Two points (x1, y1, x2, y2) or 4 total parameters (x, y, rotation, scale) assuming you're using the exact shape provided, but from the interactive egg you can free the location/angles that they used for construction so you can wind up with less egg-like 4-point eggs.
In the interactive link provided, you control the positions of 4 circle centers, each restricted along one of the axes. So you're giving 4 parameters (which also gives scale), and after including x-y offsets and rotation, it's 7 total numbers to make an arbitrary '4-point' egg anywhere in the plane
