## The curvature and torsion: how to distinguish the shape of a curve

### 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: