When the Curve is a Function

If the curve to be parameterized is given as a function, say y = f(x), then an obvious and simple parameterization is

$$x = t, \qquad y = f(t) \qquad \rightarrow \vec{r}(t) = t \hat{i} + f(t) \hat{j}.$$ (11)

The values of t can be chosen appropriately, based on the starting and ending points of the curve.

Note that we could get more complicated and choose a monotonic function of t, say g(t) and then let

$$x = g(t), \qquad y = f(g(t)) \qquad \rightarrow \vec{r}(t) = g(t) \hat{i} + f(g(t)) \hat{j}.$$ (12)

This change does not effect the shape of the curve. It merely reparameterizes the curve so as to change the speed at which the curve is traced. For the first curve,

$$\vec{v}(t) = \hat{i} + f'(t)\hat{j} \Rightarrow \vert\vert\vec{v}(t)\vert\vert = \sqrt{1 + \vert f'(t)\vert^2}.$$ (13)

For the reparameterized curve, we have

$$\vec{v}(t) = g'(t) \hat{i} + g'(t)f'(g(t)) \hat{j} \Rightarr... ...\vert\vert = \vert g'(t)\vert \sqrt{1 + \vert f'(g(t))\vert^2}.$$ (14)

The fact that g(t) should be monotonic keeps the motion from stopping before intended and turning around, or doing anything else that's too strange. A good example to see this is to take the circle $\vec{r} = \cos t \hat{i} + \sin t \hat{j}$ and let $t \rightarrow e^{-t}$.