An Euler spiral is a curve whose curvature changes linearly with its curve length (the curvature of a circular curve is equal to the reciprocal of the radius). This curve is also referred to as a clothoid or Cornu spiral. The behavior of Fresnel integrals can be illustrated by an Euler spiral, a connection first made by Marie Alfred Cornu in 1874.

The Euler spiral has applications to diffraction computations. They are also widely used in railway and highway engineering to design transition curves between straight and curved sections of railways or roads. A similar application is also found in photonic integrated circuits. The principle of linear variation of the curvature of the transition curve between a tangent and a circular curve defines the geometry of the Euler spiral:

The spiral has multiple names reflecting its discovery and application in multiple fields. The three major arenas were elastic springs ("Euler spiral", 1744), graphical computations in light diffraction ("Cornu spiral", 1874), and railway transitions ("the railway transition spiral", 1890).

Leonhard Euler's work on the spiral came after James Bernoulli posed a problem in the theory of elasticity: what shape must a pre-curved wire spring be in such that, when flattened by pressing on the free end, it becomes a straight line? Euler established the properties of the spiral in 1744, noting at that time that the curve must have two limits, points that the curve wraps around and around but never reaches. Thirty-eight years later, in 1781, he reported his discovery of the formula for the limit (by "happy chance").

Augustin Fresnel, working in 1818 on the diffraction of light, developed the Fresnel integral that defines the same spiral. He was unaware of Euler's integrals or the connection to the theory of elasticity. In 1874, Alfred Marie Cornu showed that diffraction intensity could be read off a graph of the spiral by squaring the distance between two points on the graph. In his biographical sketch of Cornu, Henri Poincaré praised the advantages of the "spiral of Cornu" over the "unpleasant multitude of hairy integral formulas". Ernesto Cesàro chose to name the same curve "clothoid" after Clotho, one of the three Fates who spin the thread of life in Greek mythology.

The third independent discovery occurred in the 1800s when various railway engineers sought a formula for gradual curvature in track shape. By 1880 Arthur Newell Talbot worked out the integral formulas and their solution, which he called the "railway transition spiral". The connection to Euler's work was not made until 1922.

Unaware of the solution of the geometry by Euler, William Rankine cited the cubic curve (a polynomial curve of degree 3), which is an approximation of the Euler spiral for small angular changes in the same way that a parabola is an approximation to a circular curve.^{[citation needed]}

The graph on the right illustrates an Euler spiral used as an easement (transition) curve between two given curves, in this case a straight line (the negative x axis) and a circle. The spiral starts at the origin in the positive x direction and gradually turns anticlockwise to osculate the circle.