Curvature

Curvature is simply a number that says how sharp a bend is: zero on a straight road, small on a gentle highway curve, big on a hairpin turn. It's what road designers use to decide how fast cars can safely take a curve. The formula $\kappa = \frac{\|\vec T\,'\|}{\|\vec r\,'\|}$ looks like a fraction of fractions, but it's just two pieces: $\|\vec T\,'\|$ (how fast your direction is turning) divided by $\|\vec r\,'\|$ (how fast you're moving). So it's "turning per distance traveled." The other formulas are faster ways to compute the same number.

In the last lesson we saw that $\vec T$ can only rotate, never stretch, and $\|\vec T\,'\|$ is how fast it rotates. Why divide by the speed? Picture driving the same curve twice, once slowly and once fast. On the fast run your direction swings around quicker, so $\|\vec T\,'\|$ is bigger, but the road hasn't changed. Dividing by the speed cancels that out, so curvature depends only on the shape, not on how fast you trace it. In the arc length parameter from the last lesson, where the speed is 1, it's just $\|d\vec T/ds\|$.

Circles are the reference. On a circle of radius $R$ you turn all the way around, $2\pi$ radians, over a distance of $2\pi R$, so the turning per distance is $\frac{2\pi}{2\pi R} = \frac1R$ everywhere. That gives every curve a circle to compare against. A curve's radius of curvature at a point is $\frac1\kappa$: the radius of the circle that best fits the curve there, called the osculating circle. It lies in the osculating plane from the T, N, B lesson.

The cross product formula is usually the easiest to compute, since it skips the messy step of differentiating $\vec T$. You just need $\vec r\,'$ and $\vec r\,''$. For a graph $y = f(x)$, plugging in $\vec r = \langle x, f(x), 0\rangle$ gives the last formula.

Highway and railway engineers limit curvature so the sideways acceleration $\kappa v^2$ stays comfortable at the design speed, and they ease curvature in gradually with transition curves.