What is the mathematical definition of a curve?

Question

I know what a curve is intuitively. I also know a straight line is also classified as a curve.

Bit I can't seem to explain that haha.

Answer 1

Let's start with two dimensions. The basic idea of a curve is very simple: it mathematises what you do when you put pencil to a sheet of paper and move it without taking the tip off the paper (i.e., draw a continuous line, in the ordinary English meaning of those words).

Now, how do we describe this mathematically? At each time $t$ between $t=a$, when you start, and $t=b$, when you stop, your pencil has a position on the paper. Hence there is a function from the line we use to describe time (the real interval $[a,b]$) to the sheet of paper (the Euclidean plane, or at least a finite rectangle in it), $\mathbf{r}(t)$, which, when fed a $t$ between $a$ and $b$, gives the position of the point (of the pencil) at time $t$. (This is the gist of the idea; obviously we've gone from a finite-sized smudge of graphite on a bumpy clump of wood fibres to an object that has no width on a flat plane, because this is easy to describe: otherwise you'd have to talk about which part of the smudge you were drawing, and was this bit done at exactly time $t$, and so on, so we work instead in an abstract setting where we have infinite precision, infinitely thin things, and so on.)

Now, if we have a set of axes in the plane, each point has a unique pair of numbers that describes it, so we can also describe the point at time $t$ using a pair of numbers, $(x(t),y(t))$. (This is essentially what two dimensions means.)

The generalisation to three dimensions is also easy to picture (and here we're going right back to the beginning, because this is supposedly how Descartes first came up with Cartesian coordinates in a smoky room in 1637): suppose we have a fly buzzing around. At each time $t$, the position of the fly may be given by a vector $\mathbf{r}(t)$, which obviously varies continuously, since flies don't teleport. Thus a continuous function from an interval $[a,b]$ to $\mathbf{r}(t)$ is a curve in three dimensions, sometimes called a space curve. Equally we could give it as three functions $(x(t),y(t),z(t))$.

(To get more technical, people do distinguish between the set of points that the path covers and the path $\mathbf{r}(t)$ itself: the former is called a curve, the latter a parametrisation of the curve ($t$ being the parameter). Since in real life most of the time paths come parametrised, I wouldn't worry too much about the distinction.)