0. Parametric curves and surfaces

Recall that a parametric curve in $\mathbb{E}^ 3$ is the image of a function    P$:~\mathbb{R}\rightarrow \mathbb{E}^ 3$, in other words,    P$(t) = \left[\begin{array}{c}f(t)\\ g(t)\\ h(t)\end{array}\right]$, or $x = f(t), y = g(t), z = h(t)$. You can think of the curve as being a distorted image of $\mathbb{R}$, produced by the function P. If only the values $a \leq t \leq b$ are used, then P$: [a,b] \rightarrow \mathbb{E}^ 3$ and the curve is a distorted image of the interval $[a,b]$, as in Figure .

Figure: A parametric curve segment as a distorted interval

book/10dir/curve.eps

A parametric surface is the image of a function P$(t,u)$ with two parameters as arguments and with values in $\mathbb{E}^ 3$. If $t,u$ are allowed to take on any values, then P$:~\mathbb{R}^ 2 \rightarrow \mathbb{E}^ 3$, and the surface is a distorted image of all of $\mathbb{R}^ 2$. If $a \leq t \leq b$ and $c \leq u \leq d$ then the domain is a rectangle and the surface is a distorted rectangle--a ``surface patch''. In either case, P$(t,u) = \left[\begin{array}{c}f(t,u)\\ g(t,u)\\ h (t,u)\end{array}\right]$, or equivalently, $x = f (t,u)$, $y = g (t,u)$, $z = h (t,u)$, as in Figure .

Figure: A parametric surface patch as a distorted rectangle

book/10dir/surf.eps

Example 1. A sphere: Regard $t,u$ as latitude and longitude and let
$x = \cos t \cos u$, $y = \cos t \sin u$, $z = \sin t$, for $-\frac {\pi}2 \leq t \leq \frac \pi 2$, $0 \leq u \leq 2 \pi$.

Example 2. The graph of a function $F$ of two variables: Let $x = t$, $y = u$, $z = F(t,u)$.

Example 3. A twisted ribbon: Let $x = -u \sin t$, $y = t$, $z = u \cos t$, for $- \pi \leq t \leq \pi$, $-1 \leq u \leq 1$.

On a computer display using wire-frame graphics, surfaces are indicated by drawing some curves in which one of $t$ and $u$ is held fixed while the other varies. These are isoparametric curves.