Oh what is there to think about? It is one of the "kid" curves our ancestors figured out, right?Although I am an Indian, most likely my ancestors too - not just Greeks - at any rate it is not as complicated as exponentials and logarithms which form the basis of decimal system - anyone remember having problems with carry in addition in their second standard? [include me as your brethern]
Of course figuring out length of the curve is the most simplest thing taught to me 13-14 years ago at which time of course I sort of cursed myself, that I didn't figure such a simple thing myself - it is just pythogoras thorem
dr^2 = dx^2 + dy^2
and of course I want to find r from dr just integrate over it, so simple ah?
Anyway to come back to main story - Yesterday after reading about classical mechanics and then just to relax I was reading Dawkins' God delusion - [A very good book BTW to give to your religious friends] and somehow my attention turned to earth's orbit and I was thinking why would people think orbit should be circle, especially if it is very improbable configuration among all ellipses. Then I wondered what is the average velocity with which we fly through the space.
It is simply our [earth's] orbit length divided by 365 days! But I didn't know orbit length.
But I knew it is an ellipse - So I wanted to calculate the circumference of the ellipse.
So I duly wrote the equation
x^2/a^2+y^2/b^2=1
and started integrating
sqrt(1+(dy/dx)^2) dx
I didn't even have a hint of what is coming. So I got a nasty expression in terms of x and y s and I thought converting into parameteric form may be easier to solve.
I set x= acos(theta)
...
to cut a long story short
I ended up at
and whatever I do I couldn't solve it. Then today I came and looked on the net [cheating] for circumference of ellipse and sure enough they stop here too and call it 'complete elliptic [obviously] integral of second kind' http://en.wikipedia.org/wiki/Ellipse
it has only an infinite series solution and an approximation by Ramanujan!
Sometimes things we take for granted turn out to be so complex!
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