Mathematical Underpinnings
3. Breaking Down Path and Trajectory with Math
Let's get a little bit mathematical (don't worry, it won't hurt too much!). A path can be described mathematically as a function that maps a single parameter (often just a distance traveled along the path) to a point in space. In two dimensions, it might look like (x(s), y(s)), where 's' is the distance along the path. In three dimensions, it would be (x(s), y(s), z(s)). The important thing is that theres no time variable here.
A trajectory, on the other hand, always involves time. It's a function that maps time to a point in space and also provides information about the velocity and acceleration at that point. In three dimensions, a trajectory might be described as (x(t), y(t), z(t)), where 't' represents time. From this function, you can derive the velocity vector (dx/dt, dy/dt, dz/dt) and the acceleration vector (dx/dt, dy/dt, dz/dt). These derivatives tell you not just where the object is at a given time, but also how its motion is changing.
To put it another way, if you took a photograph of an object moving along a path, that photo captures the path. If you took a high-speed video of that same object, that video captures the trajectory. The video reveals the temporal information that the photograph lacks.
Think about drawing a spiral on a piece of paper. The spiral itself is the path. But imagine a tiny car driving along that spiral, constantly changing its speed. The entire motion of the car along the spiral, including its speed at every point, is its trajectory. The mathematical representation allows us to precisely define and analyze both the static "where" of the path and the dynamic "how and when" of the trajectory.
