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3.1 Tangent plane and surface normal

Let us consider a curve $ u=u(t)$ , $ v=v(t)$ in the parametric domain of a parametric surface $ {\bf r}={\bf r}(u,v)$ as shown in Fig. 3.1. Then $ {\bf r}={\bf r}(t)={\bf r}(u(t), v(t))$ is a parametric curve lying on the surface $ {\bf r}={\bf r}(u,v)$ . The tangent vector to the curve on the surface is evaluated by differentiating $ {\bf r}(t)$ with respect to the parameter $ t$ using the chain rule and is given by
$\displaystyle \dot{\bf r}(t) = {\bf r}_u\dot{u} + {\bf r}_v\dot{v}\;,$     (3.1)

where subscripts $ u$ and $ v$ denote partial differentiation with respect to $ u$ and $ v$ , respectively.
Figure 3.1: The mapping of a curve in 2-D parametric space onto a 3-D parametric surface
\begin{figure}\centerline{ \psfig{file=fig/cos.eps,height=2.0in}}\end{figure}
The tangent plane at point $ P$ can be considered as a union of the tangent vectors of the form (3.1) for all $ {\bf r}(t)$ through $ P$ as illustrated in Fig. 3.2. Point $ P$ corresponds to parameters $ u_p$ , $ v_p$ . Since the tangent vector (3.1) consists of a linear combination of two surface tangents along iso-parametric curves $ {\bf r}_u$ and $ {\bf r}_v$ , the equation of the tangent plane at $ {\bf r}(u_p, v_p)$ in parametric form with parameters $ \mu$ , $ \nu$ is given by
$\displaystyle {\bf T}_p(\mu, \nu) = {\bf r}(u_p, v_p) + \mu {\bf r}_u(u_p, v_p) + \nu {\bf r}_v(u_p, v_p)\;.$     (3.2)

Figure 3.2: The tangent plane at a point on a surface
\begin{figure}\centerline{ \psfig{file=fig/tplane.eps,height=2.0in}}\end{figure}

The surface normal vector is perpendicular to the tangent plane (see Fig. 3.3) and hence the unit normal vector is given by

$\displaystyle {\bf N} = \frac{{\bf r}_u \times {\bf r}_v}{\vert{\bf r}_u \times {\bf r}_v\vert}\;.$     (3.3)

By using (3.3), the equation of the tangent plane at $ {\bf r}(u_p, v_p)$ can be written in the implicit form as
$\displaystyle ({\bf r} - {\bf r}(u_p, v_p))\cdot{\bf N}(u_p, v_p)=0\;,$     (3.4)

where $ {\bf r}$ is a point on the tangent plane.
Figure 3.3: The normal to the point on a surface
\begin{figure}\centerline{\psfig{file=fig/normal2.eps,height=2.0in}}\end{figure}

Definition 3.1.1. A regular (ordinary) point $ P$ on a parametric surface is defined as a point where $ {\bf r}_u \times {\bf r}_v \neq {\bf0}$ . A point which is not a regular point is called a singular point.

The condition $ {\bf r}_u \times {\bf r}_v \neq {\bf0}$ requires that at point $ P$ the vectors $ {\bf r}_u$ and $ {\bf r}_v$ do not vanish and have different directions, i.e. $ {\bf r}_u$ and $ {\bf r}_v$ are linearly independent. As we discussed in Sect. 1.3.6, in some design problems we need to employ triangular patches defined by parametrization over a rectangular domain. Such a degenerated patch can be generated by collapsing one boundary curve into a single point or by arranging for two partial derivatives $ {\bf r}_u$ and $ {\bf r}_v$ at one of the corners of a quadrilateral patch to be collinear. In both cases $ {\bf r}_u \times {\bf r}_v$ has zero magnitude at the degenerate corner point and (3.3) cannot be used. Conditions for the existence of surface normals at these degenerate corner points have been discussed in [116,92,453,457]. The concept of a regular surface requires additional conditions beyond the existence of a tangent plane everywhere on the surface, such as absence of self-intersections. This concept is presented fully in do Carmo [76].

There are essential and artificial singularities [444]. The essential singularities arise from specific features of the surface geometry such as the apex of a cone. The artificial singularities arise from the choice of parametrization.

Example 3.1.1. The elliptic cone can be described in a parametric form $ {\bf r} = (at\cos \theta, bt\sin \theta, ct)^T$ , where $ 0\leq \theta \leq 2\pi$ , $ 0 \leq t \leq l$ and $ a$ , $ b$ , $ c$ are constants. We have

$\displaystyle {\bf r}_{\theta}=(-at\sin \theta, bt\cos\theta,0)^T,\;\;\; {\bf r}_{t}=(a\cos \theta, b\sin\theta,c)^T\;,$      

thus
$\displaystyle \vert{\bf r}_\theta \times {\bf r}_t\vert = \vert bct\cos\theta{\... ...{\bf e}_z\vert = \sqrt{t^2(b^2c^2\cos^2\theta + a^2c^2\sin^2\theta +a^2b^2)}\;.$      

We can easily observe that the surface becomes singular only at $ t=0$ , which corresponds to the apex of the cone.

The unit normal vector for an implicit surface can be derived by considering two parametric curves $ {\bf r}_1 = (x_1(t_1), y_1(t_1), z_1(t_1))^T$ , $ {\bf r}_2 = (x_2(t_2),$ $ y_2(t_2),$ $ z_2(t_2))^T$ lying on an implicit surface $ f(x,y,z)=0$ , and intersecting at point $ P$ on the surface with different tangent directions. Thus we have the relations:

$\displaystyle f(x_1(t_1), y_1(t_1), z(t_1))=0,\;\;\;f(x_2(t_2), y_2(t_2), z(t_2))=0\;.$     (3.5)

Total differentiation of (3.5) with respect to $ t_1$ and $ t_2$ , respectively, yields
$\displaystyle f_x\frac{dx_1}{dt_1} + f_y\frac{dy_1}{dt_1} + f_z\frac{dz_1}{dt_1} = 0\;,$     (3.6)
$\displaystyle f_x\frac{dx_2}{dt_2} + f_y\frac{dy_2}{dt_2} + f_z\frac{dz_2}{dt_2} = 0\;.$     (3.7)

Now if we multiply (3.6) by $ \frac{dx_2}{dt_2}$ and subtract (3.7) multiplied by $ \frac{dx_1}{dt_1}$ , and if we multiply (3.6) by $ \frac{dy_2}{dt_2}$ and subtract (3.7) multiplied by $ \frac{dy_1}{dt_1}$ we can deduce the following relation
$\displaystyle f_x:f_y:f_z = \frac{dz_2}{dt_2}\frac{dy_1}{dt_1}- \frac{dz_1}{dt_... ...dt_1}:\frac{dx_1}{dt_1}\frac{dy_2}{dt_2}- \frac{dx_2}{dt_2}\frac{dy_1}{dt_1}\;,$     (3.8)

which indicates that vector $ \nabla f =(f_x, f_y, f_z)^T$ (also known as gradient of $ f$ ) is in the direction of the cross product of the two tangent vectors at $ P$ , i.e. in the normal direction. Thus the unit normal vector of the implicit surface is given by
$\displaystyle {\bf N} = \frac{(f_x, f_y, f_z)^T}{\sqrt{f_x^2 + f_y^2 + f_z^2}} = \frac{\nabla f}{\vert\nabla f\vert}\;,$     (3.9)

provided that $ \vert\nabla f\vert\neq 0$ .

Alternatively, we can derive (3.9) by considering an arbitrary parametric curve $ {\bf r} = {\bf r}(t)$ on an implicit surface $ f(x,y,z)=0$ , leading to the relation $ \nabla f\cdot \dot{\bf r}=0$ . Since $ {\bf r} = {\bf r}(t)$ is arbitrary, $ \nabla f$ must be perpendicular to the tangent plane, and hence it is a normal vector.

The tangent plane of an implicit surface $ f(x,y,z)=0$ at point $ P$ with coordinates $ (x_p, y_p, z_p)$ can be obtained by replacing the normal vector of parametric surface in (3.4) with (3.9), which leads to

$\displaystyle f_x(x-x_p) + f_y(y-y_p) + f_z(z-z_p) = 0\;,$     (3.10)

where $ f(x_p, y_p, z_p)=0$ and $ f_x$ , $ f_y$ $ f_z$ in (3.10) are evaluated at $ (x_p, y_p, z_p)$ .

Example 3.1.2. The elliptic cone of Example 3.1.1 has also the following implicit representation $ f(x,y,z) = (\frac{x}{a})^2 + (\frac{y}{b})^2 - (\frac{z}{c})^2=0$ . The magnitude of the normal vector $ \nabla f =(\frac{2x}{a^2}, \frac{2y}{b^2}, -\frac{2z}{c^2})^T$ , where $ (x,y,z)\in f(x,y,z)=0$ , becomes 0 only when $ x$ =$ y$ =$ z$ =0 corresponding to the apex of the cone as also derived in Example 3.1.1.

next up previous contents index
Next: 3.2 First fundamental form Up: 3. Differential Geometry of Previous: 3. Differential Geometry of   Contents   Index
December 2009