Clothoids

Clothoids, also called Euler spirals or Cornu spirals, are plane curves whose curvature increases linearly with arc length. In standard form, the curvature κ(s) at a point a distance s along the curve from a chosen origin is κ(s) = s / a^2, where a is a constant with units of length. Because curvature is the derivative of the tangent angle with respect to arc length, dθ/ds = κ(s) = s / a^2, so θ(s) = θ0 + s^2/(2a^2). The unit tangent is t(s) = (cos θ(s), sin θ(s)), and the curve r(s) is obtained by integrating t over s: r(s) = r0 + ∫0^s t(u) du. This integral yields expressions in terms of Fresnel integrals; the standard clothoid is often written in dimensionless form using t = s/a, with x(t) = a ∫0^t cos(τ^2/2) dτ and y(t) = a ∫0^t sin(τ^2/2) dτ, up to an overall rotation and translation.

Clothoids are notable for transitioning smoothly from straight lines to curved paths, making them attractive for design problems requiring gentle curvature variation. They have no simple closed-form in elementary functions, but are well described by the Fresnel integrals and numerical methods.

Applications include road and railway alignment, where gradual curvature changes reduce lateral acceleration, as well as roller coasters, autonomous vehicle paths, and computer graphics. The curve bears several names: Euler spiral in mathematics, Cornu spiral in optics, and clothoid in civil engineering, with the name reflecting its characteristic linear curvature with arc length.