Arc Length and Curvature

The central idea

For a $C^1$ curve $\mathbf r:I\to\mathbb R^3$, speed is $v=\|\mathbf r'(t)\|$ and arc length is $L=\int_a^b v(t)dt$. At a regular $C^2$ point, curvature is $\kappa=\|\mathbf T'(t)\|/\|\mathbf r'(t)\|$, or $\|\mathbf r'\times\mathbf r''\|/\|\mathbf r'\|^3$ in $\mathbb R^3$.

Definitions, hypotheses, and notation

Curvature is geometric because dividing $d\mathbf T/dt$ by $ds/dt$ produces $d\mathbf T/ds$. Tangential acceleration changes speed, whereas normal acceleration $v^2\kappa\mathbf N$ changes direction. Thus a particle may travel a highly curved path at constant speed or accelerate along a straight line with zero curvature.

The cross-product formula is specific to three dimensions; the definition through $d\mathbf T/ds$ works in any Euclidean dimension. Piecewise smooth curves require lengths to be added across pieces, and corners have no ordinary curvature value at the joining point.

Conceptual meaning

Arc length accumulates distance traveled, while curvature measures turning per unit arc length. A line has zero curvature; a circle of radius $R$ has curvature $1/R$. The unit tangent $\mathbf T=\mathbf r'/\|\mathbf r'\|$ separates direction from speed.

A dependable method and decision rule

1. State the parameter interval and confirm the curve is regular where formulas are used.
2. Differentiate componentwise to obtain velocity and speed.
3. Integrate speed for length, accounting for repeated tracing.
4. For curvature, compute either $\mathbf T'$ or $\mathbf r'\times\mathbf r''$.
5. Check dimensions: length has distance units and curvature has inverse-distance units.

Fully worked example

Common mistakes and why they fail