Parametric Surfaces

Parameterization of a surface.

A parameterization for a surface \(S\) in \({\mathbb R}^3\) is a function of two variables \(\vec{r} \colon D \rightarrow {\mathbb R}^3\) where:

• \(D\) is a region in \({\mathbb R}^2\text{.}\)
• For a given \((u,v)\) in the region \(D\text{,}\) the image \(\vec{r}(u,v)\) is interpreted as a point in \({\mathbb R}^3\text{.}\) These are the points which make up the surface \(S\text{.}\)
• As we allow \((u,v)\) to vary over all the points in \(D\text{,}\) the image \(\vec{r}(u,v)\) varies over all the points in \(S\text{.}\) Moreoever, no two points \((u,v)\) map to the same point of \(S\) under \(\vec{r}\text{:}\) i.e. \(\vec{r}\) is a one-to-one function.

Standard surface parameterizations.

Some standard surface parameterizations are the following:

• Planes. If \(\Pi\) is a plane containing the point \(P\) and the non-parallel vectors \(\vec{u}\) and \(\vec{v}\text{,}\) then the equation
  \begin{equation*} \vec{r}(s,t) =\vec{P} +\vec{u} t +\vec{v} s \end{equation*}
  is a parameterization for \(\Pi\text{.}\)

• Graphs. If a surface \(S\) is the part of the graph of a function \(z=f(x,y)\) above the region \(D\) in the \(xy\)-plane, then \(S\) can be parameterized by choosing \(x=u\text{,}\) \(y=v\) and \(z=f(u,v)\) where the \(uv\)-bounds are the same as the \(xy\)-bounds in \(D\text{.}\) That is, \(S\) is parameterized by

  \begin{equation*} \vec{r}(u,v) =u\vec{i} +v\vec{j} +f(u,v)\vec{k}. \end{equation*}

  The parameterization above can be easily modified for graphs of the type \(x=f(y,z)\) or \(y=f(x,z)\text{.}\)

• Surfaces generated by circles. If a surface \(S\) has cross sections which consist of concentric circles (or part of concentric circles) centered on the \(z\)-axis whose radii are a function of height \(f(z)\text{,}\) then \(S\) can be parameterized by letting \(z=v\text{,}\) \(x=f(v)\cos{(u)}\) and \(y=f(v)\sin{(u)}\text{.}\) That is, \(S\) is parameterized by

  \begin{equation*} \vec{r}(u,v) =f(v) \cos{(u)} \vec{i} +f(v)\sin{(u)} \vec{j} +v\vec{k}. \end{equation*}

  Since \(u\) acts as an angle of rotation and \(v\) acts as height, we can find bounds for \(u\) and \(v\) accordingly. Examples of such surfaces include spheres, cones and cylinders.

  The parameterization above can be easily modified with shifts for more general surfaces whose cross sections are concentric circles. In addition, they can also be modified to cases where the concentric circles live in planes parallel to the \(xz\)-plane or \(yz\)-plane by changing the “height” to be \(y\) or \(x\) respectively and adjusting the components which have the trigonometric functions \(f(v)\sin (u)\) and \(f(v)\cos (u)\) accordingly.

Tangent and normal vectors to parameterized surfaces.

If \(\vec{r}(u,v)\) is a parameterization for a surface \(S\) and \(P=\vec{r}(a,b)\) is a point on \(S\text{,}\) then:

• \(\vec{r}_{u}(a,b)\) and \(\vec{r}_{v}(a,b)\text{,}\) , are tangent vectors to the surface \(S\) at the point \(P\text{.}\)
• \(\vec{N} =\vec{r}_{u}(a,b) \times \vec{r}_{v}(a,b)\text{,}\) when non-zero, is a normal vector to the surface \(S\) at \(P\text{.}\)

In particular, one can use \(\vec{N}\) to find the equation for the tangent plane to \(S\) at point \(P\text{.}\)

Smooth surfaces.

A surface \(S\) parameterized by \(\vec{r}(u,v)\) is said to be smooth at a point \(P=\vec{r}(a,b)\) if \(\vec{r}\) is differentiable at \(P\text{,}\) and the normal vector \(\vec{N} (a,b)\neq \vec{0}\text{.}\) It is a smooth surface if it is smooth at all points.

Orientable surfaces.

A surface \(S\) is orientable if it is possible to choose a unit normal vector \(\vec{n}\) at every point so that \(\vec{n}\) varies continuously over \(S\text{.}\) Such a choice of normal vectors is called an orientation of \(S\text{,}\) and once an orientation of \(S\) has been specified, it is called an oriented surface.

Suppose that \(S\) is an oriented surface with orientation \(\vec{n}\) and suppose that \(\vec{r}(u,v)\) parameterizes \(S\text{.}\) Then we say \(\vec{r}\) is orientation-preserving if \(\vec{n} = \vec{N}/||\vec{N}||\) and orientation-reversing if \(\vec{n} =- \vec{N}/||\vec{N}||\text{.}\)