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Is there a formula for calculating the length of an offset path B given the length of path A and the distance between A and B, assuming the path is a closed loop?

Background: A friend and I go jogging for a 4Km loop and log our route using our phones. We have found that the person jogging on the outside is logging a greater distance than the person on the inside as expected; however, the difference is sometimes a few hundred meters which is much more than expected.

Instinct tells me that the loop, despite being irregular in shape, can be treated as a circle, and that the difference should be no more than pi * 2 * x (where x is the distance between us, around 1.5m) and should approach 0 as the length of the route increases.

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Permit me to point you toward the MO question "Shrink polygon to a specific area by offsetting":


  ShrinkQ
...although there is no question that discussion is oriented toward area rather than perimeter.

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  • $\begingroup$ I don't think my problem would have the same issues since it would be acceptable for either path A or B to cross itself or the other path. $\endgroup$ – Jens Ehrich Sep 11 '14 at 3:03

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