Chord approximation formulas
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
