Consider a particle (in two dimensions) moving as a function of time, t. Both the x-coordinate and y-coordinate vary with t, that is x = x(t) and y = y(t). The vector describing position is therefore a function of time r (t) = < x(t), y(t) > it is a vector-valued function of time!

After a little thought, one can see that the position of a particle given by

r (t) = < x(t), y(t) > can be quite complicated. In fact, it may not be the graph of a function at all! This way of describing a curve is called parameterization , where time (t) plays the role of a parameter...

One can show that the equation

r (t) = r ₀ + t v describes a straight line. (Is this obvious to you?)

Another way of writing this is to say that

x(t) = x ₀ + at
y(t) = y ₀ + bt
parametrically describes a straight line.

Eliminating t from the above equations gives y as a function of x. Therefore, in some cases, two parametric equations can be reduced to a single functional relationship. (Is this always possible?)

Last modified Mon Sep 9 21:22:37 CDT 1996