The Math Forum

Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Math Forum » Discussions » sci.math.* » sci.math

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

Topic: Blanknhorn modification of G-K and Euler methods for ellipse circumference
Replies: 0  

Advanced Search

Back to Topic List Back to Topic List

Posts: 37
Registered: 12/23/11
Blanknhorn modification of G-K and Euler methods for ellipse circumference
Posted: Oct 18, 2012 2:08 AM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

Gauss-Kummer and Euler methods for ellipse circumference calculation can be used for extremely high eccentricities:

Note for example in the Gauss-Kummer:


the fractions can be pulled out:


Note for this case, the 1+1/4+1/64+... sums to 4/pi.

You can then work with:


and when h is VERY near 1 you can obtain a workable number of significant digits in short order. This can be done similarly for Euler's method. Do note, however, the rate of convergence is nothing like that of Cayley.

With only a couple hundred terms of the series, at least 6 significant digits can be produced over all eccentricities, and with exact endpoints using this method. Since I see some people throw their own names about to gain popularity, I suppose could rename this the Blankenhorn ellipse circumference modification to the Gauss-Kummer method, haha.

Use the popular method when the ratio a/b>=0.0005 and the modified method for a/b<0.0005.

-Thomas Blankenhorn

Point your RSS reader here for a feed of the latest messages in this topic.

[Privacy Policy] [Terms of Use]

© The Math Forum at NCTM 1994-2018. All Rights Reserved.