Heart-loop torus
Geometry
The centreline is the classic heart curve, x(t) = 16 sin³t, y(t) = 13 cos t − 5 cos 2t − 2 cos 3t − cos 4t, scaled by s/17 so that its 17-unit height lands on the scale slider. Aspect stretches it sideways, and notch multiplies the three higher harmonics, deepening or softening the shallow upper dip.
C(t) = (s x/17 · aspect, s y/17, 0)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), taken along a counter-clockwise traversal so the tube always bulges away from the lobes. The loop is sampled 160 times around, which keeps the two upper lobes and the lower point smooth.
The tube radius is capped at 0.8 × the smallest centreline curvature radius, sampled over 200 positions. The classic curve carries cusps at the lower point and at the notch, where the curvature radius formally falls to zero; those two neighbourhoods are skipped, and the limit then comes from the flanks beside the notch — which is why the tube stays slim.