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For the torus :
Differential system of equations: . Writing , differential equation: , where c is a constant. 
The resolution of the above differential equation leads to elliptic integrals.
The geodesics can be wound or not; even when they are, they are not toric solenoids (except in the limit cases of meridians and the 2 principal circles).


Geodesic of the horn torus  Geodesic of the spindle torus 
See also the asymptotic lines of the torus.
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© Robert FERRÉOL 2018