Conformal Mapping: Transform a Circle to a Rectangle

[Carol_m]

Hello!

Please I need some help with this:

Is it possible to transform a circle into a rectangle? If so what would be the expressions of x' and y' in terms of x and y.

Thank you in advance!

 

[henry_m]

The answer is that it is possible (if you mean transforming the interiors). Indeed, any simply connected ("no holes") nonempty proper subset of the plane can be mapped conformally to any other. This is the Riemann mapping theorem.

Though the transformation exists, finding it explicitly is another matter entirely. According to Mathworld it can be explicitly written down using elliptic functions but I don't know how much use that is.

 

[Carol_m]

Dear henry_m,

Thank you very much for your reply. Could you please give me a reference to this so I can find the explicit formula?

Thanks again!

 

[coelho]

Have you tried using the transformation from polar coordinates to rectangular coordinates?

 

[jasonRF]

The hardest part of what you are looking for is the Schwarz–Christoffel transformation. It maps the upper half plane to a polygon. The mapping is in the form of an integral; this integral is the source of the elliptic functions henry_m mentioned.

So one approach is to map a circle to the upper half plane using a Mobius transformation, then use a Schwarz-Christoffel to map the upper half plane to rectangle. Both of these have parameters that you will need to determine for your particular problem.

Good luck!

jason

 

[Carol_m]

Basically my question is:

I have a circle an every point in the circle as (x,y). I need a transformation that changes every point in the circle to a point in a rectangle (x',y') so I could write x' in terms of x and y' in terms of y.

If you look in the attachement you can see what I mean :)

ps. I am not that concern about the base of the rectangle it could be circular so that y=y' but I am stuck with the x'

Thank you!

 

[jasonRF]

Basically my question is:

I have a circle an every point in the circle as (x,y). I need a transformation that changes every point in the circle to a point in a rectangle (x',y') so I could write x' in terms of x and y' in terms of y.

If you look in the attachement you can see what I mean :)

ps. I am not that concern about the base of the rectangle it could be circular so that y=y' but I am stuck with the x'

Thank you!

If your mapping needs to be conformal, as indicated by your post title, then my previous post should give you the words you need to search for either in Google or a complex analysis book; this should point you in the right direction so that you can figure out your mapping. By the way, the Mobius transformation is the same as the "linear fractional" transformation, which may help in your search.

 

[Carol_m]

Hi JasonRF,

It certainly needs to be conformal... Thank you for your great answer I will look into this direction, and sorry I posted my reply with the attached figure before reading yours.