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.