Curve from curvature It is possible to obtain the parameters of a curve in 2d simply by having only its curvature k(s)?
I need to obtain its parametric equations in order to reconstruct the curve but i don't have any idea how or even if its possible?
I try to search on internet but i couldn't find anything related to the problem.I even look into different differential geometry books but i couldn't found anything related to the problem
 A: If you have some initial value like the starting Point $c_0$ of the curve you can reconstruct the curve. If $c(s)$ is the curve parametrized by $s$ (for simplicity, it can be assumed that the curve is parametrized by arclength) then it holds the following equation (if $n(s)$ is the unit normal vector):
$c''(s) = k(s) n(s)$. (' is derivative by $s$)
For arclength-parametrized curves you can assume $c(s) = (cos(\theta(s)),sin(\theta(s)))$ for a function $\theta$. By using above equation you will get the differential equation:
$\frac{d}{ds} \theta(s) = k(s)$. By Integration you will obtain $c'$; integrate once again with an initial condition $c(0) = c_0$ and you have the curve.
A: The answer from @kryomaxim is the standard mathematical one. The equation that gives curvature as a function of arclength is known as the "intrinsic equation" of the curve, or sometimes the "Cesàro equation". You can start reading about this here.
But, in practice the mathematical theory by itself is not sufficient. Even with a very simple equation like $\kappa(s) = cs$ (curvature is a linear function of arclength), constructing the curve will require you to numerically compute some nasty integrals. In this particular case, you have to compute Fresnel integrals, and you'll get a curve called a Cornu spiral. These sorts of spirals are used in the design of roads and railway tracks, so they're important.
Various sorts of spirals (with simple Cesàro equations) are also used in font design. See Ralph Levien's thesis for a very nice account of all this.
A: The simple answer to your question is, yes. You can express a two-dimensional curve from the curvature as follows. I choose to use complex variables. The equation for the curve is given by
$$z(s)=\int e^{iP(s)} ds$$
where $\kappa (s)= P'(s)$, i.e., the derivative. For a more detailed description, please see my answer to a post on the Cornu spiral here: Is this Cornu spiral positively oriented or not?. As pointed out above, you need to specify an initial condition, but it will not change the shape of the curve, merely where it resides on the plane.
