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RHUMB LINE OF THE TORUS
open torus with its rhumb lines 
horn 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.

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© Robert FERRÉOL 2018