Perimeter of ellipse integral

Equation of ellipse
History of ellipse
Perimeter of ellipse

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Ramanujan's Perimeter of an Ellipse

arXiv:math/0506384v1 [math.CA] 20 Jun 2005 Ramanujan’s Perimeter of an Ellipse MARK B. VILLARINO Escuela de Matemática, Universidad de Costa Rica, 2060 San José, Costa Rica February 1, 2008 Abstract We present a detailed analysis of Ramanujan’s most accurate approximation to the perimeter of an ellipse. Contents 1 Introduction 1 2 Later History 3 3 Fundamental Lemma 3 4 Ivory’s Identity 8 5 The Accuracy Lemma 9 6 The Accuracy of Ramanujan’s Approximation 1 10 Introduction Let a and b be the semi-major and semi-minor axes of an ellipse with perimeter p and whose eccentricity is k. The final sentence of Ramanujan’s famous paper Modular Equations and Approximations to π, [5], says: “ The following approximation for p [was] obtained empirically: 3(a − b)2 √ +ε p = π (a + b) + 10(a + b) + a2 + 14ab + b2 1 (1.1) where ε is about 3ak 20 .” 68719476736 Ramanujan never explained his “empirical” method of obtaining this approximation, nor ever subsequently returned to this approximation, neither in his published work, nor in his Notebooks [3].

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Circumference of an Ellipse

And an approximation using arc sections

Collected by Paul Bourke
Corrections and contributions by David Cantrell and Charles Karney.

Update (June 2013) by Charles Karney and the AGM (Arithmetic Geometric Mean) algorithm.
Update (August 2024) new expression by Giovanni Anselmi. (15036.giovannianselmi@gmail.com)

a - Major axis radius
b - Minor axis radius
e - eccentricity = (1 - b² / a²)^1/2
f - focus = (a² - b²)^1/2
h = (a - b)² / (a + b)²
area = pi a b

The following lists and evaluates some of the approximations that can be used to calculate the circumference of an ellipse, specifically, shapes defined by the points:

To some, perhaps surprising that there is not a simple closed solution, as there is for the special case of a circle (a=b).

Anonymous

Reduces to the circumference of a circle for a=b. This performs the worst for highly eccentric ellipses, for example when b=0 the length is 3.847a rather than the expected value of 4a.

Ramanujan, first approximation

Indian mathematician Sriniva

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There’s no elementary formula for the circumference of an ellipse, but there is an elementary approximation that is extremely accurate.

An ellipse has equation (x/a)² + (y/b)² = 1. If a = b, the ellipse reduces to a circle and the circumference is simply 2πa. But the general formula for circumference requires the hypergeometric function ₂F₁:

where λ = (a − b)/(a + b).

However, if the ellipse is anywhere near circular, the following approximation due to Ramanujan is extremely good:

To quantify what we mean by extremely good, the error is O(λ¹⁰). When an ellipse is roughly circular, λ is fairly small, and the error is on the order of λ to the 10th power.

To illustrate the accuracy of the approximation, I tried the formula out on some planets. The error increases with ellipticity, so I took the most elliptical orbit of a planet or object formerly known as a planet. That distinction belongs to Pluto, in which case λ = 0.016. If Pluto’s orbit were exactly elliptical, you could use Ramanujan’s approximation to find the circumferen