The circumference of an ellipse

[deep838]

I was wondering about how to find the circumference of an ellipse.
I googled for it and found this: http://paulbourke.net/geometry/ellipsecirc/

That got me pretty amazed! I don't know high level maths, but still, can someone please explain to me why the circumference takes such a complicated form?

I mean, why shouldn't it simply be π*(a+b), where a & b are the major and minor axis.

 

[Simon Bridge]

http://www.mathsisfun.com/geometry/ellipse-perimeter.html
The best way to understand why the ellipse is so difficult to work out a formula for, try figuring out one for yourself.
Apart from that it is difficult to figure what sort of answer you are expecting: it is the property of an ellipse to be like that just like it is the property of a circle to have an irrational ratio of circumference to diameter.

 

[HallsofIvy]

I was wondering about how to find the circumference of an ellipse.
I googled for it and found this: http://paulbourke.net/geometry/ellipsecirc/

That got me pretty amazed! I don't know high level maths, but still, can someone please explain to me why the circumference takes such a complicated form?

I mean, why shouldn't it simply be π*(a+b), where a & b are the major and minor axis.

What you are really asking is "why isn't everything trivial?". To which the only reasonable answer is "why should it be?"

 

[jbunniii]

To get some intuition regarding why the circumference is not simply $\pi(a+b)$, consider a highly eccentric ellipse, say with $b >> a$. The circumference should not be very different from $4b$, because the ellipse consists of two arcs from $(0,b)$ to $(0,-b)$ (assuming appropriately chosen coordinates) which are nearly straight line segments, each of length $2b$. Thus the circumference should be close to $4b$, whereas your proposed formula gives $\pi(a+b) \approx \pi b$.

Therefore, for $b >> a$, the $\pi(a+b)$ formula would need to be multiplied by a correction factor of approximately $4/\pi \approx 1.27$.

Compare this with the "better" approximation given here, for example: http://en.wikipedia.org/wiki/Ellipse#Area

$$\pi(a+b) \left(1 + \frac{3\left(\frac{a-b}{a+b}\right)^2}{10 + \sqrt{4 - 3\left(\frac{a-b}{a+b}\right)^2}}\right)$$
We may view the expression in the large parentheses as a correction factor applied to $\pi(a+b)$. If $b>>a$ we may approximate $a \approx 0$ in that expression, and the result is
$$1 + \frac{3}{10 + \sqrt{4 - 3}} = 1 + \frac{3}{11} \approx 1.27$$

 

[deep838]

hmm... thanks to all of you... especially jbuniii ! i can now see what i wasn't seeing before! and forgive me if I'm asking too much, but can any of you provide me a link to where this/these expressions are derived?