# Parametric Surfaces

Previously, we had been describing surfaces just in terms of a single value, say, $z$ for a given pair of co-ordinates $x$ and $y$. These kinds of functions, called explict functions, only cross through each $x$ and $y$ point once and define a single $z$ value for each of the. In general, they usually always look like some infinite plane with variable depth.

However, this is not the only way to define a function. Just like a function can have multiple parameters which map to a single output, a function can also have mutiple outputs generated from a single input. Such a function is called a Vector Valued Function and such functions are usually implicit in nature.

The simplest case is a function that describes the position of some particle in 2D space as a function of time, $t$. We cannot use the standard "y as a function of x" notation here since what we are concerned with is time, and at any given time the particle could be at any $x$ and $y$ co-ordinate pair. It is not always at a unique $y$ position for its$x$ value.

Thus, we have a sort of "hidden" parameter $t$ which does not get its own axis in the output space, but still defines where a particle is over time.

$$r(t)=(\mathrm{cos}(t),\mathrm{sin}(t))$$Note that by convention we usually specify a domain for the input parameter, in this case from $0$ to $2\pi $ to prevent having to trace out the path of the particle infinitely.

$$r(t)=(\mathrm{cos}(t),\mathrm{sin}(t)),(0\le t\le 2\pi )$$It is also perfectly possible to have a vector valued function that returns three output co-ordinates. This function, for instance, wraps around a cylinder.

$$r(t)=(\mathrm{cos}(t),\mathrm{sin}(t),t),(0\le t\le 2\pi )$$Now that we have defined those what a parametric function with a single parameter looks like in two and three dimensions, consider what a parametric function with two parameters looks like in three dimensions. Such a function will actually trace out a surface. You could imagine such a parametric function tracing out a patch in space made up of an infinite number of horizontal lines, then an infinite number of vertical lines.

What is nice about parametric surfaces is that they can wrap around themselves - remember that there is no requirement that we have a unique $z$ co-ordinate for every $x$ and $y$ pair, so it is perfectly valid to start defining surfaces in terms of trigonometric functions.

The surface above defines a cylinder and protrudes out with a depth of three. But it also oscillates on the y-axis as the depth protrudes out. Can you see why?