Saddle-wave torus

A circular ring is lifted and lowered by a sinusoidal height term, so two opposite arcs rise while the other two fall — a saddle wave you read obliquely.

C(u) = (R cos u, R sin u, h sin(k u))
P(u,v) = C(u) + r (cos v N(u) + sin v B(u))

The frame is numeric: T = normalize(C(u+ε) − C(u−ε)), N = normalize(T(u+ε) − T(u−ε)), B = T × N. The tube therefore turns and rises with the ring instead of tilting against it, so it stays smooth over the crests.

k = 2 is the two-up / two-down saddle; larger k packs more waves. A taller, busier wave tightens the crests, so the tube radius is capped below the centreline's minimum radius of curvature, ρ = |C′|³ / |C′ × C″|, sampled numerically around the loop.