Orbits are predictable, usually needing only two points to compute (although more is better). Observation of two or more points also can give the velocity of the object.

Since the Earth/Moon pair has a known orbit and speed, a point of interaction with another orbit is easy to compute even without a computer program. It's been done using slide rules and even paper and pencil/quill/pen. It's a two dimensional problem, as observation and prediction should be able take into account deviations caused by the gravity of other planets, etc.