Spiral-grooved torus

The tube keeps a smooth waistline but is carved by a narrow helical groove. The mask is a Gaussian dip in sin(v + w u), so the tube radius falls to r₀(1 − d) exactly where that sine vanishes.

r(u,v) = r₀ (1 − d exp(−(sin(v + w u)/σ)²)), σ = 0.38
P(u,v) = (R cos u, R sin u, 0) + r(u,v) · (cos v (cos u, sin u) + sin v (0,0,1))

The groove centre is the helix v = −w u: as the tube sweeps once around the ring the groove creeps w times around the tube, so every groove is one continuous spiral. The cut stays narrow next to the tube's circumference and the opposite wall of the tube is untouched.

The result reads as a smooth torus with diagonal movement across it — the helix angle atan(w r₀ / R) is reported below. The outline never breaks; the grooves only shade differently as they turn.