Curvature of Curves

Definition

Suppose $\mathbf{T}(s)$ is the Unit Tangent Vector of a smooth curve, then the curvature function of the curve is $\kappa =\left|\frac{d\mathbf{T}}{ds}\right|=|{\mathbf{T}}^{\prime }(s)|=\frac{1}{|\mathbf{v}|}\left|\frac{d\mathbf{T}}{dt}\right|=\frac{|{\mathbf{r}}^{\prime }\times {\mathbf{r}}^{\prime \prime }|}{|{\mathbf{r}}^{\prime }{|}^3}$