There are many
relations in mathematics which are very useful but not
functions. However, a fair number of the tools of mathematics can only be applied to functions. Thus, it is useful to have a way to turn a nearly
arbitrary relation into a function or set of functions. There is a process, parametrization, which produces a set of parametric functions from a wide variety of relations.
For example, "xx + yy = 1".
This describes a unit circle in the x,y plane. However, it is not a function in x or y (it fails the vertical line test for each). However, this relation can be described with x and y each as the output of a function. These functions will be of a 'dummy variable', or 'parameter', t. I won't go through a formal derivation here, but "xx + yy = 1" is logically equivalent to the pair of statements "x = cos(t); y = sin(t)" (if you doubt this, use the Pythagorean identities).
Parametrizations are very useful for describing the motions of particles over a period of time (hence the use of 't'). Parametrizations are also widely used for describing the path of a line integral.
Parametrization does not need to be into one variable. Let's say we want to parametrize a sphere,
"xx + yy + zz = 1"
It is possible to write three equations in two variables that will parametrize it. Though these are somewhat complicated to print here, one can think about latitude and longitude to get the idea.
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