Drexel dragonThe Math ForumDonate to the Math Forum



Search All of the Math Forum:

Views expressed in these public forums are not endorsed by Drexel University or The Math Forum.


Math Forum » Discussions » Software » comp.soft-sys.math.mathematica

Topic: Conformal Mapping
Replies: 3   Last Post: Nov 11, 2012 1:26 AM

Advanced Search

Back to Topic List Back to Topic List Jump to Tree View Jump to Tree View   Messages: [ Previous | Next ]
Andrzej Kozlowski

Posts: 226
Registered: 1/29/05
Re: Conformal Mapping
Posted: Nov 11, 2012 1:26 AM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply


On 10 Nov 2012, at 08:07, MaxJ <maxjasper@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





Point your RSS reader here for a feed of the latest messages in this topic.

[Privacy Policy] [Terms of Use]

© Drexel University 1994-2014. All Rights Reserved.
The Math Forum is a research and educational enterprise of the Drexel University School of Education.