2.2 Principal normal and curvature

If is an arc length parametrized curve, then is a unit vector (see (2.5)), and hence . Differentiating this relation, we obtain

which states that is orthogonal to the tangent vector, provided it is not a null vector. This fact can be also interpreted from the definition of the second derivative

Figure 2.4: Derivation of the normal vector of a curve (adapted from [455])

As shown in Fig. 2.4, the direction of becomes perpendicular to the tangent vector as . The unit vector

which has the direction and sense of is called the unit principal normal vector at . The plane determined by the unit tangent and normal vectors and is called the osculating plane at . It is also well known that the plane through three consecutive points of the curve approaching a single point defines the osculating plane at that point [412].

When is moved from to , then , and form an isosceles triangle (see Fig. 2.4), since and are unit tangent vectors. Thus we have as and hence

is called the curvature , and its reciprocal is called the radius of curvature at . It follows that

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 and by the chain rule

where is the parametric speed. Taking the cross product of and we obtain

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).

Figure 2.5: Normal and tangent vectors along a 2D curve

According to this definition the unit normal vector of the plane curve is given by

and hence from (2.23) we have

For a space curve, by taking the norm of (2.23) and using (2.4), we obtain

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 (2.24) yielding

where only the sign of was used (although it is not necessary).

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 where is arc length along the implicit curve, the total derivative with respect to the arc length becomes

Now if we replace and by using (2.5) and (2.14) (+ sign), we obtain the derivative operator with respect to arc length

By applying the operator (2.29) to (2.14) (+ sign) and equating with (using (2.20) and (2.27)), we obtain

For a 3-D implicit curve, we can deduce a derivative operator [444] similar to (2.29),

where is the tangent vector of the 3-D implicit curve (see (2.15)) given by

and

By applying the derivative operator (2.31) to     we obtain

which gives

Taking the cross product of     and (2.37) yields

Thus,

A different derivation of the curvature of a 3-D implicit curve is given in Sect. 6.3.2.