Arc Length & Curvature of Space Curves

Speed, the arc-length integral, and how sharply a space curve bends.

The idea

Arc length and curvature are the two basic geometric measurements of a space curve $\mathbf{r}(t)$: how far it runs, and how sharply it bends.

The derivative $\mathbf{r}'(t)$ is the velocity of the moving point and $|\mathbf{r}'(t)|$ its speed. Distance is speed accumulated over time, so the arc length covered between $t = a$ and $t = b$ is $L = \int_{a}^{b} |\mathbf{r}'(t)|\,dt.$

Bending is a statement about direction, not speed. The unit tangent $\mathbf{T} = \mathbf{r}'/|\mathbf{r}'|$ keeps only the direction of travel: it is constant along a straight line, and it swings around as the curve turns. Curvature measures how fast $\mathbf{T}$ turns — per unit of distance rather than of time, since the shape of the curve does not depend on how fast we trace it. Dividing the turning rate by the speed makes that conversion: $\kappa = \left| \frac{d\mathbf{T}}{ds} \right| = \frac{|\mathbf{T}'(t)|}{|\mathbf{r}'(t)|}.$ A circle of radius $R$ has $\kappa = 1/R$ at every point: the larger the circle, the gentler the bend, and a straight line has $\kappa = 0$. Turned around, this gives curvature its picture: at a point $P$ of a curve, the circle that best fits the curve there — its osculating circle — has radius $1/|\kappa|$.

Ways to work on it

• Walkthrough. Speed, the arc-length integral, and why curvature is turning per unit arc length.
• Practice. Arc length of a constant-speed space curve.
• Hardest. Compute a space curve's curvature with the cross-product formula.

Not sure where to start? Take the ten-question placement test.