The parametric representation P(theta) = (acos(theta), bsin(theta)) is the foundation for most ellipse calculations. The eccentric angle theta is NOT the angle OP makes with the x-axis (a common misconception). It is the angle made by the corresponding point Q = (acos(theta), asin(theta)) on the auxiliary circle $x^2$ + $y^2$ = $a^2$. Point P on the ellipse and Q on the auxiliary circle share the same x-coordinate; PQ is vertical. The y-coordinates are related by $y_P$/$y_Q$ = $\frac{b}{a}$. This geometric interpretation explains why the ellipse is a "compressed" circle — every y-coordinate of the circle is scaled by b/a. The eccentric angle ranges from 0 to 2pi, with theta = 0 and pi at vertices, theta = pi/2 and 3pi/2 at co-vertices.