What is the equation for helical path length around a torus?

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The discussion seeks an analytical expression for the helical path length around a torus, emphasizing the need for precision rather than approximation. It suggests starting with a parameterization of the torus, defined by a cross-section radius r and a central circle radius R greater than r. Participants are advised to differentiate this parameterization to find tangent vectors and create a vector field with undetermined coefficients. The conversation hints at considering specific "pitch" for the helical path and finding integral curves based on this setup. Overall, the focus is on deriving a mathematical formula for the helical path length on a toroidal surface.
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Does anyone know the equation for the helical path length around a torus?
I need an analytical expression, not an approximation.

Thanks
 
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Hint: write a parameterization of your surface (torus of cross-section radius r centered on a circle with radius R > r). Find the tangent vectors to this surface by differentiating wrt the two parameters of your parameterization. Consider a linear combination with undetermined coefficients to obtain a vector field lying in the torus. (Did you have a particular "pitch" in mind?) Find the integral curves.
 
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