Coordinates
If the center of coordinates is taken as the center of the ellipse, the coordinates of a point P(x, y) on the ellipse satisfy the equation
where a and b are the semi-major and semi-minor axes determining the length (2a) and width (2b) of the ellipse.
The eccentric anomaly E in terms of these coordinates is given by:
The above equations can be established by drawing the auxiliary circle of radius a enclosing the elliptical path and the minor auxiliary circle of radius b inscribed within the path. The first equation is established by the definition of E. By extending a vertical line through point P to the auxiliary circle, a right triangle is formed with base that is the x-coordinate of P, and hypotenuse a, establishing the first equation. The second equation is established using the minor auxiliary circle. A horizontal line through P intersects this minor auxiliary circle of radius b, establishing another right triangle with altitude y and hypotenuse b. Labeling the adjacent angle E′:
It is next established that E′ = E. From the equation for the ellipse and the Pythagorean trigonometric identity:
establishing E′ = E.
Read more about this topic: Eccentric Anomaly