### 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:

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?