Originally Posted by

**Luc** Oops, sorry, should have clarified that a bit more.

If the initial circle's circumference is C, then the length of the *segment *from the blue dot to the red dot is C/4, and is constant (because we are considering the path of the red dot).

So I've had some ideas, but no solutions yet.

How about taking the relation

x^2/A^2 + y^2 = ((2L)/pi)^2

where L is the length along the curve from blue dot to red dot.

Since we're letting A go to infinity, we are dealing with x, y, and A as variables.

So my idea is to find x(A), y(A), then jiggle them around to get y(x).

We take L and equate it to the the integral giving the length of the curve, integrating from 0 to y(A),

so

L = [integral from 0 to y(A) of] dy x sqrt(1 + (dx/dy)^2)

then use the first relation to find the expressions for the derivatives and to get everything in terms of 2 variables...

Then you come out with

L = [integral from 0 to y(A) of] dy x sqrt(1 + (Ay)^2/(((2L)/pi)^2 - y^2))

...and that's as far as I'm willing to go for today.

PS if anyone could give me tips on how to post real math text like square roots, exponents, symbols and whatnot that would be great, since I dislike typing out all that garbage as much as you probably dislike deciphering it!