2.1 Arc length and tangent vector

to the first order approximation.

Thus as point
approaches
or in other words
, the length
becomes the differential arc
length of the curve:

Here the dot denotes differentiation with respect to the parameter . Therefore the arc length of a segment of the curve between points and can be obtained as follows (provided the function is one-to-one almost everywhere):

The vector
is called the *tangent vector*
at point
. This tangent vector has a
simple geometrical interpretation. The vector
indicates the direction from
to
. If we divide the vector by
and take the
limit as
, then the vector will converge to
the finite magnitude vector
, i.e. the tangent
vector. The magnitude of the tangent vector is derived from
(2.2) as

hence the unit tangent vector becomes

Here the prime denotes differentiation with respect to the arc length. We will keep these notations, i.e. dot is for differentiation with respect to non-arc-length parameter and prime with respect to arc length parameter throughout the book. We list some useful formulae of the derivatives of arc length with respect to parameter and vice versa:

**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
[207], if it satisfies the following:
*

- the mapping , is one-to-one,
- the vector function is of class in the interval ,
- for all .

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 [109],
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
. The
differential
of the implicit form
is zero, thus by letting
and
we have

(2.12) |

or assuming ,

Therefore the tangent vector on the implicit curve is given by , and hence the unit tangent vector is

The sign depends on the sense in which increases.

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

provided that the denominator is nonzero ( and or in other words the two surfaces are nonsingular and the surfaces are not tangent to each other at their common point under consideration). The unit tangent vector of the intersection of two implicit surfaces, when the two surfaces intersect tangentially is given in Sect. 6.4. Also here the sign depends on the sense in which increases. A more detailed treatment of the tangent vector of implicit curves resulting from intersection of various types of surfaces can be found in Chap.6.

*Example 2.1.1*
The semi-cubical parabola, which is
illustrated in Fig. 2.3, can be represented in
parametric form as the curve
[227]. 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
.