Circumference of an Ellipse (Gauss-Kummer)

Circumference of an Ellipse (Euler)

eccentricity ε =

semimajor axis a =

number of terms n =


semiminor axis b =

circumference S =

The Gauss-Kummer Series (Wolfram MathWorld) has better convergence properties than Euler's series:

Circumference S = π*(a+b)(1 + h/22+ h2/26 + h3/28 + 25h4/214 + 49h5/216 + ...)

The coefficient of hn is the square of the fractional binomial coefficient C:

coefficients Ci2
terms Ci2 hi

The convergence of the series is very slow when the eccentricity ε is almost 1:

n=10 S=4.41088659279142
n=20 S=4.410886593016652
n=30 S=4.410886593016652
n=50 S=4.410886593016652
n=85 S=4.410886593016652

There are some more series having better convergence properties.

Web Links

How Euler Did It - Arc length of an ellipse (PDF)

Circumference of an Ellipse

Specimen de constructione aequationum differentialium sine indeterminatarum separatione

(c) 2006-2016 J. Giesen