MATH 1113 PreCalculus

Day #35, Parametric Equations
Part II

A little more depth on parametric equations:

The big advantage of parametric equations is that they allow us to express, via FUNCTIONS, shapes that fail the vertical line test.

At first this may seem to defy logic, but, as we'll see, there's something hidden behind the scenes.

Consider: x = f(t), y = g(t).  What we actually have here are two SEPARATE functions of t.  Graphed separately, on an (x,t) plane and a (y,t) plane, the functions are the usual ones you've already studied.

Here's an example:

The first video is of the parametric curve:

Video:

Now, look at the separate functions graphed as x(t) and as y(t):

First (x,t)

Now, (y,t):

How are these put together?  What's actually missing is a third dimension.  In a two-dimensional world, it is possible to think of TIME as the third dimension.  So, if we let y be "up," and x be "right," and then think of the "t" axis as coming straight toward us, we see the parametric curve as an honest-to-goodness function in 3-dimensions:

Video:

Here we're standing out away from the scene to see the function traced out through time.

So what we're actually looking at, when we see the parametric curve is the projection onto the (x,y) plane of the curve.  Here's a video that rotates around to see the curve from all the different angles, finally looking end-on.  Play this one through several times.  (It was better in QuickTime format, since you could drag it back and forth; at this speed it's a bit confusing.)

Video:

In the same way, we can think of 3-d parametric curves as functions of 4-space, and 3-d parametric surfaces as functions of 5-space (two different time dimensions!).  The applications of this in computer graphics are very important.  See "uv mapping" in this Adobe Acrobat file: http://www.math.harvard.edu/archive/21a_summer_06/handouts/parametric.pdf

Page Last Modified: 2 April, 2007

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