The 1/4 of the ellipse's arc length is known (in this case) and so is the semi-major axis
. So, for each
? (Headbang)
Here is (the picture of) the equation:
http://img34.imageshack.us/img34/7319/giflatex7.gif
You don't get any "analytic" form for that. That is an example of a general type of integral called (surprize, surprize!) elliptic integrals. They cannot be done "analytically". You would need to do a numerical integration. (I remember, many years ago, seeing 20 large volumes at the University of Florida library giving values for elliptic integrals with different parameters.)
I was looking at (among other methods; I don't really know what I'm doing (Blush)) Newton's method ... but, I'm at lost as to how to use it in practice (as in above equation)?
How would you go about it, which algorithm would you use? Just a rough outline will (I hope) do.