rotating linked torri with arrow mapping I am trying to map an image of an arrow around a torus and then rotate the torus to animate it, around both its major axis and its minor axis.
I have had responses to this issue back in 2018, but now I need it to work and I do not have the means or ability to figure out why shift-r does not do anything in the blender I have. I believe blender can do what I need, but it has yet to be proven to me, as no one has yet done the animated torri with the arrow images at full bleed. If it can be done and the blender file provided to me, especially if I can animate expanding torri and the whole linked torri spinning in space as well as rotating around both axis (all four - two major and two minor), I am willing to pay money for that blender file.  I do not have the time to get up to speed with blender this time around. I appreciate the work, skill and talent, required and involved, based on my experience with Autocad mostly back in 1991 as well as the early 2000's.
I have tried using armatures in the torus and rotating the armatures both around the minor axis and the major axis of the the torus but this kinks the surface of the torus.
The single arrow, on a grid 24x24, extends to the full extent of the torus' surface, which is divided into 24x24 faces to facilitate mapping - whether in Blender or not, but in 2D animation.
Ultimately two torri are linked, each with its own arrow image, one positive and the other negative.
The positive arrow on one torrus maps into the negative hole arrow in the other torrus.
Here is a link to the entire completed animation done in 2D: https://www.sites.google.com/site/hesusjoychrist/home/hesus-joy-christ/the-spirit-of-one-animated
I would like to be able to duplicate this 2D animation in 3D Blender.
I need to be able to animate the segments of the torri so that they can expand and contract. This I will have to figure out myself I suppose.  So I think I need armatures that can influence both and each torri to fatten and thin as they rotate.
Here are some more images to illustrate the mapping: Male positive Arrow image mapping Female negative Arrow image mapping.


Here is a 2D map of the arrows, but I am using 12x12 rather than 10x10, and the arrows are full bleed in that they both extend each to the limit of the torrus mappings.
That can be something like this (if I understand the question...):

The idea is to hook a grid (1x2 proportions) then simple deform bend twice to make a torus and finally weld to remove doubles.
Then if we move the hook object (only in X/Y axis), that will rotate around the simple deform with bend option.

In short:

Hook will move all vertices (in the animation, just move the hook empty in X/Y so that the arrow will move around the torus)
Simple deform / bend 360 makes a tube
Simple deform / bend 360 makes a torus
Weld merges doubles