Arc Length in Ellipse

can anyone tell me what is the length of circumference of an ellipse? actually from symmetrical point of view between a circle and ellipse i guessed it to be pi(a+b). i tried to use arc length formula but stuck in a lengthy integral. so i need to get the correct answer in the proper way.

Note by Shubhabrota Chakraborty
4 years, 7 months ago

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There is no easy answer. The circumference has to be expressed in terms of the complete elliptic integral of the second kind \[ E(e) \; = \; \int_0^{\frac12\pi} \sqrt{1 - e^2\sin^2\theta}\,d\theta \qquad 0 < e < 1 \] and the circumference of an ellipse is \(4aE(e)\), where \(a\) is the semimajor axis and \(e\) the eccentricity. A good approximation is \[ \pi(a+b)\left(1 + \frac{3h}{10 + \sqrt{4-3h}}\right) \qquad h = \tfrac{(a-b)^2}{(a+b)^2} \] so your guess was a good approximation for not too eccentric ellipses.

Mark Hennings - 4 years, 7 months ago

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thanks for letting me know

Shubhabrota Chakraborty - 4 years, 7 months ago

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