1.1.3 Properties of paths
Let be a path and the corresponding plane curve that is the image of .
- If only when and (or perhaps never at all) then the path is said to be an injective parameterization of the curve . Note this is a slight tweak of the definition of injective we used in Linear Algebra.
- A curve that encloses area in the plane and has no visible endpoints is said to be a closed curve. Usually a parameterization of a closed curve has the property .
- A curve has a self-intersection if it loops back on itself. A parameterization of such a curve will usually fail to be injective. A simple curve is one that has no self-intersections. Note that a closed curve such as a circle can still be considered simple if it does not have crossing points.
- If the component functions of are differentiable across the domain the resulting image curve is said to be a differentiable curve.
- Name the component functions and . If the component functions are differentiable and there is no value so that then is said to be regular.