[gfcsbbjz] 3D guilloche

Generalize guilloche patterns to 3 dimensions.

Fairly easy is to continue the idea of a moving point on the edge of one circle becoming the center for the next, but with the epicyclical circles being able to be oriented at any angle in 3D.  An orthogonal projection of this would be the same as a 2D guilloche pattern generalized to allow ellipses.

Instead of each epicycle being a tilted circle, each one could be a Lissajous curve filling a box, or tilted parallelepiped.

Start with a point traveling along a straight line.  Attach a radial arm orthogonal to the line at the moving point, then spin the radial arm as its anchor point moves along the straight line.  The end of the arm traces out a helix.  Generalize this idea allowing the starting path to be a curve instead of a straight line, then recursively apply the same idea, another spinning radial arm, to the new curve.  Starting from a circle, the end result is a very fancy torus.  Tricky problems: What happens if the parent curve hits a cusp?  Can we algebraically determine whether a curve will have a cusp?  What is the initial orientation of the radial arm?

Create an animation of a 3D guilloche pattern rotating in space.  This may be computationally very expensive if the curve has a long period.

3D print the complicated mess of the single curving fiber.

Project the pattern onto the surface of a sphere.  The curve must avoid the center of the sphere: how can this be determined algebraically from the equation of the curve?

Even stranger things might be possible in 4 dimensions (followed by some dimension reduction), because of the richness of how the dimensions can be split 2+2.