So you want a transformation from (x,y) to (u,v) s.t u and v represent a circle, you might as well write it down as:
sowe get: x^2+y^2=L^2(1+W^2-2Wcos((n+1)*theta))
Now also DC+AB=1.
There are multiple choices to choose from, but watch also for the fact that W and theta are dependent on eachother, to find this dependence, you need to eqaute u^2+v^2=L^2.
Now ofcourse my approach is valid if W and theta are indeed dependent in the question, if not then you might not get a radius L from this transformation.