The curvature and torsion: how to distinguish the shape of a curve 
Curvature and Torsion
Definition: Let \(f:\, I\rightarrow\mathbb{R}^{3}\) be a \(C^{2}\) regular curve parameterized by arc length \(s\in I\). The curvature of \(f\) is the function \(k\) given by \[k(s)=\left|f'(s)\right|\]
Important observation about the curvature...
Definition: Let \(f:\, I\rightarrow\mathbb{R}^{3}\) be a \(C^{2}\) regular curve parameterized by arc length \(s\in I\). The torsion of \(f\) is the function \(\tau\) defined on the points where \(f''(s)\neq0\) (that is, where the curvature is strictly positive), such that \[B'(s)=-\tau(s)N(s).\]
