Asymptotic lines of the torus

open torus
horn torus spindle torus

Toric differential equation: , for the torus .
Toric equation: , with if a > b (open torus) and if a < b (spindle torus).

In the case of the horn torus (b = a), this elliptic integral can be computed, and we get
Toric equation: .
Polar equation of the projection on xOy: .
Cartesian parametrization:
In the case of the horn torus, the projection of the asymptotic line is a spiral with an asymptotic point.

Note that since the asymptotic lines only pass through hyperbolic points, they only exist in the internal part of the torus.