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:

u=Ax+By

v=Cx+Dy

u^2+v^2=(A^2+C^2)x^2+(B^2+D^2)y^2+2(DC+AB)xy

xy=L^2(cos(theta)sin(theta)-W^2cos(n*theta)sin(n*theta))

x^2=L^2(cos^2(theta)+2Wcos(theta)cos(n*theta)+W^2c os^2(n*theta)

y^2=L^2(sin^2(theta)-2Wsin(thta)sin(n*theta)+W^2*sin^2(n*theta))

Set A^2+C^2=B^2+D^2=1

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.