📂Geometry

Tangents, Normals, and Curvature of Plane Curves

Definitions ¹

Let us assume that a unit speed plane curve $\alpha : (a,b) \to \mathbb{R}^{2}$ is given.

1. The tangent (vector field) is defined as $t (s) := \alpha^{\prime} (s)$.
2. The unique vector field $n(s)$ that makes $\left\{ t(s), n(s) \right\}$ the counterclockwise basis of $\mathbb{R}^{2}$ is defined as normal (vector field).
3. The plane curvature is defined as $k(s) := \left< t^{\prime}(s) , n (s) \right>$.

Basic Properties

• [1] $$ \begin{align*} \alpha (s) =& \left( x(s) , y(s) \right) \\ t(s) =& \left( x^{\prime}(s) , y^{\prime}(s) \right) \\ n(s) =& \left( -y^{\prime}(s) , x^{\prime}(s) \right) \end{align*} $$

• [2] If $t(s)$ is differentiable $$ \begin{align*} t^{\prime}(s) =& \left< t^{\prime}(s) , t(s) \right> t(s) + \left< t^{\prime}(s) , n(s) \right> n(s) \\ =& 0 \cdot t(s) + k(s) n(s) \\ =& k(s) n(s) \end{align*} $$

• [3] If $n(s)$ is differentiable $$ n^{\prime}(s) = - k(s) t(s) $$

• [4] In the Frenet-Serret Apparatus $$ \begin{align*} t(s) =& T(s) \\ n(s) =& \pm N (s) \qquad , \text{if } \exists N(s) \\ \left| k (s) \right| =& \kappa (s) \end{align*} $$

Explanation

Although it is similar to the Frenet-Serret apparatus, one can see that there are newly defined terms, as it pertains to a plane.

Especially with curvature, different from the local curve theory, it does not necessarily have to be positive. If $k > 0$, the curve tends to approach the direction of $n$, and if $k<0$, it tends to move away from $n$.

The reason for considering such plane curves is that one cannot vaguely think about ’turning’ when considering the global geometry of three-dimensional curves.