1.1.3 Properties of paths
Let be a path and the corresponding plane curve that is the image of .
- A curve that encloses area in the plane and has no visible endpoints is said to be a closed curve. Note that this definition does not depend on parameterization. This is a curve property.
- Recall: a function is said to be injective on its domain if implies always. We will modify this definition slightly for paths by saying the path is said to be injective if it is injective on . This allows us to describe the standard parameterization of the unit circle as injective. Since this definition relies on a parameterization this is a path property, not a curve property. In other words, the same curve might have injective and non-injective parameterizations.
- If fails to be injective on , we say the path (and the corresponding curve) has a self-intersection. That is there is a point on so that for two distinct values and in .
- A simple curve is one that can be parameterized on some closed interval without self-intersections. Note that even though this definition references a parameterization this is still a curve property. In other words, we can identify a simple curve without specifying a parameterization.
- If the component functions of are differentiable across the domain then the path is a differentiable and the resulting image curve is said to be a differentiable curve. In fact any curve for which a differentiable parameterization can be found is said to be a differentiable curve, making differentiable both a path and curve property.
- Name the component functions and . If the component functions are differentiable and there is no value so that then is said to be regular. This is a path property, not a curve property.