Input Data

P := RandomGoodPoly(30):
N := 300:
a := 0.5:

EllipseProperties(P, alpha = a, scaling = "eval"):
AnimateConvergence(P, N, alpha = a, scaling = "eval");

Output Data

Under eigenvalue rescaling, the limiting ellipse will have normal form equation:
0.29 x^2 + 0.31 xy + 0.20 y^2 = 1
which has:
semi-major axis length = 3.4158
semi-minor axis length = 1.5609
eccentricity = 0.88948
with the semi-major axis tilted at an angle of 52.96 degrees counter-clockwise from the x-axis.