ATRACTOR

Circular helices

parameterized by \[\gamma_{r,a}(t)=\left(r\cos(t),r\sin(t),at\right),\] where \(r>0\) and \(a\in\mathbb{R}\), have constant positive curvature \(k(s)=k_{0}>0\), and constant torsion \(\tau(s)=\tau_{0}\), so they are specified, alternatively, by equations \[k(s)=\frac{r}{r^{2}+a^{2}},\: s\in\mathbb{R}\] and \[\tau(s)=\frac{a}{r^{2}+a^{2}},\: s\in\mathbb{R}.\]

On the other hand, any curve \(\gamma\) in \(\mathbb{R}^{3}\), parameterized by arc length \(s\), with constant positive curvature, \(k(s)=k_{0}>0\), and constant torsion, \(\tau(s)=\tau_{0}\), in every point is, by the Fundamental Theorem of Curves, up to some rigid motion of \(\mathbb{R}^{3}\), a circular helix \(\gamma_{r,a}\) such that \[k_{0}=\frac{r}{r^{2}+a^{2}}\] and \[\tau_{0}=\frac{a}{r^{2}+a^{2}},\] that is, with \[r=\frac{k_{0}}{k_{0}^{2}+\tau_{0}^{2}}\] and \[a=\frac{\tau_{0}}{k_{0}^{2}+\tau_{0}^{2}}.\]

Therefore, as the circle is the plane curve characterized by having constant curvature, the circular helix is the space curve characterized by having constant curvature and torsion.