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2.1 Arc length and tangent vector

Let us consider a segment of a parametric curve $ {\bf r} = {\bf r}(t)$ between two points $ P$ ( $ {\bf r}(t)$ ) and $ Q$ ( $ {\bf r}(t+\Delta t)$ ) as shown in Fig. 2.1. Its length $ \Delta s$ can be approximated by a chord length $ \vert\Delta{\bf r}\vert = \vert{\bf r}(t+\Delta t) - {\bf r}(t)\vert$ , and by means of a Taylor expansion we have
$\displaystyle \Delta s \simeq \vert\Delta{\bf r}\vert = \vert{\bf r}(t+\Delta t... ...Delta t)^2\right\vert\simeq \left\vert\frac{d{\bf r}}{dt}\right\vert\Delta t\;,$     (2.1)

to the first order approximation.

Figure 2.1: A segment $ \Delta {\bf r}$ connecting two point $ P$ and $ Q$ on a parametric curve $ {\bf r}(t)$
\begin{figure}\centerline{ \psfig{figure=fig/dArcLength.eps,height=3.5in} }\end{figure}

Thus as point $ Q$ approaches $ P$ or in other words $ \Delta t\rightarrow 0$ , the length $ \Delta s$ becomes the differential arc length of the curve:

$\displaystyle ds = \left\vert\frac{d{\bf r}}{dt}\right\vert dt = \vert\dot{\bf r}\vert dt = \sqrt{\dot{\bf r}\cdot \dot{\bf r}} dt\;.$     (2.2)

Here the dot $ \dot{}$ denotes differentiation with respect to the parameter $ t$ . Therefore the arc length of a segment of the curve between points $ {\bf r}(t_o)$ and $ {\bf r}(t)$ can be obtained as follows (provided the function $ t \in [t_0,t]$ $ \rightarrow$ $ {\bf r}(t)$ is one-to-one almost everywhere):
$\displaystyle s(t)= \int_{t_o}^{t}ds = \int_{t_o}^{t}\sqrt{\dot{\bf r}\cdot\dot{\bf r}} dt=\int_{t_o}^{t}\sqrt{\dot{x}^2(t) + \dot{y}^2(t) +\dot{z}^2(t)}dt\;.$     (2.3)

The vector $ \frac{d{\bf r}}{dt}$ is called the tangent vector at point $ P$ . This tangent vector has a simple geometrical interpretation. The vector $ {\bf r}(t+\Delta t) - {\bf r}(t)$ indicates the direction from $ {\bf r}(t)$ to $ {\bf r}(t+\Delta t)$ . If we divide the vector by $ \Delta t$ and take the limit as $ \Delta t\rightarrow 0$ , then the vector will converge to the finite magnitude vector $ \dot{\bf r}(t)$ , i.e. the tangent vector. The magnitude of the tangent vector is derived from (2.2) as

$\displaystyle \vert\dot{\bf r}\vert = \frac{ds}{dt}\;,$     (2.4)

hence the unit tangent vector becomes
$\displaystyle {\bf t} = \frac{\dot{\bf r}}{\vert\dot{\bf r}\vert} = \frac{\frac{d{\bf r}}{dt}}{\frac{ds}{dt}} = \frac{d{\bf r}}{ds} \equiv {\bf r}'\;.$     (2.5)

Here the prime $ '$ denotes differentiation with respect to the arc length. We will keep these notations, i.e. dot $ \dot{}$ is for differentiation with respect to non-arc-length parameter $ t$ and prime $ '$ with respect to arc length parameter $ s$ throughout the book. We list some useful formulae of the derivatives of arc length $ s$ with respect to parameter $ t$ and vice versa:
$\displaystyle \dot{s} = \frac{ds}{dt} = \vert\dot{\bf r}\vert = \sqrt{\dot{\bf r}\cdot\dot{\bf r}}\;,$     (2.6)
$\displaystyle \ddot{s} = \frac{d\dot{s}}{dt} = \frac{\dot{\bf r}\cdot\ddot{\bf r}} {\sqrt{\dot{\bf r}\cdot\dot{\bf r}}}\;,$     (2.7)
$\displaystyle \stackrel{\cdots}{s} = \frac{d\ddot{s}}{dt} = \frac{(\dot{\bf r}\... ...f r} \cdot \ddot{\bf r})^2} {(\dot{\bf r} \cdot \dot{\bf r})^{\frac{3}{2}} }\;,$     (2.8)
$\displaystyle t' = \frac{dt}{ds} = \frac{1}{\vert\dot{\bf r}\vert} = \frac{1}{\sqrt{ \dot {\bf r} \cdot \dot {\bf r}}}\;,$     (2.9)
$\displaystyle t'' = \frac{dt'}{ds} = - \frac{\dot{\bf r}\cdot\ddot{\bf r}} {(\dot{\bf r}\cdot\dot{\bf r})^2}\;,$     (2.10)
$\displaystyle t''' = \frac{dt''}{ds} = - \frac{(\ddot{\bf r}\cdot\ddot{\bf r} +... ...ot{\bf r}\cdot\ddot{\bf r})^2}{(\dot{\bf r}\cdot\dot {\bf r})^{\frac{7}{2}}}\;.$     (2.11)

Definition 2.1.1. A regular (ordinary) point $ P$ on a parametric curve $ {\bf r}={\bf r}(t)= (x(t), y(t), z(t))^T$ is defined as a point where $ \vert\dot{\bf r}(t)\vert\neq 0$ . A point which is not a regular point is called a singular point.

Definition 2.1.2. A parametrization $ {\bf r}={\bf r}(t)= (x(t), y(t), z(t))^T$ of a curve defined in the interval $ I$ is called an allowable representation of class $ r$ [207], if it satisfies the following:

  1. the mapping $ {\bf r}: I \rightarrow {\bf R}^3$ , $ t\mapsto {\bf r}(t)= (x(t), y(t), z(t))^T$ is one-to-one,
  2. the vector function $ {\bf r} = {\bf r}(t)$ is of class $ r \geq 1$ in the interval $ I$ ,
  3. $ \vert\dot{\bf r}(t)\vert\neq 0$ for all $ t\in I$ .

A parametric curve satisfying Definition 2.1.2 is also referred to as a regular curve. The magnitude of the tangent vector $ \frac{ds}{dt}$ can be interpreted as a rate of change of the arc length $ s$ with respect to the parameter $ t$ and is called the parametric speed. If we assume the curve $ \dot{\bf r}(t)$ to be regular, then by definition $ \vert\dot{\bf r}(t)\vert$ is never zero and hence $ \frac{ds}{dt}$ is always positive. When $ \frac{ds}{dt}=1$ , 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 $ t_0,t_1,\cdots,t_N$ corresponding to a uniform increment $ \Delta t= t_k - t_{k-1}$ , 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 ($ PH$ ) 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 $ s(t)$ of the parameter $ t$ . We provide a further review of Pythagorean hodograph curves and surfaces in Sect. 11.4.



Figure 2.2: When parametric speed does not vary significantly, points with uniformly spaced parameter values are nearly uniformly spaced along a parametric curve
\begin{figure}\centerline{ \psfig{file=fig/prm.eps,height=1.0in}}\end{figure}

Definition 2.1.3. A point $ (x_0, y_0)$ of a planar irreducible implicit curve $ f(x,y)=0$ is said to be singular if $ f(x_0,y_0)=f_x(x_0, y_0)=f_y(x_0, y_0)=0$ .

The unit tangent vector for implicit curves can also be derived as follows. First we start with the planar curve $ f(x,y)=0$ . The differential $ df$ of the implicit form $ f=0$ is zero, thus by letting $ f_x=\frac{\partial f}{\partial x}$ and $ f_y=\frac{\partial f}{\partial y}$ we have

$\displaystyle df = f_xdx + f_ydy = 0\;,$     (2.12)

or assuming $ f_y\neq0$ ,
$\displaystyle \frac{dy}{dx}=-\frac{f_x}{f_y}\;.$     (2.13)

Therefore the tangent vector on the implicit curve is given by $ \pm (f_y, -f_x)^T$ , and hence the unit tangent vector is
$\displaystyle {\bf t}=\pm\frac{(f_y, -f_x)^T}{\sqrt{f_x^2 + f_y^2}}\;.$     (2.14)

The sign depends on the sense in which $ s$ increases.

As shown in Table 1.1, an implicit space curve is defined as the intersection of two implicit surfaces, $ f(x,y,z)=0$ and $ g(x,y,z)=0$ . As we will see in Sect. 3.1, the normal vectors of these two implicit surfaces are $ \nabla f$ and $ \nabla g$ , respectively, where the symbol $ \nabla$ represents the gradient vector operator which is of the form $ \nabla =\left(\frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z}\right)^T$ .

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

$\displaystyle {\bf t}=\pm\frac{\nabla f\times \nabla g}{\vert\nabla f\times \nabla g\vert}\;,$     (2.15)

provided that the denominator is nonzero ( $ \nabla f \neq {\bf0}$ and $ \nabla g \neq {\bf0}$ 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 $ s$ 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 $ {\bf r}(t) = (t^2,t^3)^T$ [227]. The parametric speed is evaluated as $ \vert\dot{{\bf r}}(t)\vert = \sqrt{t^2(4+9t^2)}$ . It becomes zero when $ t=0$ , 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 $ f(x,y)= x^3 - y^2=0$ . We can also observe that $ f(0,0)=f_x(0,0)=f_y(0,0)= 0$ .

Figure 2.3: A singular point occurs on a semi-cubical parabola in the form of a cusp
\begin{figure}\centerline{ \psfig{file=fig/cub_par_curv.eps,height=3.0in}}\end{figure}

next up previous contents index
Next: 2.2 Principal normal and Up: 2. Differential Geometry of Previous: 2. Differential Geometry of   Contents   Index
December 2009