ATRACTOR

\(\require{color}\newcommand{\arule}[1]{{\color{#1}\Rule{3em}{1ex}{0ex}\;}}\)

Let \[\begin{array}{rl} f:[0,4\pi] & \rightarrow\mathbb{R}^{3}\\ t & \rightarrow\frac{3}{8}\left(\sqrt{7}\cos(t),\,\sqrt{7}\sin(t),\,\frac{t}{3}\right) \end{array}\]

Then \[\arule{JungleGreen}=f'(t)=\frac{3}{8}\left(-\sqrt{7}\sin(t),\,\sqrt{7}\cos(t),\,\frac{1}{3}\right);\] \[\definecolor{castanho}{rgb}{0.6,0.4,0.2} \arule{castanho}=N(t)=\frac{f''(t)}{\left|f''(t)\right|}=\left(-\cos(t),\,-\sin(t),\,0\right);\] \[\arule{blue}=B(t)=T\times N=\frac{1}{8}\left(\sin(t),-\cos(t),\,3\sqrt{7}\right)\] and the trace of curve \(f\) is given by the helix:

Click on the image to see the animation