Thus as point
or in other words
, the length
becomes the differential arc
length of the curve:
is called the tangent vector
. This tangent vector has a
simple geometrical interpretation. The vector
indicates the direction from
. If we divide the vector by
and take the
, then the vector will converge to
the finite magnitude vector
, i.e. the tangent
vector. The magnitude of the tangent vector is derived from
Definition 2.1.1. A regular (ordinary) point on a parametric curve is defined as a point where . A point which is not a regular point is called a singular point.
Definition 2.1.2. A parametrization of a curve defined in the interval is called an allowable representation of class , if it satisfies the following:
A parametric curve satisfying Definition 2.1.2 is also referred to as a regular curve. The magnitude of the tangent vector can be interpreted as a rate of change of the arc length with respect to the parameter and is called the parametric speed. If we assume the curve to be regular, then by definition is never zero and hence is always positive. When , the curve is said to be arc length parametrized or to have unit speed. If the parametric speed does not vary significantly, points of the curve obtained at parameter values corresponding to a uniform increment , will be nearly evenly distributed along the curve, as illustrated in Fig. 2.2. It is well known that every regular curve has an arc length parametrization , however, in practice it is very difficult to find it analytically, due to the fact that (2.3) is hard to integrate analytically. Pythagorean hodograph ( ) curves, introduced by Farouki and Sakkalis [108,110], form a class of special planar polynomial curves whose parametric speed is a polynomial. Accordingly, its arc length is a polynomial function of the parameter . We provide a further review of Pythagorean hodograph curves and surfaces in Sect. 11.4.
Definition 2.1.3. A point of a planar irreducible implicit curve is said to be singular if .
The unit tangent vector for implicit curves can also be derived
as follows. First we start with the planar curve
of the implicit form
is zero, thus by letting
As shown in Table 1.1, an implicit space curve is defined as the intersection of two implicit surfaces, and . As we will see in Sect. 3.1, the normal vectors of these two implicit surfaces are and , respectively, where the symbol represents the gradient vector operator which is of the form .
Since the tangent vector to the
intersection curve is orthogonal to the normals of the two implicit
surfaces, the unit tangent vector is given by
Example 2.1.1 The semi-cubical parabola, which is illustrated in Fig. 2.3, can be represented in parametric form as the curve . The parametric speed is evaluated as . It becomes zero when , hence it is singular at the origin and forms a cusp, which is illustrated in Fig. 2.3. The curve can be also represented implicitly . We can also observe that .