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 cul-de-sac.
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 50-ft 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 50-ft 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+0-frac{arc^3}{24R^2}[/tex]
4th degree:
[tex]chordapprox 0+arc+0-frac{arc^3}{24R^2}+0[/tex]
Higher-degree 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 5th-degree approximation is accurate to 0.01 ft, for the given 50-ft radius?
- Doug
Given The Standard HP 48 Data Collector
Chord length for any combinations is but a few strokes.
Paul in PA
Granted
I agree, Paul, and I wondered about that at the time. Or maybe he had a cheapie calculator with no trig functions?
I was just giving the context of the original question as a lead-in to a more interesting question.
- Doug
Correction
I messed up the accuracy statements for the original approximations.
For the given 50-ft radius, this approximation
[tex]chordapprox arc[/tex]
doesn't attain 0.01 ft error until the arc length increases to 8.5 ft.
And this approximation
[tex]chordapprox arc - frac{arc^3}{24R^2}[/tex]
doesn't attain 0.01 ft error until the arc length increases to 41.4 ft.
- Doug
Correction
Nice work with TeX, Doug. As for your actual questions, I'm afraid that the coffee hasn't quite taken hold yet, but I'll bet that one of Jeff Lucas' acolytes will be along shortly to insist that surveyors don't need to know any mathematics. Don't ask me what's wrong with those folks.
Equal time
Kent,
Hmm, I wouldn't want to be considered one-dimensional with just the math, so give me a minute and I'll post a legal question, too.
- Doug
Equal time
We don't need no stinkin math........
Correction
> I messed up the accuracy statements for the original approximations.
>
> For the given 50-ft radius, this approximation
> [tex]chordapprox arc[/tex]
> doesn't attain 0.01 ft error until the arc length increases to 8.5 ft.
>
>
> And this approximation
> [tex]chordapprox arc - frac{arc^3}{24R^2}[/tex]
> doesn't attain 0.01 ft error until the arc length increases to 41.4 ft.
>
> - Doug
Friends don't let friends derive drunk 🙂
Surveillance?
In a strange coincidence, I am actually drinking coffee out of a beer mug right now.
{uneasily scans room for cameras}
- Doug
LC=2R sin½Delta: Which also can be written as 2R Sin Deflection Angle.
½ Delta is the deflection Angle.
Higher-degree approximations
Here are some higher-degree polynomial chord approximations, along with accuracy limits.
0th degree:
[tex]chordapprox 0[/tex]
1st degree: (when R=50, chord error < 0.01 for arc < 8.5)
[tex]chordapprox 0+arc[/tex]
2nd degree:
[tex]chordapprox 0+arc+0[/tex]
3rd degree: (when R=50, chord error < 0.01 for arc < 41.4)
[tex]chordapprox 0+arc+0-frac{arc^3}{24R^2}[/tex]
4th degree:
[tex]chordapprox 0+arc+0-frac{arc^3}{24R^2}+0[/tex]
5th degree: (when R=50, chord error < 0.01 for arc < 90.9)
[tex]chordapprox 0+arc+0-frac{arc^3}{24R^2}+0+frac{arc^5}{1920R^4}[/tex]
6th degree:
[tex]chordapprox 0+arc+0-frac{arc^3}{24R^2}+0+frac{arc^5}{1920R^4}+0[/tex]
7th degree: (when R=50, chord error < 0.01 for arc < 149.4)
[tex]chordapprox 0+arc+0-frac{arc^3}{24R^2}+0+frac{arc^5}{1920R^4}+0-frac{arc^7}{322560R^6}[/tex]
8th degree:
[tex]chordapprox 0+arc+0-frac{arc^3}{24R^2}+0+frac{arc^5}{1920R^4}+0-frac{arc^7}{322560R^6}+0[/tex]
9th degree: (when R=50, chord error < 0.01 for arc < 213.1)
[tex]chordapprox 0+arc+0-frac{arc^3}{24R^2}+0+frac{arc^5}{1920R^4}+0-frac{arc^7}{322560R^6}+0+frac{arc^9}{92897280R^8}[/tex]
- Doug