## 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 $k(s)=\left|f'(s)\right|$

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 $B'(s)=-\tau(s)N(s).$