How to prove there always exists a two variables or one variable equation corresponding to a curve drawn on a 2D plane? How to prove there always exists a two variables or one variable equation corresponding to a curve drawn on a 2D plane?
For example, a circle 's equation is $x^2+y^2=r^2$.

Since Victor didn't specify what he means by "a curve drawn ...", I'll assume that
this is a parametric curve $p = \gamma(t)$, $0 \le t \le 1$, with $\gamma: [0,1] \to \mathbb R^2$ continuous.  The image $C = \gamma([0,1])$ is then a compact set in the plane, and $F(p) = \text{dist}(p, C)$ (i.e. the distance from $p$ to $C$) is a continuous function on $\mathbb R^2$ such that $C = \{p: F(p) = 0\}$, i.e. 
$F(p)= 0$ can be taken to be the "equation" of the curve.
