Parametrization of curves

I've never seen this on a textbook, but it's possible to think on the process of parametrization of curves in terms of projected shadows. Suppose you have a moving object describing some curve on the plane. If we could have a projection of its shadow over a wall, one projection for each axis, we could see the shadow moving in one dimension, right or left over a straight line. Now the shadow in each wall would move faster or slower depending on the object's trajectory. That rate of change in position is associated to a function. With the motion on each axis being different it's natural to assume that there is a different function for each coordinate.

One may think on the graph of each function for each coordinate and try to find a relationship between the graph of each function and the curve. While it may be possible to do that, it's hard to make the association. With functions we can imagine the graph when the do composition, products and sum of functions. However, doing the same for curves doesn't work well because we'd be trying to make an association between graphs which are plotted with an independent and a dependent variable with a graph that is plotted with two independent variables (each coordinate is linearly independent).