We have dealt extensively with vector equations for curves, ${\bf r}(t)=\langle x(t),y(t),z(t)\rangle$. A similar technique can be used to represent surfaces in a way that is more general than the equations for surfaces we have used so far. Recall that when we use ${\bf r}(t)$ to represent a curve, we imagine the vector ${\bf r}(t)$ with its tail at the origin, and then we follow the head of the arrow as $t$ changes. The vector "draws'' the curve through space as $t$ varies.

Suppose we instead have a vector function of two variables, $${\bf r}(u,v)=\langle x(u,v),y(u,v),z(u,v)\rangle.$$ As both $u$ and $v$ vary, we again imagine the vector ${\bf r}(u,v)$ with its tail at the origin, and its head sweeps out a surface in space. A useful analogy is the technology of CRT video screens, in which an electron gun fires electrons in the direction of the screen. The gun's direction sweeps horizontally and vertically to "paint'' the screen with the desired image. In practice, the gun moves horizontally through an entire line, then moves vertically to the next line and repeats the operation. In the same way, it can be useful to imagine fixing a value of $v$ and letting ${\bf r}(u,v)$ sweep out a curve as $u$ changes. Then $v$ can change a bit, and ${\bf r}(u,v)$ sweeps out a new curve very close to the first. Put enough of these curves together and they form a surface.

Example 16.6.1 Consider the function ${\bf r}(u,v)=\langle v\cos u,v\sin u, v\rangle$. For a fixed value of $v$, as $u$ varies from 0 to $2\pi$, this traces a circle of radius $v$ at height $v$ above the $x$-$y$ plane. Put lots and lots of these together,and they form a cone, as in figure 16.6.1. Alternately, we can fix $u$, and as $v$ ranges from $0$ to infinity, ${\bf r}(u,v)$ traces out a line; examples of these lines can be seen in the wall of the cone, or alone in the third graph of the figure. $\square$