If I understand, your question is about the building of the formation (arrival of the enemies from outside the screen into a "space-invaders like" formation (aka end-formation)).
Having played this game a lot, here is how I think they have done it:
- They have a handful of "incoming trajectories" which come from out of the screen to an end-point roughly in the middle of the screen, just under the end-formation.
- Those trajectories are likely to be pre-computed, and stored as lists of x,y coordinates (regarding the age of the game, they probably have been drawn on paper and "digitized" by hand).
- Each enemy has a "terminal" position in the end-formation (some of them don't; that's the ones that will try to dive on you)
- All enemies move following the chosen trajectory until the end-point, and then they continue to their assigned position in the end-formation more or less in straight line
To make it smoother, you may give each enemy a different exit point for the trajectories (that would be the index of the last point of the trajectory before going to its end-position). It may make the transition more natural.
Also the diving enemies will leave the trajectory a lot earlier, and probably re-use the algorithm that guide missiles to the player ship.
Also, the end-formation is slowly moving horizontally. When the enemy will go in straight line from the end-point toward its end-position, you need to go to the position where the end-point will be at the time of arrival (aim in advance); or you may try to aim the current position of the end point; it could make the trajectory more natural, but you'll have to try to be sure.
Bottom-line: This method was common in the 80s. On today's hardware, you can probably model the trajectories as mathematical functions; that would allow you to have a smother movement, and less pre-computed data; also you may use some polynomial function instead of "straight line" for the last part of the arrival. The rest of the algorithm doesn't change (go to trajectory end-point, then go to your end-position)
EDIT: I forgot the last part of your question: I'm not sure there is a "simple" mathematical function for those trajectories, but you may obtain something close by stitching ellipses arcs together.
Here is an attempt to visualize it:
Here the trajectory is decomposed in 3 ellipse arcs (first the black, then the red and the green. The "Pinkish" part is the last part of the arrival (the straight line).