The clothoid, also called radioid of arcs, Euler spiral (after Leonhard Euler) or Cornu spiral (in honor of Marie Alfred Cornu), is a curve tangent to the axis of the abscissa at the origin and whose radius of curvature decreases inversely proportional to the distance traveled on it. That is why at the origin point of the curve, the radius is infinite.

The usual mathematical expression is:

being.

The Cornu spiral, also known as clothoid, is the curve whose parametric equations are given by S(t) and C(t), the Fresnel integrals. Since:

In this parameterization the tangent vector has unit length and t is the arc length measured from (0,0) (and including sign), from which it follows that the curve has infinite length.

The Cornu spiral has the property that its curvature at any point is proportional to the distance along the curve measured from the origin. This property makes it useful as a "transition curve" in the layout of highways or railways, since a vehicle that follows said curve at a constant speed will have a constant angular acceleration. Thus, this curve is used for planimetric agreements in road and, especially, railway layouts (except for the use of the previous esplanade for which other agreement curves are used), in order to avoid discontinuities in the centripetal acceleration of the vehicles. The resulting transition curve has an infinite radius at the point tangent to the straight part of the layout, and radius R at the point of tangency with the uniform circular curve, in this way the type of curve on roads is straight-clotoid-circular-clotoid-straight section except for circular curves with a radius greater than five thousand meters (< 5,000 m) on roads of Groups 1 and 2 and for circular curves with a radius greater than two thousand five hundred meters (< 2,500 m) on roads of Group 3.

Likewise, sections of this clothoid spiral are commonly used in roller coasters, so some complete turns are known as "clothoid" loops.

In the geometric design of roads, normally used in horizontal layout.

On early railways, due to the low speeds and large radii used in curves, it was possible to ignore any type of transition between curve and straight line. It is from the century onwards that increases in speed give rise to the need for curves that gradually change their curvature. In 1862 Rankine in his work "Civil Engineering" already cites some curves that could be used, among them a proposal from 1828 or 1829 called the "sine curve" by William Gravatt and William Froude's adjustment curve from 1842 that approximated an elastic curve. The current equation given by Rankine is that of a cubic curve, or polynomial of degree 3.