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This is actually an applied engineering situation, not a homework problem. I hope this is the right forum to ask for help.

I have a cable with a minimum bend radius of 10 cm. I need to coil it around a cylinder to make a helix shape. I understand that if I coil with a small pitch (near 90°, based on this diagram) the cylinder will need to have a 10 cm radius. But if I coil the cable with a large pitch, the cylinder can be much smaller.

I've looked into the equations for the helix, but the radius in those equations is the cylinder radius, not the radius of the coil. I'm trying to find the relationship that tells me the pitch if I have a 5 cm cylinder and the 10 cm minimum radius constraint. Or conversely, if I want 10 cm vertical coil spacing, and have the 10 cm radius constraint, what is the minimum cylinder radius?

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An angled slice through a cylinder of radius $r$ will give you an ellipse, with minor radius equal to the cylinder $b=r$ and major radius of $a=\frac{r}{\cos \theta}$. The radius of curvature $c$ for cable wrapping at such an angle is $c=\frac{a^2}{b} = \frac{r}{\cos^2\theta}$ and the pitch $p$ is $p=2\pi r\tan \theta$

Thus starting from the pitch and known radius, you can calculate the angle of wrap from $\theta = \tan^{-1}\frac{p}{2\pi r}$ and get the coiling radius as above.

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    Is it true that the radius of curvature at a point on the helix is equivalent to the radius of curvature on the short axis vertex of an ellipse formed by an angled slice through the cylinder at the same angle as the helix at that point? – 2018-12-08
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    @kreeser1 if I understand you correctly, then yes, it's the largest radius of curvature on the ellipse. – 2018-12-09