Input Data

P := RandomGoodPoly(30):
N := 300:
a := 0.3:

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

Output Data

Under norm rescaling, the limiting ellipse will have normal form equation:
105.11 x^2 + 194.65 xy + 105.11 y^2 = 1
which has:
semi-major axis length = 0.3583
semi-minor axis length = 0.0703
eccentricity = 0.98057
with the semi-major axis tilted at an angle of 45.00 degrees counter-clockwise from the x-axis.