Suppose you want to wrap Christmas lights around a tree trunk that we can approximate by a cylinder of radius r.
You want to wrap lights around the tree in a helix, going up a distance h every time you go around the tree once. What length of lights do you need to make n turns around the tree?
You can model the lights as a parametric curve with equation
(r cos t, r sin t, ht/2π)
If t ranges from 0 to T, the corresponding curve length is
(r² + h²/4π²)1/2 T
and you can set T =2πn if you want to find the length of n turns.
How much does h matter? Not much if h is less than r, as is often the case when wrapping a tree trunk with Christmas lights. My daughter Allison discovered this while wrapping lights around our pine tree, and then I wrote this post adding the math details.
If we expand (r² + h²/4π²)1/2 as a function of h in a Taylor series centered at 0 we get
(r² + h²/4π²)1/2 = r + h² / 8π²r + O(h4).
For example, suppose a tree is r = 10 inches in diameter and move h = 4 inches vertically with each turn. To complete one turn we let T = 2π. The exact length of one turn is
(r² + h²/4π²)1/2 T = (10² + 4²/4π²)1/2 (2π) = 62.96 inches.
If we ignore the h term we get
rT = 10 (2π) = 62.83 inches.
In short, the length of n turns around the tree is 2πrn, the same as 10 circles around the tree. The difference in length between a helix with n turns and n circles is negligible, provided h is smaller than r. Even if h = r = 10 in the example above, we’d get a length of 63.6 inches and our approximation would still be off by less than an inch per turn.
The heart of this calculation pops up frequently in various contexts. For example, the same calculation appears in the post It doesn’t matter much if the tape is straight.