page_title: Fresnel integral
section_title: Euler spiral
hierarchical_section_title: Fresnel integral / Euler spiral
context_page_description: The Fresnel integrals S(x) and C(x) are two transcendental functions named after Augustin-Jean Fresnel that are used in optics and are closely related to the error function. They arise in the description of near-field Fresnel diffraction phenomena and are defined through the following integral representations:
  The simultaneous parametric plot of S(x) and C(x) is the Euler spiral. Recently, they have been used in the design of highways and other engineering projects.
context_section_description: The Euler spiral, also known as Cornu spiral or clothoid, is the curve generated by a parametric plot of  against . The Cornu spiral was created by Marie Alfred Cornu as a nomogram for diffraction computations in science and engineering.
  From the definitions of Fresnel integrals, the infinitesimals  and  are thus:
  Thus the length of the spiral measured from the origin can be expressed as
  That is, the parameter  is the curve length measured from the origin , and the Euler spiral has infinite length. The vector  also expresses the unit tangent vector along the spiral, giving . Since t is the curve length, the curvature  can be expressed as
  Thus the rate of change of curvature with respect to the curve length is
  An Euler spiral has the property that its curvature at any point is proportional to the distance along the spiral, measured from the origin.  This property makes it useful as a transition curve in highway and railway engineering: If a vehicle follows the spiral at unit speed, the parameter  in the above derivatives also represents the time. Consequently, a vehicle following the spiral at constant speed will have a constant rate of angular acceleration.
  Sections from Euler spirals are commonly incorporated into the shape of roller-coaster loops to make what are known as clothoid loops.
caption: Euler spiral 
    
      
        
          (
          x
          ,
          y
          )
          =
          (
          C
          (
          t
          )
          ,
          S
          (
          t
          )
          )
        
      
      {\displaystyle (x,y)=(C(t),S(t))}
    
  . The spiral converges to the centre of the holes in the image as 
    
      
        
          t
        
      
      {\displaystyle t}
    
   tends to positive or negative infinity.