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.

In the GeoGebra applet below you can select from several different curves and see which of these properties apply.

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