Pointwise coloring the plane or a region or curve

Is there a way to color the plane or a region or a curve (e.g., a locus) pointwise?

The only way I know is to create a curve that essentially covers the plane (e.g., a tight spiral, r = theta/300), then to place a point on that curve with the point dynamically colored; now animate the point slowly while its trace is on, preferably with small point size for greater precision.

The troubles with the above workaround are: (1) it is very slow; (2) I know of no way to save the coloration in GeoGebra [though I can save and export the image to e.g. "Paint", or via a printscreen]; (3) it seems imperfect in the sense that if I re-start the animation + tracing, there appears an artificial fuzzy boundary between my original colored animation and the new one [I suppose this is expected if I change the speed or point-size, but it even appears with no change in those parameters].

Is there a way to save such a trace image in GeoGebra? [Another application when this would be desirable is when one wants to see the envelope of a line dependent upon some varying parameter.] Perhaps I can save the image via "Paint" or printscreen, then reload that image into GeoGebra, but I'm not sure how to accurately place the image (translucently) onto my pre-existing GeoGebra construction. One example of one goal: Suppose the traced coloration produces an apparent ellipse (or other curve) whose equation I don't know; I could at least get an approximation to the ellipse by placing 5 independent points onto the colored trace image, etc.