coordinates for
this animation and its parametric form
(source) :for all points P on the surface of a horn torus with fixed radius
r is validx = r·(1 − cosφ)·cosω y = r·(1 − cosφ)·sinω z = r·sinφ the 'unrolling line', indicated in the animation, is divided into two parts (referring to the standard dynamic horn torus as unit): r > 1 and r < 1 case r > 1: r and ω increase with φ, starting with , accordingφ_{1} = 2πr = φ / 2π (↝ and ω = r·φ = φr)_{1} = 1^{2} / 2π
(↝ , so we haveω)_{1} = 2πx = (1 − cosφ)·cos(φ ^{2}/2π)·φ/2πy = (1 − cosφ) · sin(φ ^{2}/2π)·φ/2πz = sinφ · φ/2π (helical lines - at φ = const. each - not to scale), see also case r < 1: |