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I need some help in calculating the length of a spiral helix wrapping a torus at a given angle. Real-world application: wrapping tape around a hula-hoop, calculating the length of tape used, given the thickness of the hoop, the circumference of the outside of the hoop, and the angle of wrapping.

Assume 0-degrees is the poloidal angle, and 90-degrees is the angle along the equator of the hoop.

Is this equation correct?

$L$ = length
$C$ = outer circumference of torus/hoop
$t$ = thickness of hoop
$ϕ$ = angle of wrapping helix

$$L = {C-{2 π(t/2)} \over \sin(ϕ)}.$$

EDIT: To clarify, the application is not attempting to completely cover the surface of the hoop with tape. Imagine a decorative tape that wraps the hoop at an angle (say $30°$), leaving a gap between each helictical go-round. At $90°$ degrees, the amount of tape used is equal to the hoop's outer circumference. At $0$, the amount of tape used is equal to the hoop material's thickness. Have I found the correct equation to calculate tape-length at various other angles?

Sebastiano
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Bryan Smith
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  • Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. – Community Jun 22 '25 at 00:49
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    For your real world application you need a length of tape (of given width) that has the same area as the torus. Buy extra tape, of course, to account for bunching and overlap. The width of the tape will determine the angle. – Ethan Bolker Jun 22 '25 at 01:03
  • Winding tape around a torus is not a helix. You need to describe with mathematical rigour what is the line integral you're trying to calculate. I suspect, for exactness, it's much more complicated than your equation. Please show how you derived your equation. – Simon Goater Jun 22 '25 at 16:54
  • related? https://math.stackexchange.com/questions/1812229/length-of-a-regular-coil-on-torus – DrM Jun 22 '25 at 17:20
  • I have no background in mathematics, and I don't know how to read or use integral mathematics. My equation was derived from Claude 4.0 Sonnet. I think it is wrong though, because according to it, increase of angle yeilds longer (helix coil?) tape. That is wrong because a 90 degree wrap is the circumference of the hoop/torus, and wrapped-tape length should get longer as you decrease the wrapping angle, until it reaches the asymptote of 0 degrees (at which point tape length = tube thickness).

    So I've turned to human help. I'm not a mathematician, just someone looking for a practical use.

     – Bryan Smith Jun 25 '25 at 02:13

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