Editing Elliptic orbit (section)

====Using XY Coordinates====
This can be done in cartesian coordinates using the following procedure:

The general equation of an ellipse under the assumptions above is:

:<math> \sqrt{ \left(f_x - x\right)^2 + \left(f_y - y\right)^2} + \sqrt{ x^2  + y^2 } = 2a  \qquad\mid  z=0</math>

Given:
:<math>r_x, r_y \quad</math> the initial position coordinates
:<math>v_x, v_y \quad</math> the initial velocity coordinates

and
:<math>\mu = Gm_1 \quad</math> the gravitational parameter

Then:
:<math>h = r_x v_y - r_y v_x \quad</math> specific angular momentum
		
:<math>r = \sqrt{r_x^2 + r_y^2} \quad</math> initial distance from F1 (at the origin)

:<math>a = \frac{\mu r}{2\mu - r \left(v_x^2 + v_y^2 \right)} \quad</math> the semi-major axis length


:<math>e_x = \frac{r_x}{r} - \frac{h v_y}{\mu} \quad</math> the [[Eccentricity vector]] coordinates
:<math>e_y = \frac{r_y}{r} + \frac{h v_x}{\mu} \quad</math>


Finally, the empty focus coordinates
:<math>f_x = - 2 a e_x \quad</math>
:<math>f_y = - 2 a e_y \quad</math>


Now the result values ''fx, fy'' and ''a'' can be applied to the general ellipse equation above.