Input Data

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

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:
167.62 x^2 + -319.89 xy + 167.62 y^2 = 1
which has:
semi-major axis length = 0.3609
semi-minor axis length = 0.0553
eccentricity = 0.98821
with the semi-major axis tilted at an angle of 135.00 degrees counter-clockwise from the x-axis.