Answer
In short, curvature is the rate of change of the unit tangent vector $\mathrm{T}$ with respect to $\mathrm{s}$, (the curve being parametrized by its arc length $s$ ) $\kappa(s)=\left\|\frac{d T}{d s}\right\|=\left\|r^{\prime \prime}(s)\right\|$ Geometrically, it defines the "sharpness" of bending.
Work Step by Step
In short, curvature is the rate of change of the unit tangent vector $\mathrm{T}$ with respect to $\mathrm{s}$, (the curve being parametrized by its arc length $s$ ) $\kappa(s)=\left\|\frac{d T}{d s}\right\|=\left\|r^{\prime \prime}(s)\right\|$ Geometrically, it defines the "sharpness" of bending.
