Curve whose signed curvature is a function Let $f:[a,b]\rightarrow\mathbb{R}$ be a function. Is it always possible to find a curve whose signed curvature is the function $f$? I know if $f$ is smooth then it can be possible. But I don't know any result for any arbitrary function. Please tell me is there any results. Any type of results of this type will be very helpful.
Thank you. 

Supposing that $s\in [0,a]$ is the arclength parameter of the curve you seek, you want to solve the Frenet equation
$$\frac{d\mathbf T}{ds} = \kappa(s)\mathbf N(s),$$
where $\mathbf T$ is the unit tangent and $\mathbf T(s),\mathbf N(s)$ always form a positively oriented ("right-handed") orthonormal basis for the plane. Once you specify an initial condition (say $\mathbf T(0) = (1,0)$), this system of ordinary differential equations will have a unique solution. You then integrate 
$$\frac{d\alpha}{ds} = \mathbf T(s)$$
(with an initial condition, say $\alpha(0)=(0,0)$) to get the curve.
