### The Curvature (plane case)

Let \(f:\, I\rightarrow\mathbb{R}^{2}\)
be a differentiable parametrized curve. For each \(t\in I\) with \(f'(t) \neq 0\), there is a unique line that passes through point \(f(t)\) and has the direction of the vector
\(f'(t)\). This is the *tangent line* to the curve \(f\) at \(t\).
In the study of curves, it is usual (and convenient) to assume the existence of the tangent at every point of the curve:

**regular**if \(f'(t) \neq 0\) for every \(t \in I\).

**arc length**of a regular parametrized curve \(f:\, I\rightarrow\mathbb{R}^{2}\), from point \(t_{0} \in I\), is the function \[s(t)=\int_{t_{0}}^{t}\left|f'(t)\right|dt.\]

Note that \(s'(t)=\left|f'(t)\right|\).

A regular parametrized curve is said to be parametrized by arc length if \(\left|f'(t)\right|=1\) for every \(t \in I\).

Assuming that a curve \(f\) determines the position of a certain particle, the fact that the curve is parametrized by arc length means that the particle is moving at constant speed \(1\).

Furthermore: the *tangent vector*
\(T=\left(T_{1},T_{2}\right)\) to the curve at a certain \(t\) is given by \(f'(t)\) and the *normal vector* \(N=\left(N_{1},N_{2}\right)\) is
given by \[\frac{f''(t)}{\left|f''(t)\right|}.\]

The definition of curvature that we present below is only valid for curves parametrized by arc length. This restriction is not really significative: given any smooth curve, it is possible to determine a differentiable curve parametrized by arc length which has the same trace as the given smooth curve.

**curvature (with sign)**of \(f\) is given by \[k(s)=§\left|f'(s)\right|\] where \(§\) is defined in the following way

Value of \(§\) | Condition | "Scheme" |
---|---|---|

\(§=1\) | \(T_{1}\times N_{2}-T_{2}\times N_{1} >0\) | |

\(§=-1\) | \(T_{1}\times N_{2}-T_{2}\times N_{1} <0\) |