Direction of Motion

A parametric curve is a path, and a path has a direction of travel built into it by the parameter t. The curve traced by (x(t), y(t)) might look identical to another curve traced by (x(s), y(s)), same shape, same points, but the particles can move through those points in opposite orders. That is why two parametrizations can look the same when graphed but describe completely different motions.

To find direction at a specific moment, check whether x(t) and y(t) are increasing or decreasing right there. If x is increasing, the particle moves right; if decreasing, it moves left. If y is increasing, the particle moves up; if decreasing, it moves down. Combine the two to get a direction arrow, like up-and-right or down-and-left. You can confirm this numerically: compute x at two nearby t-values and subtract to see if the x-coordinate grew or shrank, then do the same for y.

This connects directly to the idea of reversing a parametrization. Replacing t with -t everywhere keeps every point on the curve exactly where it was, since the set of outputs does not change, but it runs the clock backward. Whatever was increasing becomes decreasing and vice versa, so every direction arrow flips. This same reasoning shows up again when comparing two different parametrizations of the same conic or line, and when matching a graph to its correct parametric equations based on which way it is traced.