Still the curvature (I)

Imagine a car, moving at constant speed 1, that describes the following path:

The car initially moves straight; then turn left and turn right and then backs to move in a straight line. Then:

time \(t_{0}\)  \(t_{1}\)  \(t_{2}\)  \(t_{3}\)  \(t_{4}\)
car direction   front   left   right   front  
sign of curvature   0   +   -   0  

Therefore, the curvature is given by a graph of the following type:

To find out if you can identify the curvature function of a given curve, play the following games:

easy game

hard game

In fact, any smooth curve has its curvature function. To construct it, assume a particle ("our car") is describing the curve with constant speed 1 and observe how the curve changes with respect to its successive tangent lines.

Is the reciprocal also true? Given an arbitrary curvature function, is it possible to identify a curve whose curvature function is precisely the given function? And if it exists, is it unique?