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Curvature — Definition, Formula & Examples

Curvature is a measure of how quickly a curve changes direction at a given point. A straight line has zero curvature everywhere, while a tight circle has high curvature.

For a smooth, parameterized curve r(t)\mathbf{r}(t), the curvature κ\kappa at a point is defined as the magnitude of the rate of change of the unit tangent vector T\mathbf{T} with respect to arc length ss: κ=∥dTds∥\kappa = \left\|\frac{d\mathbf{T}}{ds}\right\|. Equivalently, κ=∥r′(t)×r′′(t)∥∥r′(t)∥3\kappa = \frac{\|\mathbf{r}'(t) \times \mathbf{r}''(t)\|}{\|\mathbf{r}'(t)\|^3}.

Key Formula

κ=∣f′′(x)∣(1+(f′(x))2)3/2\kappa = \frac{|f''(x)|}{\left(1 + (f'(x))^2\right)^{3/2}}
Where:
  • κ\kappa = Curvature at the point $x$
  • f′(x)f'(x) = First derivative of the curve $y = f(x)$
  • f′′(x)f''(x) = Second derivative of the curve $y = f(x)$

How It Works

Curvature quantifies bending by comparing how the tangent direction rotates relative to distance traveled along the curve. A circle of radius RR has constant curvature κ=1/R\kappa = 1/R, so smaller circles curve more sharply. At any point on a general curve, the osculating circle (the best-fit circle at that point) has radius ρ=1/κ\rho = 1/\kappa, called the radius of curvature. For a plane curve given by y=f(x)y = f(x), a convenient formula is κ=∣f′′(x)∣(1+(f′(x))2)3/2\kappa = \frac{|f''(x)|}{(1 + (f'(x))^2)^{3/2}}.

Worked Example

Problem: Find the curvature of y=x2y = x^2 at the point x=1x = 1.
Find derivatives: Compute the first and second derivatives of f(x)=x2f(x) = x^2.
f′(x)=2x,f′′(x)=2f'(x) = 2x, \quad f''(x) = 2
Evaluate at x = 1: Substitute x=1x = 1 into each derivative.
f′(1)=2,f′′(1)=2f'(1) = 2, \quad f''(1) = 2
Apply the formula: Plug into the curvature formula for a plane curve.
κ=∣2∣(1+22)3/2=2(5)3/2=255=2525\kappa = \frac{|2|}{(1 + 2^2)^{3/2}} = \frac{2}{(5)^{3/2}} = \frac{2}{5\sqrt{5}} = \frac{2\sqrt{5}}{25}
Answer: The curvature at x=1x = 1 is κ=2525≈0.179\kappa = \frac{2\sqrt{5}}{25} \approx 0.179. The radius of curvature there is ρ≈5.59\rho \approx 5.59.

Why It Matters

Curvature is essential in physics for describing particle motion along curved paths, where the normal component of acceleration depends directly on κ\kappa. Engineers use curvature to design roads and roller coasters, ensuring turns are safe at given speeds. In differential geometry and general relativity, the concept generalizes to surfaces and spacetime.

Common Mistakes

Mistake: Using κ=∣f′′(x)∣\kappa = |f''(x)| without the denominator (1+(f′)2)3/2(1+(f')^2)^{3/2}.
Correction: The second derivative alone only equals curvature when the slope f′(x)=0f'(x) = 0. The denominator accounts for how arc length stretches relative to xx; omitting it gives incorrect values wherever the curve is not nearly flat.