Rhumb line of the torus

open torus with its rhumb lines

horn torus spindle torus

See this exam paper TPE 96 maths 1

1st method, in toroidal coordinates.
Differential equation: , for the torus , being the angle that form the rhumb lines and the parallels.
For the open torus (a > b): .
The curve is rational (and, in this case, closed) when is rational, and when m = 1, i.e. for or , we get the Villarceau circles.
For the horn torus (a = b): .
For the spindle torus (a < b) : .
2nd method in cylindrical coordinates.
Differential equation: , for the torus .
For the open torus (a > b): (, ); the projections on xOy are therefore the polygasteroids.
For the horn torus (a = b): (inverse of the Galilean spiral).
For the spindle torus (a < b): with ,.

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