# 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.

$t=$

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.

$r\left(t\right)=\left(\mathrm{cos}\left(t\right),\mathrm{sin}\left(t\right)\right)$

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.

$r\left(t\right)=\left(\mathrm{cos}\left(t\right),\mathrm{sin}\left(t\right)\right),\left(0\le t\le 2\pi \right)$

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.

$r\left(t\right)=\left(\mathrm{cos}\left(t\right),\mathrm{sin}\left(t\right),t\right),\left(0\le t\le 2\pi \right)$
$t=$

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.

$s\left(u,v\right)=\left(4{u}^{2},3v,uv-{v}^{2}+\left(u-\frac{3}{2}{\right)}^{2}\right),\left(0\le u\le 1,0\le v\le 1\right)$

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.

$s\left(u,v\right)=\left(\mathrm{cos}\left(2\pi u\right),\mathrm{sin}\left(2\pi u\right)+\mathrm{sin}\left(4\pi v\right),3v\right),\left(0\le u\le 1,0\le v\le 1\right)$

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?