Draw the Equation

So far we watch the balls and see where they went. With one planet, we can do better: work out the whole path the moment the ball leaves the muzzle.

The orbit's numbers

A ball's orbit around one planet is always the same kind of curve. Only its size and stretch change. Two numbers pin it down:

1. p, the orbit's width at the planet. Go a quarter turn round from the closest point, and the ball is p away.
2. e, how stretched it is. 0 is a circle. Between 0 and 1 is an ellipse. 1 is a parabola, and above 1 is a hyperbola.

Both come straight from where the ball starts, r, and how it's moving, v. You don't need to work these by hand. The × and the little hat on r̂ are 3D arrow maths, and the game does them for you:

energy               ε = v²/2 − GM/r          below 0, the ball comes back
spin                 h = r × v                angular momentum
stretch              e = (v × h)/GM − r̂       a vector pointing at the closest point
width                p = h²/GM
the path             r(ν) = p / (1 + e·cos ν) ν is the angle from the closest point

The picture above is the speed-12 shot. The formula puts the far point at 28.2857. The game's own step flies the ball out to 28.2876. They agree to three decimal places.