Any point on x^2/a^2 + y^2/b^2 = 1 is P(theta) = (acos(theta), bsin(theta)), where theta is the eccentric angle. theta is NOT the angle OP makes with x-axis (unless a = b). It is the angle that the corresponding point on the auxiliary circle makes. The auxiliary circle is x^2 + y^2 = a^2; for point P on ellipse, the point Q on the auxiliary circle directly above/below P has angle theta. The eccentric angle ranges from 0 to 2*pi.