Incomplete Elliptic Integrals of the 2nd Kind, E(k,phi) first created 11/11/06 - last modified xx/xx/xx Page Author: Ty Harness

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There are many textbooks that cover elliptic integrals [1] and tables [2] of the following equation can be used to calculate the arc length. I guess all students are probably ok with incomplete elliptic integrals of the second kind to find the arc length around an ellipse.

$ E(k,phi) = int_0^phi sqrt(1- k^2 * sin^2(theta)) d theta $ where $ 0 <= phi <= pi/2 $ and $ 0 < k < 1 $ Equation 1.

Only joking, equation 1 was derived to torture students. We need to get the equation of an ellipse into the same form. Firstly it's worth looking at the equation of a circle.

$R^2 = x^2 + y^2$ where $x = R*cos(theta)$ and $y = R*sin(theta)$

$R^2 = R^2*cos^2(theta) + R^2*sin^2(theta)$ and dividing all the terms by $R^2$ gives the trigonometrical identity

$1 = cos^2(theta) + sin^2(theta)$ or $cos^2(theta) = 1 - sin^2(theta)$

and the circumference ($0 .. 2pi$) of a circle can be calculated from the integral:

$int_0 ^ (2*pi) R*d theta = [R*theta]_0 ^(2*pi) = 2*pi*R$

With the above method in mind we can rewrite the $R^2$ for an ellipse:
$R^2 = a^2 cos^2(theta) + b^2 sin^2(theta) $

$R^2 = a^2*(1 - sin^2(theta)) + b^2 sin(theta) $

$R^2/a^2 = 1 - sin^2(theta) + b^2/a^2 sin(theta) $

$R^2/a^2 = 1 + (-1 + b^2/a^2) sin^2(theta) = 1 - (1 - b^2/a^2) sin^2(theta) $

$R = a*sqrt(1 - (1 - b^2/a^2) sin^2(theta)) $

Equation 1 only allows us to have $phi = pi/2$ as a maximum but the perimeter of a full ellipse would be 4 times greater.

$S = int_0 ^(pi/2) = R d theta = a* int_0 ^ (pi/2) sqrt(1 - (1 - b^2/a^2) sin^2(theta)) $



$k^2 = 1 - b^2/a^2 $ and therefore $ k = sqrt( 1 - b^2/a^2 )$

Worked Example

$a = 200; b = 100$ therefore $k = sqrt( 1- 100^2/200^2) = 0.866$ to 3dp

which importantly for formula x that k is between 0 and 1. Some math text books write this integral

$S = a*E(phi,alpha)$ where $k = sin(alpha)$ hence $alpha = arcsin(0.866) = 60 deg$

and then using tables [3 p72]

$E(90deg,60deg) = 1.2111$

and finally

$S = 1.2111*a = 242.22$ which is a quarter of the ellipse perimeter.

In this day and age you can have the function calculated online by Wolfram Research but you insert $m = k^2$ and use the EllipticE function.

EllipticE[z, m] = EllipticE[1.571, 0.75] Function Evaluation Created by webMathematica ? 1.21116 http://functions.wolfram.com/

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References (1) Jahnke, E., Emde, F., Tables of Functions with Formulae and Curves, Dover, , New York: (1943), . p72

(2) Stephenson, G., Mathematical Methods for Science Students, Longman, 2nd Ed., London: (1973), 188 - 194.

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