TITLE: What is parameterization?
QUESTION [35 upvotes]: I am struggling with the concept of parameterizing curves. I am not even sure if I know what it means so I tried to look some things up.
On Wikipedia it says:

Parametrization is... the process of finding parametric equations of a curve, a surface, or, more generally, a manifold or a variety, defined by an implicit equation. The inverse process is called implicitization.

Since I didn't know what a parametric equation was I also looked that up:

In mathematics, parametric equations of a curve express the coordinates of the points of the curve as functions of a variable, called a parameter. For example,
\begin{align} x&=\cos t\\ y&=\sin t \end{align} .

So, basically I have some curve and I want to make up some equation that describes that curve? But doesn't this get very difficult if you don't have simple circles or spirals? 
Maybe someone can explain to me what's going on here.
Thanks in advance.

REPLY [15 votes]: Often a curve $\gamma$ in the plane is defined as the set of points $(x,y)$ satisfying a certain geometric or algebraic condition. An example is
$$\gamma:=\bigl\{(x,y)\>\bigm|\>{x^2\over a^2}+{y^2\over b^2}=1\bigr\}\ ,\tag{1}$$
whereby the values of $a>0$ and $b>0$ are given. Such a description is implicit; it just provides a quick test whether a trial point $(x,y)$ is lying on $\gamma$ or not.
When we really want to geometrically analyze the curve $\gamma$, which means calculating its length or the enclosed area, etc.,  then we need a parametric representation. This is a production scheme that produces all points of $\gamma$ in a systematic and analytically manageable way. In this way the points of the curve $\gamma$ in $(1)$ are produced by the vector-valued function
$$t\mapsto\left\{\eqalign{x(t)&:=a\cos t\cr y(t)&:=b\sin t\cr}\right.\qquad(0\leq t\leq 2\pi)\tag{2}$$
in a $1:1$ way, whereby $t=0$ and $t=2\pi$ produce the same point.
Since an implicit representation of a curve $\gamma: \>F(x,y)=0$ does not determine a parametric representation (a "timetable") $t\mapsto\bigl(x(t),y(t)\bigr)\>$ in an unique way there is no automatic procedure (like multiplication of two polynomials, or calculating a derivative) that generates a parametric representation from an equation $F(x,y)=0$; whence experience and intuition is asked for. Sometimes you can solve such an equation for $y$ and obtain a representation as a graph in the form $x\mapsto\bigl(x, y(x)\bigr)\>$.
That deep mathematics is involved here can be seen from the following simple example: The innocent looking equation $x^2+y^2=1$ determines a curve $\gamma$ that encloses an area of $\pi$ and whose length is $2\pi$.