
Chord approximation formulas
A couple of years ago on some other surveying message board, someone started a thread asking about formulas relating chord length to arc length.
(Digression: Sorry about the vagueness, but that message board no longer exists, and I couldn’t find that thread cached in Google.)
The questioner was well aware of the exact formula relating arc length and chord length for a given radius,
[tex]chord = 2Rsinfrac{arc}{2R}[/tex]
but he was interested in finding any “faster” approximate formulas (fewer calculator keystrokes) for this relationship. In particular, he was interested in approximations that could be used in a standard 50′ radius culdesac.
Someone suggested that, for very short distances,
[tex]chordapprox arc[/tex]
Indeed, this approximation is accurate to better than 0.01 ft for arc lengths of less than 7 ft, for the given 50ft radius.
Then someone else suggested the improved approximation,
[tex]chordapprox arc – frac{arc^3}{24R^2}[/tex]
which is accurate to better than 0.01 ft for arc lengths of less than 28 feet, again for the given 50ft radius.
At the time, no one mentioned that these two approximations are members of a family of Maclaurin polynomial sums. Listed below are the first several members of that family:
0th degree:
[tex]chordapprox 0[/tex]1st degree:
[tex]chordapprox 0+arc[/tex]2nd degree:
[tex]chordapprox 0+arc+0[/tex]3rd degree:
[tex]chordapprox 0+arc+0frac{arc^3}{24R^2}[/tex]4th degree:
[tex]chordapprox 0+arc+0frac{arc^3}{24R^2}+0[/tex]Higherdegree Maclaurin sums bring improved accuracy to the approximation. Let’s now ignore the original poster’s criterion of fewer keystrokes (because this post is largely a transparent excuse to try out the board’s TeX functions). What is the next equation in this family? And what is the maximum arc length for which this 5thdegree approximation is accurate to 0.01 ft, for the given 50ft radius?
– Doug
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