Rounded-triangle torus
Geometry
A smooth tube is swept around a softly rounded triangle. The centreline is a polar superellipse, ρ(u) = R (|cos θ|^n + |sin θ|^n)^(−1/n) with θ = 0.75 (u − φ) − π/8 and C(u) = (ρ cos u, ρ sin u, 0).
The argument advances three quarters of a turn per revolution, so the curve closes with threefold symmetry — the m = 3 case of the superformula. At n = 2 it is a circle; larger n draws material out into three broad corners. Phase turns the triangle without changing its size or its corners.
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 cross-section stays circular and perpendicular to the centreline. The tube radius is capped at 0.8 × the smallest centreline curvature radius over 200 samples — the tightest point is always a corner.