### Plane Curves

A (parameterized) plane curve is a map \[\begin{array}{cl} f: & I\rightarrow\mathbb{R}^{2}\\ & t\rightarrow\left(f_{1}(t),f_{2}(t)\right) \end{array}\] where \(I\) is an interval of \(\mathbb{R}\).

Usually, one thinks of variable \(t\) as representing time and of the image
\(f(t)\) as the position of a particle moving in space at instant \(t\). Therefore,
at \(t=t_{0}\) the particle is at point \(f(t_{0})=\left(f_{1}(t_{0}),f_{2}(t_{0})\right)\).
When \(f\) is differentiable, we may take its first derivative \(f'(t_{0})=\left(f'_{1}(t_{0}),f'_{2}(t_{0})\right)\),
which represents the vector tangent to the curve at point \(f(t_{0})\). This
is usually called the velocity vector of the particle at time \(t\). The *velocity*
is the length of the velocity vector. It is given by \[v(t)=\left|f'(t)\right|=\sqrt{\left(f'_{1}(t)\right)^{2}+\left(f'_{2}(t)\right)^{2}}.\]

The image \(f(I)\), a subset of \(\mathbb{R}^{2}\), is called the *trace*
of \(f\).

Some examples: