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][*]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]Note that this definition does not depend on parameterization. This is a [i]curve[/i][b] [/b]property.[/*][*]Recall: a function is said to be [b][color=#ff0000]injective[/color][/b] on its domain if [math]f\left(x_1\right)=f\left(x_2\right)[/math] implies [math]x_1=x_2[/math] always. We will modify this definition slightly for paths by saying the path [math]\vec{c}[/math] is said to be injective if it is injective on [math] \left[a,b\right) [/math]. This allows us to describe the standard parameterization of the unit circle [math]\vec{c}\left(t\right)=\left(\cos t,\sin t\right),t\in\left[0,2\pi\right][/math] as injective. Since this definition relies on a parameterization this is a [i]path[/i] property, not a [i]curve[/i] property. In other words, the same curve might have injective and non-injective parameterizations.[/*][*]If [math]\vec{c}[/math] fails to be injective on [math] \left[a,b\right) [/math], we say the path (and the corresponding curve) has a [b][color=#ff0000]self-intersection[/color][/b]. That is there is a point [math]\left(x,y\right)[/math] on [math]C[/math] so that [math]\vec{c}\left(t_1\right)=\left(x,y\right)=\vec{c}\left(t_2\right)[/math] for two distinct values [math]t_1[/math] and [math]t_2[/math] in [math] \left[a,b\right) [/math]. [/*][*]A [b][color=#ff0000]simple[/color][/b] curve is one that can be parameterized on some closed interval [math]\left[a,b\right][/math] without self-intersections. Note that even though this definition references a parameterization this is still a [i]curve[/i] property. In other words, we can identify a simple curve without specifying a parameterization.[/*][*]If the component functions of [math]\vec{c}[/math] are differentiable across the domain [math]\left[a,b\right][/math] then the path is a [b][color=#ff0000]differentiable[/color] [/b]and the resulting image curve is said to be a [color=#ff0000][b]differentiable[/b] [/color]curve. In fact any curve for which a differentiable parameterization can be found is said to be a differentiable curve, making differentiable both a [i]path[/i] and [i]curve[/i] property.[/*][*]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]. This is a [i]path[/i] property, not a [i]curve[/i] property (though I will sometimes refer to a curve as differentiable or regular meaning that there exists a differentiable or regular parameterization of that curve).[/*][/list][br]In the GeoGebra applet below you can select from several different curves and see which of these properties apply.