Section 9.1 Parametric Representation of Surfaces
As noted in the introduction to the present chapter, a surface, \(S\text{,}\) is a deformed plane domain, \(D\subset\mathbb R^2\text{,}\) lying in space. Hence to describe \(S\) mathematically we need to say which point in \(D\) corresponds to what point on \(S\text{,}\) just as a map of the world (represented by \(D\)) shows the geographic locations in the real world (the surface \(S\)). This correspondence defines a (vector valued) one-to-one function \(\vect g\) from \(D\) to \(\mathbb R^3\) whose image is \(S\text{.}\) The function \(\vect g\colon D\to\mathbb R^3\) is called a parametrisation of \(S\text{.}\) Some surfaces cannot be represented by one single parametrisation, or it is not convenient to do so. (To represent the world we often use a collection of maps, an atlas, and not a single map of the whole world.) As with parametrisations of a curve, a parametrisation of a surface is never unique! We call a representation by means of a parametrisation a parametric representation of a surface.
The above class of surfaces is too big. We only want to work with smooth (or at least piecewise smooth) surfaces. To define exactly what we mean by a smooth surface let us assume that \(\vect g\colon D\to\mathbb R^3\) is the parametrisation of (part of) a surface. Suppose that \(\vect a=(a_1,a_2)\in D\text{,}\) so that \(\vect g(\vect a)\) is the corresponding point on \(S\text{.}\) Then the map
\begin{equation*}
x_1\to\vect g(x_1,a_2)
\end{equation*}
is the parametrisation of a curve on \(S\) through \(\vect g(\vect a)\text{.}\) Similarly, the map
\begin{equation*}
x_2\to\vect g(a_1,x_2)
\end{equation*}
is the parametrisation of another curve on \(S\) through \(\vect g(\vect a)\text{.}\) In order for \(S\) to be smooth we assume that these curves are smooth. More precisely this means that the above are regular parametrisations, so in particular we require that
\begin{equation*}
\vect v_i:=\frac{\partial}{\partial x_i}\vect g(\vect a)\neq\vect 0
\end{equation*}
for
\(i=1,2\text{.}\) For
\(S\) to be smooth we require more. If we are very unlucky, the image of
\(D\) is only a curve, and not a surface. In that case the vectors
\(\vect v_1\) and
\(\vect v_2\) are parallel. To prevent this we require in addition to the above that
\(\vect v_1\) and
\(\vect v_2\) not be parallel. Both of the above requirements are satisfied if and only if the area of the parallelogram spanned by
\(\vect v_1\) and
\(\vect v_2\) is nonzero. In
Theorem 1.17 we derived a formula to compute the area of a parallelogram. We have to form the matrix with columns
\(\vect v_1\) and
\(\vect v_2\text{,}\) which in our present case is the Jacobian matrix
\(J_{\vect g}(\vect a)\) of
\(\vect g\) at
\(\vect a\text{.}\) According to
Theorem 1.17 the surface area spanned by
\(\vect v_1\) and
\(\vect v_2\) is
\begin{equation*}
\sqrt{\det\bigl(\bigl(J_{\vect g}(\vect y)\bigr)^T J_{\vect g}(\vect y)\bigr)}\text{.}
\end{equation*}
Because of its significance in the theory of surfaces, or more generally in differential geometry, the the square root has a name.
Definition 9.1.
Suppose that \(\vect g\colon\mathbb R^k\to\mathbb R^N\) is a differentiable map. Then the expression
\begin{equation*}
\sqrt{\det\bigl(\bigl(J_{\vect g}(\vect y)\bigr)^T
J_{\vect g}(\vect y)\bigr)}
\end{equation*}
is called the Jacobian of the parametrisation \(\vect g\) at \(\vect y\text{.}\)
Definition 9.2.
Suppose that \(\vect g\colon D\to\mathbb R^3\) is the parametrisation of (part of) a surface. We say that \(\vect g\) is a regular parametrisation of (part of) \(S\text{,}\) if the Jacobian of \(\vect g\) is nowhere zero. Finally, \(S\) is called a smooth surface, if it can be fully described by one (or several) regular parametrisation. If several parametrisations are required then their images on \(S\) must not overlap by more than just a line.
We also sometimes look at piecewise smooth surfaces which are finite unions of smooth surfaces.
Example 9.3. Jacobian of a sphere.
Suppose that
\(S\) is the sphere of radius
\(R\) centred at the origin in
\(\mathbb R^3\text{.}\) We can describe every point on that sphere by indicating two angles, one from the vertical, the other from the horizontal. (Called latitude and longitude on a map of the world). According to the considerations in
Subsection 6.4.1 a parametrisation is given by
\begin{equation*}
\vect g(\theta,\varphi)
:=\bigl(R\cos\varphi\sin\theta,R\sin\varphi\sin\theta,R\cos\theta\bigr),
\end{equation*}
where
\((\theta,\varphi)\in D:=[0,\pi]\times [0,2\pi)\text{.}\) This is the same as
Subsection 6.4.1, except that we keep the radius constant. We next show that
\(\vect g\) is a regular parametrisation. To do so we compute the Jacobian matrix
\begin{equation*}
J_{\vect g}(\theta,\varphi)=R
\begin{bmatrix}
\cos\varphi\cos\theta & -\sin\varphi\sin\theta \\
\sin\varphi\cos\theta & \cos\varphi\sin\theta \\
-\sin\theta & 0
\end{bmatrix}\text{.}
\end{equation*}
Now we compute the product
\begin{align*}
\bigl(J_{\vect g}\bigr)^T J_{\vect g}
&=R^2
\begin{bmatrix}
\cos\varphi\cos\theta & \sin\varphi\cos\theta & -\sin\theta \\
-\sin\varphi\sin\theta & \cos\varphi\sin\theta & 0
\end{bmatrix}
\begin{bmatrix}
\cos\varphi\cos\theta & -\sin\varphi\sin\theta \\
\sin\varphi\cos\theta & \cos\varphi\sin\theta \\
-\sin\theta & 0
\end{bmatrix}\\
&=R^2
\begin{bmatrix}
1 & 0 \\
0 & \sin^2\theta
\end{bmatrix}
\end{align*}
and thus
\begin{equation*}
\det\bigl(\bigl(J_{\vect g}\bigr)^T J_{\vect g}\bigr)
=R^4\det
\begin{bmatrix}
1 & 0 \\
0 & \sin^2\theta
\end{bmatrix}
=R^4\sin^2\theta\text{.}
\end{equation*}
As \(\sin\theta\neq 0\) for \((\theta,\varphi)\in (0,\pi)\times[0,2\pi)\text{,}\) the Jacobian is nonzero except for the points at the point on top and the bottom of the sphere (\(\theta=0,\pi\)). For most purposes these exceptional points are irrelevant.
