How do I rotate to animate, a torus around its minor axis?

Question

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.

Either I can rotate the mapped image, or I can rotate the torus? Can I create an array of a cylinder and map the image of the arrow onto the array?

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.

Here is a link to the entire completed animation done in 2D: enter link description here

Here is a .gif of the completed animation done in 2D. I would like to be able to duplicate this 2D animation in 3D Blender.

Here are some more images to illustrate the 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

https://sites.google.com/site/vid932008/animation/blender#TOC-3D-Computer-Animation-is-unable-to-resolve-the-rotation-of-the-Spirit-of-One

Answer 1

An alternative is to animate a texture.

Here an arrow is mapped on top of the material of the torus. Animating the coordinates for the texture makes the arrow move around the surface:

Answer 2

If you wanted to animate an arrow going round a torus, this is one way of how I would do it:

Model your arrow, then shear the arrow and warp it to make it coil.

Create a lattice and make sure the arrow fits inside. I used ten divisions, less might still be ok. Give the arrow a lattice deform. Parent the arrow to the lattice.

Then use warp again, this time on the lattice. It makes sure the arrow bends around the torus.

Now you just have to animate the lattice in a circle and animate the arrow with local rotation. The lattice will deform the arrow while it rotates.

Here's the finished result:

If you want a snug fit, use a shrinkwrap modifier on top with just a smidge of offset. It might even make the lattice unnecessary.

Of course there's more than one way of doing this. If you want to turn the torus in itself, here's how:

Create a torus and an armature. Doesn't have to be low poly, it's just a bit easier. Put the cursor in the middle of the top edgeloop.

Move the bone in Edit mode to the cursor. Then put the cursor in the middle and duplicate the bone. You can use SHIFTR to do t quickly.

Create a ninth bone and give all other bones Copy Rotation Constraints to that bone, local coordinates to local coordinates. Parent the torus to the armature using envelopes. You can also use weight paint or any other method, just make sure that every bone has only one edgeloop under his control. No kinks whatsoever.

Finishing touches. I moved the ninth bone to the center, gave all of them a custom shape and rotated the ninth bone so that it's relevant axis points towards the screen. When I rotate the main bone clockwise, all others rotate inwards.

And the finished result: