The curvature and torsion: how to distinguish the shape of a curve 
Example 1
Let \[\begin{array}{rl} f:[0,2\pi] & \rightarrow\mathbb{R}^{3}\\ t & \rightarrow\left(\cos(t),\,\sin(t),\,2\right) \end{array}\]
Then \[\arule{JungleGreen}=f'(t)=\left(-\sin(t),\,\cos(t),\,0\right);\] \[ \definecolor{castanho}{rgb}{0.6,0.4,0.2} \arule{castanho}=N(t)=\frac{f''(t)}{\left|f''(t)\right|}=\left(-\cos(t),\,-\sin(t),\,0\right);\] \[\arule{blue}=B(t)=T\times N=\left(0,0,\cos^{2}(t)+\sin^{2}(t)\right)=(0,0,1)\] and the trace of the curve \(f\) is given by the circle:
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