Tapered torus

A circular centreline of radius R carries a tube whose thickness changes as it travels around the loop: it swells, narrows and returns smoothly, with no step or seam.

r(u) = r0 (1 + s1 sin(u + φ1) + s2 sin(2u + φ2))
P(u,v) = (R cos u, R sin u, 0) + r(u)(cos v (cos u, sin u, 0) + sin v (0,0,1))

s1 is the slow swell and s2 a faster secondary rhythm; the two are held below 0.55 together so the tube never pinches shut. The cross-section still stays circular — only its radius breathes.

A restrained gradient follows the thickness: the fill colour is scaled by 1 + 0.18 (r(u) − r0)/r0, so the swollen passages read slightly lighter than the thin ones.