Long time listener, first time caller. I'm making a little hobby game in XNA, its about transport ships in space, analogous to container ships at sea. There are to be AI ships and player controlled ships, and I need to be able to predict the encounter between a Ship and a planet/moons gravitational sphere of influence in a restricted 2D environment so that AI can arrive reliably at destinations and that players can have information to help them decide courses. The positions at time of the Ship and planet/moon, Body for short, are determined from keplerian orbital elements. The Ship and Body both orbit the same centre of attraction.
The approach I have devised so far is first to do some preliminary checks on the apoapsis and periapsis (farthest and closest points from centre of attraction) to see if an encounter is possible. Between checks such as this and if the Ship's orbit is open (hyperbolic, I approximate the parabola case to a hyperbola), it can rule out many scenarios where there could not be an encounter.
If these checks determine an encounter is possible, I determine the minimum and maximum distance from centre of attraction that the Ship is eligible for an encounter. I then get the intersection points of the ships orbit with the two circles defined by that minimum and maximum. This results in zero, two, or four, points on the Ship's orbit, defining zero, one, or two periods where it could encounter the sphere of the Body. At this point if there are zero intersections it is possible the whole of the Ship orbit is in the encounter zone, this is probably an uncommon extreme case but would need to be covered.
I can get the times that the Ship will pass those points on it's orbit, giving one or two windows of time to check for encounter, but from there my best solution is searching the time span by dividing it into steps, calculating the Body's position at those times, and then testing for encounter.
The trouble with that approach is knowing the size to make the steps to efficiently find the encounter. Getting the Body's position at time is somewhat expensive so I'd rather do it as little as possible, but steps too large could potentially miss the encounter.
Are there any properties of confocal conic shapes that could help reduce the search space? Or are there other ways to predict encounter/collision between what is effectively a point moving along a conic path and a circle moving along an ellipse sharing a focal point.
I decided on this approach as, while it is more complex in some ways and less realistic, I would hope it would be less computationally expensive than projecting an integrated path for each ship as it considered different trajectories.