Flattened pebble torus
Geometry
The ring is squashed vertically and its polar radius is nudged by two gentle harmonics, so the opening stays small and slightly irregular — a river stone with a hole.
ρ(u) = R (1 + ε1 sin(2u + φ1) + ε2 sin(3u + φ2))C(u) = (ρ cos u, ρ sin u, 0), N(u) = normalize(Ty, −Tx)P(u,v) = C(u) + a cos v N(u) + b sin v (0,0,1)
N is the outward ring normal taken from the finite-difference tangent T = normalize(C(u+ε) − C(u−ε)) and sign-checked against the radial direction. The cross-section is an ellipse flattened to b ≤ a, which is what removes the tube's roundness and leaves a low, wide band.
The finish is deliberately matte: the gloss slider scales the specular term, so at low settings the form is read almost entirely from the diffuse gradient — ε1 = 0.6ε and ε2 = 0.4ε at fixed phases (φ1 = 0.9, φ2 = −0.5) keep the irregularity gentle.