Curvature of a Plane Curve

Signed curvature: turning rate of the unit tangent.

The idea

The curvature of a plane curve measures how sharply the curve bends at each point.

Parametrize the curve by arc length and follow its unit tangent vector. Along a straight line the tangent never turns; around a circle it turns at a constant rate, faster for a smaller circle. The signed curvature $\kappa$ at a point is the rate of change of the tangent's direction angle per unit of arc length.

Measuring per unit of arc length, rather than per unit of an arbitrary parameter, makes $\kappa$ a property of the curve: it is invariant under reparametrization and under translations and rotations of the plane. Reversing the direction of travel, or reflecting the plane, flips its sign. The magnitude $|\kappa|$ gives the sharpness of the bend, and $1/|\kappa|$ is the radius of the osculating circle, the circle that best fits the curve at a point $P$; a circle of radius $R$ has $|\kappa| = 1/R$.

For a curve given as a graph $y = f(x)$, $\kappa = \frac{f''(x)}{\bigl(1 + f'(x)^{2}\bigr)^{3/2}}.$ The numerator is the concavity and carries the sign; the denominator corrects for the fact that $x$ is not arc length, since a steep stretch of graph covers more distance than $x$-interval.

Ways to work on it

• Walkthrough. The graph formula, sign of bending, and a circle's curvature.
• Practice. Curvature of a graph at a horizontal-tangent point.
• Hardest. Parametric curvature where the arc-length rescaling matters.

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