So recently I've been playing around with some ideas in my head and wondering whether there are loci in which more than 1 point is movable and others are fixed. For example, I started with a circle centered at the origin and a line intersecting it (also passing through the origin). Obviously the locus of the midpoint of the two intersections when one of the points of the line is fixed and the other moves around the circle is a smaller circle passing through the center and with half the radius of the larger circle. Here's a pic for clarity:
The locus is the thick red circle formed by point M (currently located at the origin) when Q (in the 1st quadrant) moves along the blue, larger circle. A similar circle predominantly in the 1st quadrant would be formed had we moved P and kept Q fixed.
My question is whether we have the mathematical tools to describe what happens when both P and Q move at the same time. Usually loci are thought out as being "traced out" instantaneously, but in this case if P and Q trace out the locus while moving around the circle at the same rate it is just the point M at the origin. I've done some experimenting, and when P and Q move in opposite directions at different speeds the result is actually quite nice:
And also when they move in the same direction:
It's no doubt that they're very beautiful. Perhaps even more interesting is the fact that the loci obtained are the same when rotating two parallel lines by two points (one for each) on two congruent circles and seeing the locus of the midpoint of the two points. (I won't add a figure as the question is already much too long.
So, to sum up, are these really "loci" in the full sense of the word since they are produced with two moving points? Also do we have the mathematical tools to describe these curves and predict their properties? Has this been written about before? Is this just a dead-end application of geometry or could this be something bigger?
Thanks!