2.3 Binormal vector and torsion

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

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

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:

(2.49) |

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

(2.50) |

and therefore

(2.51) |

Taking the dot product with (2.37) we obtain

(2.52) |

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.

- ...
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

(2.46)

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

(2.47)