full classification has never been carried out.
The study of curves as algebraic equations is not the whole story. Many curves
such as catenarys, cycloids (curves traced out by a point on a revolving wheel)
and spirals are not easisly expressible as algebraic equations.
A definition
What mathematicians were after was a definition of a curve itself, not just
specific examples. Camille Jordan proposed a theory of curves built on the
definition of a curve in terms of variable points.
Here’s an example. If we let x = t^2 and y = 2t then, for different values of t,
we get many different points that we can write as coordinates (x, y). For
example, if t = 0 we get the point (0, 0), t = 1 gives the point (1, 2), and so on.
If we plot these points on the x–y axes and ‘join the dots’ we will get a parabola.
Jordan refined this idea of points being traced out. For him this was the
definition of a curve.
A simple closed Jordan curve
Jordan’s curves can be intricate, even when they are like the circle, in that they