# Definition:Cornu Spiral

## Definition

The Cornu spiral is the locus $C$ of the equation expressed in intrinsic coordinates as:

$s = a^2 \kappa$

where:

$s$ denotes the length of arc at a point of $C$ from the origin
$\kappa$ denotes the curvature of $C$ at that point.

## Also presented as

The Cornu spiral can also be presented in the form:

$s = a^2 \dfrac {\d \psi} {\d s}$

where $\psi$ is the turning angle of $C$ at the point where the length of arc from the origin is $s$.

## Also known as

Other names for the Cornu spiral include:

Euler spiral (for Leonhard Paul Euler)
klothoid or clothoid

## Source of Name

This entry was named for Marie Alfred Cornu.

## Historical Note

The Cornu spiral is used primarily for:

analysis of intensities of diffraction patterns
designing curves on railway lines and roller coasters so as to provide a smooth curvature transition