2.3 Binormal vector and torsion

Figure 2.6: The tangent, normal, and binormal vectors define an orthogonal coordinate system along a space curve

In Sects. 2.1 and 2.2, we have introduced the tangent and normal vectors, which are orthogonal to each other and lie in the osculating plane. Let us define a unit binormal vector such that form a right-handed screw, i.e.

which is shown in Fig. 2.6. The plane defined by normal and binormal vectors is called the normal plane and the plane defined by binormal and tangent vectors is called the rectifying plane (see Fig. 2.6). As mentioned before, the plane defined by tangent and normal vectors is called the osculating plane. The binormal vector for the arbitrary speed curve with nonzero curvature can be obtained by using (2.23) and the first equation of (2.40) as follows:

The binormal vector is perpendicular to the osculating plane and its rate of change is expressed by the vector

where we used the fact that .

Since is a unit vector , we have . Therefore is parallel to the rectifying plane ( ), and hence can be expressed as a linear combination of and :

Thus, using (2.42) and (2.43), we obtain

The coefficient is called the torsion and measures how much the curve deviates from the osculating plane. By taking the dot product with , we obtain the torsion of the curve at a nonzero curvature point

where (2.20) is used and is a triple scalar product. ^2.1

The torsion for an arbitrary speed curve is given by

The evaluation of torsion when curvature vanishes is discussed in Sect. 6.2.

While the curvature is determined only in magnitude, except for plane curves, torsion is determined both in magnitude and sign. Torsion is positive when the rotation of the osculating plane is in the direction of a right-handed screw moving in the direction of as increases. If the torsion is zero at all points, the curve is planar.

The binormal vector of a 3-D implicit curve can be obtained from (2.38) as follows:

The torsion for a 3-D implicit curve can be derived by applying the derivative operator (2.31) to (2.38) [444], which gives

and therefore

Taking the dot product with (2.37) we obtain

from which we calculate . An alternative approach for evaluating the torsion of 3-D implicit curves is presented in Sect. 6.3.3.

Example 2.3.1 A circular helix in parametric representation is given by . Figure 2.7 shows a circular helix with , for . The parametric speed is easily computed as , which is a constant. Therefore the curve is regular and its arc length is

We can easily reparametrize the curve with arc length by replacing by yielding . The first three derivatives are evaluated as

The curvature and torsion are evaluated as follows:

Note that the circular helix has constant curvature and torsion and when , it is a right-handed helix while when , it is a left-handed helix.

Footnotes

... product^2.1

A triple scalar product is numerically equal to the volume of the parallelepiped having the edge vectors , and , and is given by

Also a cyclic permutation maintains the value of the triple scalar product: