# Relating Angles in Elliptical Orbit

I've been trying to model the orbit of a comet of arbitrary mass around our sun from having a known initial condition with $r_0, v_0$ with angle $\alpha$ at $t_0$ around our sun. I've been able to derive the radius in the form

$r=$ ${c}\over{\epsilon \cos(\phi + \delta)+1}$

With c, and $\epsilon$ corresponding to their meanings in conic sections (effective radius and eccentricity).

My problem is almost done, I just wanted to know if there was a geometric relationship between $\delta$ and $\alpha$ I could exploit? There is an energetics way of solving this but a geometric method to find the tangent line as a function of angle from foci seems like something that should be solvable.

The complement of $\alpha$ is known as the heading angle ($\psi$). The supplement of $-\delta$ is known as the true anomaly ($\nu$).
$\psi = \frac{\pi}{2} - \alpha$
$\nu = \pi + \delta$.
The heading angle as a function of the true anomaly and eccentricity is given by $$\psi = \arctan\left( \frac{ e \sin(\nu) }{ 1 + e \cos(\nu)} \right)$$