Euclidean geometry and complex plane

  • #1
Can someone please describe to me how Euclidean Geometry is connected to the complex plane? Angles preservations, distance, Mobius Transformations, isometries, anything would be nice.

Also, how can hyperbolic geometry be described with complex numbers?
 

Answers and Replies

  • #2
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Perhaps you should read a good complex analysis book. Try to read "visual complex analysis" by Needham. A very easy book, but with a lot of intuition and connections to geometry.
 
  • #3
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Multiplying one complex number by another scales it and rotates it. Adding complex numbers give you translation. So, complex numbers can give you any dilation. So, if two figures are the similar, you can get from one to the other by multiplying and adding complex numbers to it.

There's a model for the hyperbolic plane which is the unit disc in the complex plane. Mobius transformations that map the unit disc to itself act as the rigid motions of the hyperbolic plane.
 
  • #4
mathwonk
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representing the hyperbolic plane by complex numbers z with real part > 0, one obtains that maps defined by [az+b]/[cz+d/ with a,b,c,d real and ad-bc >0, are exactly the hyperbolic isometries. I think. oops that's what the previous post said.
 
  • #5
1,765
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representing the hyperbolic plane by complex numbers z with real part > 0, one obtains that maps defined by [az+b]/[cz+d/ with a,b,c,d real and ad-bc >0, are exactly the hyperbolic isometries. I think. oops that's what the previous post said.
I referred to the complex numbers of modulus less than 1. That's the disk model.

Mathwonk is referring to the upper-half plane model, which is an equivalent way to think about it.
 

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