If you compute the curvature it starts flat, does a few bump and then flattens again (this curvature is not a very smooth function however). An n-ellipse is not

If you compute the curvature it starts flat, does a few bump and then flattens again (this curvature is not a very smooth function however). An n-ellipse is not quite the right curve. Apple used a bunch of Bezier curves. I think their design could be made a little more aesthetic if you used some smoothed piecewise bump (like a Gaussian nearly). You can numerically solve for the curve given the curvature and then approximate it.