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Re: Conformal Mapping

• To: mathgroup at smc.vnet.net
• Subject: [mg128626] Re: Conformal Mapping
• From: Andrzej Kozlowski <akozlowski at gmail.com>
• Date: Sun, 11 Nov 2012 01:26:47 -0500 (EST)
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On 10 Nov 2012, at 08:07, MaxJ <maxjasper at shaw.ca> wrote:

> Thanks a lot to all you folks for your superb & enlightening comments 
and solutions.
>
> QUESTION:
>
> Based on your solutions, shall I conclude that the lens region can 
only be mapped into half of unit disk?
>
> Thanks.
>
> Max.
>

It depends on what sort of mapping you are asking for. The lens can be 
mapped conformally onto the unit disk, of course. In fact, by the 
Riemann mapping theorem any open simply connected region (except the 
entire complex plane)  can be mapped conformally onto the unit disk. 
However, such a mapping cannot be a Moebius transformation (which is the 
same as a "linear fractional transformation" and which is what you 
explicitly asked for in your question). A Moebius transformation mapps 
(generalized) circles to circles and hence it clearly cannot map a lens 
onto a disk. But of course you can map a lens into a unit disk be means 
of a Moebius transformation in lots of different ways. In fact, since a 
Moebius transformation is completely determined by choosing three 
distinct points in the source space and three distinct points as their 
images in the target space, the method I posted makes it possible to 
find all ways of mapping a lens into the unit disk by a Moebius 
transformation. I chose the Moebius map that sends the lens onto a half 
disk only because it seemed the most "attractive" of an infinitely many 
"into" Moebius mappings.
Of course none of these mappings is onto. The the biholomorphic mapping 
which I constructed in another post is a rational map of degree 2 - 
hence not a Moebius map.

Andrzej Kozlowski

• References:
  • Re: Conformal Mapping
    • From: MaxJ <maxjasper@shaw.ca>