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

### Curvature / Torsion

If, in the case of plane curves, a simple example is the car, in the three-dimensional case the most obvious is the motion of an airplane. The aircraft performs the same movements as a car - for example, when it is on the landing strip, the plane can move in a straight line or in circles like a normal car - to which are roughly added the movements of taking off and landing so that it can actually fly.

Consider first the situation where the airplane remains in the same plane as, for example, on the landing strip. In this situation, all properties previously studied remain valid as, for example:

• straight lines have zero curvature;
• circles have constant curvature.

A line and a circle are both plane, but there is a big difference between them: a circle belongs only to a single plane while a straight line belongs to an infinite number of planes.

Since the circle is contained in a unique plane (never "twisting" to try to escape to the plane), mathematicians say that it has no torsion or has zero torsion. Although the straight line is also a plane curve, it has no defined torsion - due to its property of being contained in a multitude of planes.

Imagine an airplane in a steady upward motion and assume that we turn his "steering wheel" to the left and keep it fixed. What is the track of this plane?

Answer: the airplane will make a helical path.

See this applet to check if you know how to "drive" a plane...

But the tracks of a plane are not only plane curves or helices...