As shown in Fig. 2.4, the direction of
becomes perpendicular to the tangent
. The unit vector
is moved from
form an isosceles triangle (see Fig. 2.4), since
are unit tangent
vectors. Thus we have
The vector is called the curvature vector, and measures the rate of change of the tangent along the curve. By definition is nonnegative, thus the sense of the normal vector is the same as that of .
The curvature for arbitrary speed (non-arc-length parametrized) curve
can be obtained as follows.
First we evaluate
by the chain rule
For the planar curve, we can give the curvature a sign by defining the normal vector such that form a right-handed screw, where as shown in Fig. 2.5. The point where the curvature changes sign is called an inflection point (see also Fig. 8.3).
According to this definition the unit normal vector of the plane curve
is given by
The normal vector for the arbitrary speed curve can be obtained from , where is the unit binormal vector which will be introduced in Sect. 2.3 (see (2.41)).
The unit principal normal vector and curvature for implicit curves
can be obtained as follows. For the planar curve the normal vector can
be deduced by combining (2.14) and
We will introduce a derivative operator with respect to arc length
so that the derivation becomes simple. If we rewrite the plane
implicit curve as
is arc length along the
implicit curve, the total derivative with respect to the arc length
For a 3-D implicit curve, we can deduce a derivative operator  similar to (2.29),
By applying the derivative operator (2.31) to