How Sharply a Car Must Steer

A self-driving car follows a planned path, and at every point it must turn exactly as sharply as the path bends. Curvature measures that bend as one over the radius of the best-fitting circle. It sets the steering angle, and how slow the car must go.

How it works

Curvature

In one sentence

For a curve $y = f(x)$ , the curvature is $\kappa = |f''|/(1 + f'^2)^{3/2}$ , the reciprocal of the radius of the osculating circle; for a vector function, $\kappa = \lVert\mathbf T'(t)\rVert/\lVert\mathbf r'(t)\rVert$ .

General formula

{\htmlData{sym=0}{\kappa}} = \frac{|{\htmlData{sym=1}{y''}}|}{\left(1 + ({\htmlData{sym=2}{y'}})^2\right)^{3/2}}, \qquad {\htmlData{sym=3}{R}} = \frac1{\htmlData{sym=0}{\kappa}}, \qquad \tan{\htmlData{sym=4}{\delta}} = {\htmlData{sym=0}{\kappa}} {\htmlData{sym=5}{L}}

The second derivative measures bending, but steep stretches look stretched out, so dividing by $(1 + f'^2)^{3/2}$ measures bend per unit of distance actually traveled.

What the symbols mean

$\kappa$
Curvature
How sharply the path bends: 1 over the turning radius.
$y''$
Second derivative
How fast the path's slope changes.
$y'$
Slope
The path's direction at the point.
$R$
Turning radius
Radius of the best-fitting circle.
$\delta$
Steering angle
How far the front wheels turn.
$L$
Wheelbase
Distance between the car's axles.

At each point, the osculating circle is the circle that best hugs the path: same slope and same bend.
Its radius $R = 1/\kappa$ comes from the first two derivatives: a sharper bend means bigger $|y''|$ and a smaller circle.
A car with wheelbase $L$ follows a circle of radius $R$ when its steering angle satisfies $\tan\delta = L/R = \kappa L$ .
To keep sideways acceleration $v^2\kappa$ below the tires' grip, the car must slow down where $\kappa$ is large.

Learn the math

Tangent, Normal and Binormal VectorsCalc III · Lesson 12, Three-Dimensional Space CurvatureCalc III · Lesson 14, Three-Dimensional Space