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


Example 2

Let \[\begin{array}{rl} g:[0,4\pi] & \rightarrow\mathbb{R}^{2}\\ t & \rightarrow\left(\cos\left(\frac{t}{2}\right),\,\sin\left(\frac{t}{2}\right)\right) \end{array}\]

Then \(g'=\left(-\frac{1}{2}\sin\left(\frac{t}{2}\right),\,\frac{1}{2}\cos\left(\frac{t}{2}\right)\right)\); \(v(t)=\left|g'(t)\right|=\frac{1}{2}\) and the trace of this curve is:

Note that although the functions of examples 1 and 2 have the same trace, they are different (the velocities at which each circle is covered are different in both cases).