I have searched in Internet and found many sites, but all these sites give the whole perimeter (circumference) length of an ellipse with series expansion, so I calculate till the accuracy that I want (as in image below).
But I would like to calculate the length of an elliptic arc with series expansions till wanted accuracy, I can not deduce so formula.
Ellipse arc length with series expansion
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sequences-and-series
elliptic-integrals
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0See this [**journal article**](http://www.sciencedirect.com/science/article/pii/S0377042711001270), the pdf is free to download – 2017-02-06
1 Answers
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Plenty of accurate approximations are known for the ellipse perimeter.
If we denote the $p$-th power mean through
$$ M_p(a,b) = \sqrt[p]{\frac{a^p+b^p}{2}} \tag{1}$$
due to the results of Muirhead, Alzer and Qiu, the perimeter $L(a,b)$ of an ellipse with semi-axis $a$ and $b$ fulfills
$$ M_{\alpha}(a,b) \leq \frac{L(a,b)}{2\pi}\leq M_{\beta}(a,b),\qquad \alpha=\frac{3}{2},\;\beta=\frac{\log 2}{\log(\pi/2)}.\tag{2} $$
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0At this website: http://what-when-how.com/the-3-d-global-spatial-data-model/example-of-bk2-transformationvincentys-method-same-point-geometrical-geodesy-the-3-d-global-spatial-data-model-part-1/ how can I more coefficients calculate, to wanted accuracy.... I would like meridian arc length not the whole perimeter in terms of semi-axis major and minor or eccentricity with series expansion method. Thanks in advance – 2017-02-07
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0@user412889: you already know the Taylor series of the elliptic perimeter (or arc length) in terms of $h$, what else do you need? It can be computed with a variation of the AGM mean (that has quadratic convergence) by all means. – 2017-02-07