# Math Help - Ellipse: extract "minor axis" (b) when given "arc length" and "major axis" (a)

1. ## Ellipse: extract "minor axis" (b) when given "arc length" and "major axis" (a)

The 1/4 of the ellipse's arc length is known ( $3$ in this case) and so is the semi-major axis $a$. So, for each $a$ that would be plugged in, how would you get $b$?

Here is (the picture of) the equation:

2. 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.)

3. I was looking at (among other methods; I don't really know what I'm doing ) 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.