Fluted torus

A ring of radius R carries a tube whose cross-section is not a constant circle: the tube radius breathes with the angle v around the tube, so the surface turns into evenly spaced rounded ribs.

r(v) = r₀ (1 + d cos(m v))
P(u,v) = (R cos u, R sin u, 0) + r(v) · (cos v (cos u, sin u) + sin v (0,0,1))

Every rib is a band of constant v, so a rib runs the whole way around the ring rather than along it, and m of them are spaced around the tube. d sets how pronounced each rib is.

Depth stays shallow — capped at 0.4 — so the ribs read as a finish rather than a feature. The outline remains a clean torus while the surface ripples, and the 144 × 64 grid resolves the flutes smoothly at every m.