On 1/12/10, Fred lunnon <fred.lunnon@gmail.com> wrote:

I haven't tested this, but it looks straightforward enough ...

Using standard schoolroom stuff,

And heading rapidly towards the bottom of the class ...

curvature = dt / ds , where t = tangent angle, s = arc length; and tan t = sin t / cos t = dy / dx .

Missed out a step there: the point is, tan t = dy/dx, and ds^2 = dx^2 + dy^2 ; so dx/ds = cos t, dy/ds = sin t .

Since your curvature is a polynomial function f(s) of arc length, integrating t = g(s) , where dg/ds = f ;

where g(s) is also a polynomial, OK so far ...

integrating again (and quietly losing an irrelevant sign), x = sin g(s), y = cos g(s) .

Nooo! In fact, all we can say is x = \int cos g(s) ds, y = \int sin g(s) ds, which in general won't be integrable in terms of elementary functions. At this point, I'd suggest reaching for a numerical integrator. But if you want to press on, there's things called Fresnel and Lommel functions ... WFL