Find wrapping angle of helix on a torus with constant torsion

Niculae George

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Find wrapping angle of helix on a torus

I need some help in calculating the wrapping angle of a spiral helix wrapped on a torus with constant angle against all the meridians of the torus.
The wrapping angle (or the angle measured around and/or against the torus circular cross-section [See here][1]) always remains constant as the helix curve spirals around the torus. In layman's terms means it spirals with the same angle always. if I were to know the arc length of one such turn, my solution will be trivial: Arc len=(2*pi*R2)/cos(wrapping_angle), From here i can find my wrapping angle, where R2 is torus minor radius. This formula is valid only for 1 turn spiral helix and ONLY if I have constant wrapping angle. The thing is I don't know the arc length of one turn, but I do know R1, R2 and the azimuth angle of 1 turn (or 1 turn step-angle or the angle measured around torus central axis [see it here][2]). I want to find the arc length of 1 turn of the helix as a function of step-angle and using such formula in combination with the one I already have, I can find my wrapping angle. I don't know how to integrate Can you help?

[1]: https://i.stack.imgur.com/WVkOD.jpg
[2]: https://i.stack.imgur.com/pRzID.jpg
 
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