Thread: Approximations of the Perimeter of an Ellipse

1. Approximations of the Perimeter of an Ellipse

Why is there no equation for the perimeter of an ellipse‽ - YouTube
Stand-Up Maths, by Matt Parker

Mentions several approximations.

For semi-axis lengths a, b, with a >= b:

Eccentricity:
$e = \frac{\sqrt{a^2-b^2}}{a}$

Relative difference squared:
$h = \left( \frac{a-b}{a+b} \right)^2$

He mentioned these approximations:

Arithmetic mean:
$\pi (a + b)$

Geometric mean:
$2\pi \sqrt{a b}$

Root mean square:
$2\pi \sqrt{\frac{a^2 + b^2}{2}}$

Srinivasa Ramanujan has two formulas:
$\left( 3(a+b) - \sqrt{(3a+b)(a+3b)} \right)$
(best linear + sqrt(quadratic))

$\pi (a+b) \left( 1 + \frac{3h}{10 + \sqrt{4-3h}} \right)$
(best of this form)

Matt Parker also mentioned some asymmetric formulas that have slightly erroneous small-eccentricity limits:
$\pi \left( \frac{6}{5} a + \frac{3}{4} b \right)$

$\pi \left( \frac{53}{3} a + \frac{717}{35} b - \sqrt{269 a^2 + 667 a b + 371 b^2} \right)$

The exact formula is a complete Jacobi elliptic integral of the second kind
$4 a E(e) \ / \ 4 a E(e^2)$

Using both argument conventions:
$E(k) = \int_0^{\pi/2} \frac{d\phi}{\sqrt{1 - k^2 \sin^2 \phi}}$
$E(m) = \int_0^{\pi/2} \frac{d\phi}{\sqrt{1 - m \sin^2 \phi}}$

I used this page to test my formulas:
Interactive LaTeX Editor

2. Elliptic integrals have some interesting mathematical properties, along with their mathematical relatives, elliptic functions.

I remember from some decades back, when I was fond of the book Abramowitz and Stegun: Handbook of Mathematical Functions It got rather worn, and I ended up throwing it out, because I don't want to have to lug its bulk around. Fortunately, someone has scanned most of its pages, and that link is to online copies of those scanned pages. Those scans have lots of definitions and relations and approximations, though they omit most of the tables.

Here are several references on elliptic integrals:

have a relatively simple relationship with elliptic integrals. and are alternate formulations, though are a different sort.

The usual form of the elliptic integrals is the The is an alternate way of expressing elliptic integrals that can be convenient for calculation.

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