## Curvature integration

Hello all.
in a technical application I've to calculate the path of a curve starting from its curvature. In mathematical terms:

(X''(L)).^2+(Y''(L)).^2 = (C(L)).^2

L is the curvilinear abscissa , C(L) is the curvature function

After a bit of thinking I've reached a partial result imposing that:

(X'(L)).^2+(Y'(L)).^2=1

that is, the modulus of the tangent vector is 1, as it should be if L is the curvilinear abscissa.

By this assumptions, I've obtained:

F1'(L)=C(L)*sqrt(1-F1(L)^2);
F2'(L)=F1(L)

where:

F2(L)=Y(L)

Once the system is solved (by a numerical methods for ODE systems),
I can get X'(L)

X'(L)=sqrt(1-Y'(L))

and integrating it X(L)

When I put these equations in a numerical routine for the solution of ODE systems, troubles arise from the square roots (which sign?) and from the condition Y'(L)=1, which leads to Y''(L)=0, a condition in which the system does not evolve any more.

The final question is: is there a way to express the system so that these pitfalls are avoided? Are there softwares/mathematical routines/internet links/ documents... etc, which, in a more general way, deals with curvature integration?

Thank you very much!
John