Section 9.2 Implicit and Explicit Representations of Surfaces
Quite often, surfaces are given as the set of points \(\vect x=(x_1,x_2,x_3)\in\mathbb R^3\) such that \(f(\vect x)=0\) for some given function \(\vect f\colon\mathbb R^3\to\mathbb R\text{.}\) We call this an implicit representation of a surface. In some cases it is possible to solve \(f(\vect x)=0\) for one of the variables. For instance we can solve for \(x_3\text{.}\) If we can solve with one single function then \(S\) is the graph of a function. Graphs of a function \(h\) defined on a subset \(D\subset\mathbb R^3\) are the simplest possible surfaces. We call \(x_3=h(x_1,x_2)\text{,}\) \((x_1,x_2)\in D\) an explicit representation of the surface. In many cases, however, solving for one variable will only be possible locally, even for very simple surfaces like the sphere.
Example 9.5. Implicit representation of a sphere.
Suppose that \(S\) is a sphere of radius \(R\text{.}\) By definition \(S\) is the set of points in \(\mathbb R^3\) with distance \(R\) from the origin. Hence \(S\) is the set of all points \(\vect x=(x_1,x_2,x_3)\in\mathbb R^3\) such that
\begin{equation*}
f(\vect x):=x_1^2+x_2^2+x_3^2-R^2=0,
\end{equation*}
providing an implicit representation of the sphere. We can solve the above equation for \(x_3\) to get
\begin{equation*}
x_3=\pm\sqrt{R^2-x_1^2-x_2^2}\text{.}
\end{equation*}
Note that we need two different functions to describe the upper and the lower half of the sphere. The above provides an explicit representation of \(S\text{.}\)
Not for every function
\(f\) does the equation
\(f(\vect x)=0\) describe a smooth surface. For instance the only solution to
\(x_1^2+x_2^2+x_3^2=0\) is
\((0,0,0)\text{,}\) that is, a single point. Hence we need to find criteria which guarantee that the solutions of
\(f(\vect x)=0\) form a surface. We have the following theorem, which is a generalised form of the implicit function
Theorem 4.33, where also the ideas for a more general result are indicated.
Theorem 9.6. Implicit function theorem.
Suppose that \(f\colon\mathbb R^3\to\mathbb R\) is a function with continuous gradient. Moreover, suppose that \(\grad f(\vect x)\neq\vect 0\) for all \(\vect x\in\mathbb R^3\) satisfying \(f(\vect x)=0\text{.}\) Then the set of points \(\vect x\in\mathbb R^3\) for which \(f(\vect x)=0\) forms a smooth surface.
In the above example of the sphere we have \(\grad f(\vect x)=(2x_1,2x_2,2x_3)\text{,}\) which is clearly nonzero if \(R>0\) and \(x_1^2+x_2^2+x_3^2=R\text{.}\)
As mentioned already smooth surfaces, in principle, can be represented by all three means. We consider some examples.
Example 9.7. Representations of a Sphere.
We already saw three different representations of the sphere with radius \(R\) centred at \(\vect 0\text{:}\)
Example 9.8. Graph of a function.
Suppose that \(h\colon D\to\mathbb R\) is continuously differentiable on the open set \(D\subset\mathbb R^2\text{.}\) Then the graph
\begin{equation*}
S:=\{(x_1,x_2,h(x_1,x_2))\colon (x_1,x_2)\in D\}
\end{equation*}
forms a smooth surface in \(\mathbb R^3\text{.}\) Then the three possible representations are as follows:
explicit representation is \(x_3=h(x_1,x_2)\text{,}\) \((x_1,x_2)\in D\text{;}\)
implicit representation is \(f(\vect x):=x_3-h(x_1,x_2)=0\text{;}\)
parametric representation is \(\vect g(x_1,x_2)=(x_1,x_2,h(x_1,x_2))\) for \((x_1,x_2)\in D\text{.}\)
To check that \(S\) is smooth just note that \(\grad f(\vect x)=(-\grad h(x_1,x_2), 1)\text{,}\) which is always nonzero. Hence by the above theorem the surface is smooth.
Example 9.9. Implicit representation of a plane.
Most commonly, a plane in \(\mathbb R^3\) is given in implicit form by the equation
\begin{equation*}
ax+by+cz+d=0\text{.}
\end{equation*}
At least one of \(a,b,c\) is nonzero. If for instance \(a\neq 0\) we get the explicit representation
\begin{equation*}
x=-\frac{1}{a}\Bigl(by+cz+d\Bigr)\text{.}
\end{equation*}
A parametric representation can be obtained as follows. Suppose that \(\vect v_1\) and \(\vect v_2\) are linearly independent vectors, lying on the plane \(ax+by+cz=0\) (the original plane translated to the origin). If \(\vect a\) is a point on the original plane, then
\begin{equation*}
\vect g(s,t):=\vect a+s\vect v_1+t\vect v_2,\qquad (s,t)\in\mathbb R^2
\end{equation*}
is a parametric representation of the plane.
