How Can I Map Cassini's Oval to the Unit Disk?

[T-O7]

Okay, so I'm having problems figuring out how exactly i can map the interior of Cassini's oval ($$|z^2-a^2|<r^2 , 0<a<r$$) onto the unit disk ($$|w|<1$$), so that the axis of symmetry are preserved. Anyone know how to do this?
(In this case, the cassini oval is a peanut shaped domain, i think)

 

[ReyChiquito]

Tuff question... Interesting though. Ill do some research and see what i can come up with. Meanwhile I've found some Cassinis ovals http://astronomy.swin.edu.au/~pbourke/surfaces/egg/

 

[M98Ranger]

To map Cassini's oval onto the unit disk, we can use a Mobius transformation. First, let's define the oval as z = x + iy, where x and y are real numbers. We can then rewrite the equation as (x^2 - a^2)^2 + y^2 = r^4.

Let's now consider the mapping w = (z^2 - a^2) / (z^2 + a^2). This is a Mobius transformation that maps the interior of Cassini's oval onto the unit disk. To see this, let's plug in our values for z into the equation for w:

w = ((x + iy)^2 - a^2) / ((x + iy)^2 + a^2)
= ((x^2 - y^2) + 2ixy - a^2) / (x^2 + y^2 + a^2)
= [(x^2 - a^2)^2 + y^2] / [(x^2 + a^2)^2 + y^2]
= (x^2 - a^2)^2 / [(x^2 + a^2)^2 + y^2] + y^2 / [(x^2 + a^2)^2 + y^2]
= (x^2 - a^2)^2 / [(x^2 - a^2)^2 + (x^2 + a^2)^2] + y^2 / [(x^2 - a^2)^2 + (x^2 + a^2)^2]
= (x^2 - a^2)^2 / [(x^2 - a^2)^2 + (x^2 + a^2)^2] + y^2 / [(x^2 - a^2)^2 + (x^2 + a^2)^2]
= (x^2 - a^2)^2 / [(x^2 - a^2)^2 + (x^2 + a^2)^2] + (x^2 + a^2)^2 / [(x^2 - a^2)^2 + (x^2 + a^2)^2]
= (x^2 - a^2)^2 / [(x^2 - a^2)^2 + (x^2 + a^2)^2] + (x^2