Recommend Geometric Animation Software

I want to do some animation of geometric forms with complicated color coding.  Looking around, the market is saturated with animation intended to produce dramas: cartoons or battling superheros.  There is so much that I couldn't find a single candidate.  Perhaps some of these packages will do what I want, but there is no way to tell from the sales literature.

Here's a concrete example:  A dozen or so rotating toruses drawn via moving colored dots.  The dots follow an orbit on the surface of the torus and change color in a certain way as they do so.  The toruses are of different sizes, and are superimposed on one another to form a sphere.

No way to do this by hand, I need to be able to program the thing with mathematical formulae.  But the basic capabilities are simple:  just drawing colored discs appearing at various angles, or maybe lines.  Here's an example.  Gifs animados de Graficos, animaciones de Graficos

Being able to produce GIFs is nice, but what I really want to do is something I can upload to Youtube to promote a book.  Such a vid would have a voiceover.

Surely there are plenty of packages that can do this.  Any suggestions?

Even if you don't know, maybe there are some keywords I can search on.  All I could find were packages to produce cartoons and superhero epics.  I couldn't get anywhere near what I was looking for.  Mathematica, maybe...

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