The fundamental difference is that a path describes how you trace out a shape over time, while a curve is simply the shape itself – the set of points visited.
According to the provided reference, a path is formally defined as a function x : R → Rm, where R represents time and Rm is the space the path exists in (like 2D or 3D). This function tells you the position of a point at any given time t. Think of the path as the journey, including the speed and direction at each moment.
The reference further states, "The curve is the image of this path, that is, a subset of Rm." The curve is the collection of all the points that the path visits during its journey. It's the shape traced out, independent of how fast or in what direction it was traced.
Understanding the Distinction
Imagine drawing a circle.
- The Path: This would be the action of your pen moving from a starting point, around the circle, and back to the start. The speed at which you move your pen, whether you go clockwise or counterclockwise, and even if you go around the circle multiple times – these are all aspects of the path. The path is the function describing your pen's position over time.
- The Curve: This is the actual circle drawn on the paper. It's the geometric shape, the collection of all the points that make up the circle's circumference. The curve is the image of the path.
As the reference notes, you could have a path of a point that moves around the same circle, but the point is moving twice as fast. Even though the path (the function describing the movement) is different because the speed changes, The curve is the same for both functions. This highlights that the curve is about the shape itself, not the dynamics of tracing it.
Key Differences Summarized
Here's a table to quickly grasp the main distinctions:
| Feature | Path | Curve |
|---|---|---|
| Nature | A function or mapping | A set of points or geometric shape |
| Defines | How a point moves (position over time) | The shape traced out |
| Includes | Speed, direction, starting/ending points | Only the collection of points |
| Formally | Function x : R → Rm |
Image of the function x(R) ⊆ Rm |
| Example | Moving around a circle twice as fast | The circle itself |
In essence, a path provides the dynamic description of movement, while a curve is the static geometric result of that movement. Different paths can trace out the exact same curve.
