I need to calculate the area of a conformal mapping closed shape on the complex plane that gives the perimeter. I have the following parameters:
α0 = 0.2; α1 = 1.0; α2 = -α1 - α0;
u0 = 0.5; u1 = 0.2; u2 = -0.4;β = 0.0;
ax = 1.0; τ = 1.0 I; ay = τ*ax;
And the following conformal map:
Z[u_] :=
α0*WeierstrassZeta[u - u0, WeierstrassInvariants[{ax, ay}]] +
α1*WeierstrassZeta[u - u1, WeierstrassInvariants[{ax, ay}]] +
α2*WeierstrassZeta[u - u2, WeierstrassInvariants[{ax, ay}]] + β;
with this I draw the following shape:
B1 =
ParametricPlot[{Re[Z[u*I + ax]], Im[Z[u*I + ax]]}, {u, -Im[ay], Im[ay]},
PlotRange -> All]
So I have the perimeter of the enclosed shape, which is the conformal map. How can I extract the area of the shape on the complex plane, from the given perimeter numerically?