This game has the One True way of managing vector movement in two dimensions (Ad Astra Game's Attack Vector has the One True way for three dimension).
The game is played on a hexgrid map covered with a piece of clear plastic or laminated. Grease pencil or other erasable markers are used to draw vectors on the map. A "vector" is a line with an arrow at the end, starting and ending in the center of a hex. The cardboard counter representing the spacecraft is placed on the last arrow.
A spacecraft which moves from hex A to hex B in turn 1 will move to hex C in turn 2, provided that it does not accelerate due to gravity or burning fuel. This is good ol' Newton's First Law: a moving spacecraft will move for the rest of eternity at the same vector unless acted upon by some force. For instance, slaming into an asteroid.
To determine the spacecraft's future position next turn (hex C) examine the last vector (in this case, from hex A to hex B), start at the spacecraft's current position (hex B) and move in the same way as the last vector (three hexes to the right).
Triplanetary spacecraft can generally only burn one unit of fuel per turn. This changes the vector by one hex. Look at the diagram above. If the craft does nothing, next turn it will wind up in hex C at the end of the dotted green arrow. By burning one unit of fuel, the craft can change its vector's end point to D1, D2, D3, D4, D5 or D6.
Say it burns one unit and chooses D5 as the new end point. Draw the new vector from B to D5. This is the new "last vector".
Now that wasn't hard, was it?
Gravity isn't much harder. The six black arrows around a planet are the "gravity hexes". On turn two, the spacecraft moves from A to B, passing through two of Venus' gravity hexes. (Note that one does not count any gravity hexes at the start of the vector, i.e., if there was a gravity hex in hex A it would be ignored)
On turn three, one would expect the ship to move to hex C. But the gravity passed through on turn two takes its toll. The first gravity hex moves the vector endpoint to hex D (that is, one hex in the same direction as the first gravity arrow), and the second moves it to hex E. Draw the new vector from B to E. Notice that the spacecraft moves through a third gravity hex.
In the same way, on turn four one would expect the ship to move to hex F, but the third gravity hex changes the vector end-point to G. Draw the new vector from E to G.
Please note that during all this the ship has burnt no fuel. All the change in course was due to the influence of gravity.
And now for the shining gem of elegance in this movement system: orbiting a planet.
In all other games, planetary orbits have to be taken care of by an ad hoc rule. But in the Triplanetary system orbits occur as a natural consequence of the existing rules.
Say the spacecraft is moving from hex A to hex B. Note it passed through one gravity hex (in hex B, you ignore any gravity hexes at vector start, remember?). In turn two, instead of moving to hex C, gravity alters the vector end point to hex D. Draw the new vector from B to D.
If you keep this up, you will realize that the spacecraft is in a one hex per turn orbit around Venus with no fuel or ad hoc rule required!
Programmer James McNeill in his blog has some notes on creating a computer algorithm to calculate paths using the Triplanetary movement system. You can find his notes here, here, and here.