Let [math]\vec{c}:\left[a,b\right]\to\mathbb{R}^2[/math] be a path and [math]C[/math] the corresponding plane curve that is the image of [math]\vec{c}[/math].[br][list][*]If [math]\vec{c}\left(t_1\right)=\vec{c}\left(t_2\right)[/math] only when [math]t_1=a[/math] and [math]t_2=b[/math] (or perhaps never at all) then the path is said to be an [b][color=#ff0000]injective[/color][/b] parameterization of the curve [math]C[/math]. 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 [color=#ff0000][b]closed curve[/b][/color].[b] [/b]Usually a parameterization of a closed curve has the property [math]\vec{c}\left(a\right)=\vec{c}\left(b\right)[/math].[/*][*]A curve has a [b][color=#ff0000]self-intersection[/color][/b] if it loops back on itself. A parameterization of such a curve will usually fail to be injective. A [b][color=#ff0000]simple[/color][/b] 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 [math]\vec{c}[/math] are differentiable across the domain [math]\left[a,b\right][/math] the resulting image curve is said to be a [color=#ff0000][b]differentiable[/b] [/color]curve.[/*][*]Name the component functions [math]x\left(t\right)[/math] and [math]y\left(t\right)[/math]. If the component functions are differentiable [i]and[/i] there is no value [math]t_0\in\left[a,b\right][/math] so that [math]x'\left(t_0\right)=y'\left(t_0\right)=0[/math] then [math]\vec{c}\left(t\right)[/math] is said to be [b][color=#ff0000]regular[/color][/b].[/*][/list][br]In the GeoGebra applet below you can select from several different curves and see which of these properties apply.