit: Apr 20th, 2009, 5:20pm by trusure » IP Logged
Eigenray
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Re: point of symmetry  
« Reply #1 on: Apr 20th, 2009, 8:11pm »
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Probably not, if you don't show what you did.  Solving
 
(z, 0; i, -i) = (f(z), 1; 2i, -2i)
 
for f(z) should work.
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trusure
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Re: point of symmetry  
« Reply #2 on: Apr 20th, 2009, 8:56pm »
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does it matter the order of the points while using the cross ratio ?!!
« Last Edit: Apr 20th, 2009, 9:17pm by trusure » IP Logged
Eigenray
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Re: point of symmetry  
« Reply #3 on: Apr 20th, 2009, 9:26pm »
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No, as long as you put the points in the same order as their images.  E.g.,
 
(-i, 0; z, i) = (-2i, 1; f(z); 2i)
 
would also work, but not
 
(z, 0; i, -i) = (f(z), 2i; -2i, 1)
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