Equation for circle in the complex plane...

[cosine]

Hey, it help me alot if you could give me an equation solved for a complex [math]z[/math] in the complex plane union with infinity (sorry I don't know the latex for that). I'm looking for an equation of the form z = an expression made of the 3 complex numbers that determine it. I've been trying to find something like this for a few days, any help is appreciated.

 

Edit: By the way this is in Mobius geometry. The name aludes me of the plane, but basically the plane is the union of the complex plane and the point at infinity. Again, thanks for any help.

[Dave]

I'm fairly sure you're talking about projective 2 or 3-space, but I can't remember which. I do remember that if you squash the boundary of a Mobius strip to a point, you get P². Can't remember the defining equation though.

[cosine]

Yeah my professor gave me a parametrication based on the theorem that the the transformation of z is real iff it is on the cline (also called a m-circle, or just a circle in Mobius geometry) of z_1, z_2, and z_3.

[cosine]

Hm my professor explained to me how to make a good parametrication, since we proved that The transformation on the cline (circle in mobius geometry) is real for a given point z if it is on the cline determined by the three points. So setting the transformation equal to a real number will provide an parametrication.