The curvature and torsion: how to distinguish the shape of a curve

Curvature

Let's imagine ourselves driving a car. If we do not turn the steering wheel to the left or to the right, what is the path?

Obviously the car goes straight in front.

We say that a line (or a line segment) has no curvature or has zero curvature (since to keep this path we have not turned the wheel).

Now suppose we turn the wheel to the left and keep it in that position. What will be the path of our car?

After some time we are again at the starting point, after travelling along a circular path.

We say that a circle has constant curvature (we keep that path if we "curve" always at the same pace).

And if at the beginning we turn the wheel further to the left than before?

As in the previous case, we would again go along a circular path.

But this new circle has a smaller radius: as it is necessary to "bend" more to follow this new circular path, we say that the second circle has more curvature than the first.

If we had initially turned the wheel to the right, we would have a perfectly similar situation and the circles would be equal to the previous ones.

The value of the curvature in both situations is the same: the circular paths are equal (independently of the side to which we turn the wheel: left or right).

But then how can we distinguish the two situations? The solution is to impose that the absolute value of the curvature is the same in both cases and to distinguish them by the sign: the curves that bend to the left have positive curvature while the ones that bend to the right have negative curvature.

In summary, the sign of the curvature encodes the information about the side to which the curve bends, while its absolute value measures the intensity of the bending. The following properties hold:

1) Straight lines have null curvature.

2) Circles have constant (nonzero) curvature.

3) Circles travelled counterclockwise have positive curvature.

4) Circles travelled clockwise have negative curvature.

5) The smaller the radius of the circle, the greater the absolute value of its curvature.

See this steering wheel to confirm these properties.