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Circumference of an ellipse
By Anonymous on Tuesday, October 12, 1999 - 08:17 pm:

What is the formula for the Circumference of an ellipse?
Could it be (a+b) p when a and b represent axis of an ellipse?

By Alex Barnard (Agb21) on Friday, October 15, 1999 - 12:53 pm:

Unfortunately there is no nice formula for the circumference of an ellipse. The first guess of (a+b)× p is unfortunately not right. Mathematicians have invented functions called elliptic functions to do things like this [they're called elliptic because they were first discovered in order to try and do this problem].

Before you say that inventing functions is a bit of a cheat remember that functions like sine and cosine were invented to try and solve problems to do with the circle. It is only because they are easy to use and have been around for ages that they have become regarded as common. It is possible that in a thousand years elliptic functions may be so common that A-level students could be being taught them!!!

If you are interested I can try and find out more about elliptic functions... just let me know.

AlexB.

By Anonymous on Monday, October 18, 1999 - 11:59 am:

Can you not treat the ellipse as parametric equations and by integration find the length of the upper right quadrant:


ó
õ		   æ
Ö
æ
è		  dy
da
 ö
ø		 2
 
 + æ
è		  dy
db
 ö
ø		 2
 
 
 dy
???
Or is this integral not generalisable for
a = M sin y
b= N cos y

???
By Alex Barnard (Agb21) on Monday, October 18, 1999 - 03:15 pm:

You will find that you can't do the integration using functions that you know about. This integral is in fact one of the original definitions of the elliptic functions. Although nowadays they are defined in a more abstract way.

AlexB.

By Panna Lal Patodia (M152) on Friday, November 5, 1999 - 01:12 am:

Circumference of an ellipse:
===========================
It is true that there is no simple formula to compute circumference of an ellipse. Indian Mathematician, S. Ramanujan gave a wonderful formula which gives a very good approximate value of circumference of an ellipse without using Integration or Summation of Series.

Ramanujan's Formula:
t = (a-b)2 / (a+b)2
S = p (a+b) (1+3t / (10 + sqrt(4 - 3t)))

Below we give some results which will give an idea about the
accuracy of Ramanjuan's Formula:

1) a=5, b=4:
Correct Value = 28.36166788897448463135586391
Ramanujan = 28.36166788897429520546584673
2) a=4, b=3:
Correct Value = 22.10349216070950504528558646
Ramanujan = 22.10349216070766187370358244
3) a=3, b=2:
Correct Value = 15.86543958929058979133166302
Ramanujan = 15.86543958925123398181034084
4) a=2, b=1:
Correct Value = 9.688448220547676198428503196
Ramanujan = 9.688448216130084165990476817

I have developed an algorithm which can give the circumference correct to thousands places and it computes quite fast. This has been derived from Ramanujan's work only. But, it is too complex to be given. I may be submitting this as an article, so look for it, if you want that kind of accuracy. However, for all practical purpose, Ramanujan's formula is sufficient.

P L Patodia, Bangalore, India