Three primitive functions or operations are defined and compared: completion, stellation, and faceting. These "primitive" operations may be applied to any convex polytope of two or more dimensions. In the animations above, the three processes are applied to a regular convex 17-gon.
Perhaps the simplest is faceting; here, the vertices of the 17-gon are fixed, and new sides are constructed, connecting every other vertex, every third vertex, and so on, until no further possibilities exist. A series of star polygons arises, all with the same vertices.
In stellation, the sides of the 17-gon are produced, until they intersect in new vertices. Then they are still further produced, and a new set of 17 vertices is found. Again, a series of star 17-gons arises, with an ever-growing circumradius.
Zonogonal completion begins with a convex polygon which need not be, itself, a zonogon (a zonogon is a centrally-symmetrical polygon whose sides fall into equal opposite pairs). Let the center of the parent polygon be the origin, then every side of that polygon defines two vectors (the endpoints of the side). Construct the parallelogram determined by these two vectors (when the parent polygon is regular, the parallelograms are always rhombs). Repreat this process for each side of the polygon. Now, half of the edges of the parallelograms fall into coincidence in pairs, but the other half are free, forming the exterior, non-convex hull. Discard the interior, coincident edges, and retain the exterior hull edges.
The next step is to identify all the concave locations (vertices) on this hull, and the two edges flanking each such vertex. Construct parallelograms based upon these flanking sides, and so fill each concavity.
Repeat this process until a convex polygon closes up. This final polygon is always itself a zonogon. Suppose the parent polygon to be regular, as in the animation above. Then if the parent be a zonogon of an even number of sides, say, a j-gon, then the final zonogon is a regular j-gon of edge 2 (each side is made of two collinear segments). But if the parent polygon is not a zonogon and has an odd number of sides, say, a k-gon, then the final zonogon is a 2k-gon.
These three processes are rather simple, when applied to regular polygons in the plane. In three dimensions things become vastly more interesting and complex. For instance, there are many million different stellations of Kepler's Rhombic Triacontahedron; and the entire series of zonohedral completions has not yet even been worked out for the Archimedean solids.
If you have Mathematica, you may download my polygonal completion notebook, in which the above animations were made.