Curvature

How fast the unit tangent vector turns per unit of arc length, $\kappa=|\mathbf{r}'\times\mathbf{r}''|/|\mathbf{r}'|^3$ .

Curvature $\kappa = |\mathbf{T}'(t)|/|\mathbf{r}'(t)|$ measures how fast $\mathbf{T}$ turns per unit of arc length rather than per unit of time — a fact about the curve's shape alone, independent of how quickly anything moves along it. The equivalent cross-product form $\kappa = |\mathbf{r}'\times\mathbf{r}''|/|\mathbf{r}'|^3$ is what actually gets computed, since it skips finding $\mathbf{T}(t)$ first. Where $\kappa$ is large, the tightest circle hugging the curve there (the osculating circle) has a small radius $1/\kappa$ ; where $\kappa\to0$ , that circle flattens toward a straight line.

Examples

The helix $\mathbf{r}(t) = \langle 3\cos t, 3\sin t, 4t\rangle$ has constant curvature $\kappa=\tfrac{3}{25}$ .
A tight hairpin turn needs a big turn of the steering wheel per mile driven, so $\kappa$ is large there; a straight stretch has $\kappa=0$ .