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 itsx 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(t)=(cos(t),sin(t))

Note that by convention we usually specify a domain for the input parameter, in this case from 0 to 2π to prevent having to trace out the path of the particle infinitely.

r(t)=(cos(t),sin(t)),(0t2π)

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)=(cos(t),sin(t),t),(0t2π)
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(u,v)=(4u2,3v,uvv2+(u32)2),(0u1,0v1)

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(u,v)=(cos(2πu),sin(2πu)+sin(4πv),3v),(0u1,0v1)

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?