as animation parametric form this coarse graphic shows the 'unrolling line' of an increasing horn torus selected between ratios v(φ) : v(ω) = 10 : 1 and v(φ) : v(ω) = 1 : 1000 but not to scale and using rather rough differentials for the single steps, (otherwise the lines would lay very densely, according to the following:) case r < 1: 1/r and φ increase with ω, starting with ,
accordingω_{1} = 2π1/r = ω / 2π (↝ and φ = ω / r = ωr)_{1} = 1^{2} / 2π
(↝ , so we haveφ)_{1} = 2πx = (1 − cos(ω ^{2} / 2π))·cosω·2π/ωy = (1 − cos(ω ^{2} / 2π))· sinω·2π/ωz = sin(ω ^{2} / 2π)· 2π/ω the parametric forms show mirrored conditions, compare case r > 1: |