Petal torus
Geometry
The ring carries five or six gently rounded lobes, like a flower with an open centre. The centreline is a polar curve, ρ(u) = R + A cos(k u), drawn as C(u) = (ρ cos u, ρ sin u, 0). Since k is a whole number the curve closes on itself after one turn with k evenly spaced petals.
P(u,v) = C(u) + r cos v N(u) + r sin v (0,0,1)
N is the numerical outward normal, N = normalize(Ty, −Tx) with T = C(u+ε) − C(u−ε), so the tube keeps a circular cross-section square to the centreline and the lobes stay shallow and flowing rather than moulded. Depth stays well below the base radius, which keeps the opening broad.
The tube radius is capped at 0.8 × the smallest centreline curvature radius, sampled over 200 positions. That minimum always sits in a valley between two petals and falls quickly as the petal depth A grows, so the radius slider shortens by itself.