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→Circumference: Arc length in terms of Cartesian coordinates Tags: Mobile edit Mobile app edit Android app edit |
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The errors in these approximations, which were obtained empirically, are of order <math>h^3</math> and <math>h^5,</math> respectively.
More generally, the [[arc length]] of a portion of the circumference, as a function of the angle subtended (or
:<math>y=b\sqrt{1-\frac{x^{2}}{a^{2}}}.</math>
Then the arc length <math>s</math> from <math>x_{1}</math> to <math>x_{2}</math>
:<math>s=-b\int_{\arccos \frac{x_{1}}{a}}^{\arccos \frac{x_{2}}{a}} \sqrt{1-\left(1-\frac{a^{2}}{b^{2}}\right)\sin ^{2}z} \, \mathrm dz.</math>
This is
:<math>s=-b\Biggr[E\left(z \, \Biggr| \, 1-\frac{a^{2}}{b^{2}}\right)\Biggr]^{\arccos \frac{x_{2}}{a}}_{\arccos \frac{x_{1}}{a}}</math>
where <math>E(z \, | \, m)</math> is the incomplete elliptic integral of the second kind with parameter <math>m=k^{2}.</math>
{{See also|Meridian arc#Meridian distance on the ellipsoid}}
The [[inverse function]], the angle subtended as a function of the arc length, is given by the [[elliptic functions]].{{Citation needed|date=October 2010}}
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