where the number  is called the torsion of the curve C at .

We also suspect that torsion measures the degree of twisting that the curve exhibits near appoint, or the degree to which the curve fails to be planar. It may take positive or negative values depending on the right-handedness or left-handedness of the twisting. (see the circular helix)

To get a more concrete and mathematical treatment to this number , we will proof that

where  is the angle between  and  which is in other words saying that the extent to which the unit binormal  changes direction with respect to a given change in arc length is given by the magnitude of torsion  at that point.

From the definition of torsion, we will take the magnitude of both sides of the equation.

Since we know that the magnitude of the unit normal is always 1. We have here now an equation involving the unit binormal in its derivative form. Next, we shall rewrite this in its limits form, as this will greatly facilitate its interpretation.

Let us now have a geometrical representation of the unit binormal and arc length.

Since we are dealing with a small change in s, we thus need to write the unit binormal in terms of . We also introduce the quantity  which is the angle between the vectors  and . Now, recognize that

We now rewrite the limit as

where  is the angle described above. Here comes some good old limits ‘trickery’ as some people call it. Let’s focus on the  and the . Recall that, the arc length subtended by an angle is given by . In this case, since  is unit length, the arc length subtended by  is simply  or . And so, I can argue that as ,  will tend closer towards , and so the length of , the chord, will be equal to , the arc length, or equivalently, their ratios would be 1. Thus

As required.

This equation then tells us that the magnitude of  measures the degree to which the direction of the unit binormal changes at a certain point, as the point moves along the curve.