as animation parametric form
this coarse graphic shows the 'unrolling line' of an increasing horn torus
selected between ratios v(φ) : v(ω) = 4 : 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 ω1 = 2π, according
1/r = ω / 2π (↝ r1 = 1) and φ = ω / r = ω2 / 2π (↝ φ1 = 2π), so we have
x = (1 − cos(ω2 / 2π))·cosω·2π/ω
y = (1 − cos(ω2 / 2π))· sinω·2π/ω
z = sin(ω2 / 2π)· 2π/ω
the parametric forms show mirrored conditions, compare
case r > 1: