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

Graphical or geometric meaning

The helix climbs one vertical unit for each radian while circling a unit cylinder. Its tangent tilts equally between horizontal motion and vertical motion. The osculating circle at a point has radius $1/\kappa=2$, capturing local bending rather than cylinder radius.

Common mistakes and why they fail

Verification and reasonableness checks

• Verify curvature is unchanged by a regular reparametrization.
• Compare with known line or circle cases.
• Confirm cross-product numerator and speed-cubed denominator have compatible units.

Curvature separates turning from speed

For a regular curve, arc length accumulates speed: $s=\int\|\mathbf r'(t)\|dt$. Curvature measures change of the unit tangent per unit distance, $\kappa=\|d\mathbf T/ds\|$, so it is unaffected by how quickly the parameter traverses the path. The computational formula $\kappa=\|\mathbf r'\times\mathbf r''\|/\|\mathbf r'\|^3$ requires $\mathbf r'\ne\mathbf0$. A straight line has zero curvature; a circle of radius $R$ has constant curvature $1/R$, providing scale checks. If speed is constant, tangential acceleration vanishes and acceleration points in the normal direction. Before using a formula, verify regularity and consistent units: curvature has reciprocal-length units, while arc length has length units.

Practice

1. Find the length of $\langle3t,4t,0\rangle$, $0\le t\le2$.
2. What is curvature of a circle of radius $5$?
3. What must hold at a regular point?

Connections and next steps

• Space Curves
• Derivatives of Vector Functions
• Vector-Valued Functions