Euler spiral

Euler spiral
A double-end Euler spiral.

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). Euler spirals are also commonly referred to as spiros, clothoids or Cornu spirals.

Euler spirals have applications to diffraction computations. They are also widely used as transition curve in railroad engineering/highway engineering for connecting and transiting the geometry between a tangent and a circular curve. 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:

  • Its curvature begins with zero at the straight section (the tangent) and increases linearly with its curve length.
  • Where the Euler spiral meets the circular curve, its curvature becomes equal to that of the latter.



Track transition curve

An object traveling on a circular path experiences a centripetal acceleration. When a vehicle traveling on a straight path approaches a circular path, it experiences a sudden centripetal acceleration starting at the tangent point; and thus centripetal force acts instantly causing much discomfort.

On early railroads this instant application of lateral force was not an issue since low speeds and wide-radius curves were employed (lateral forces on the passengers and the lateral sway was small and tolerable). As speeds of rail vehicles increased over the years, it became obvious that an easement is necessary so that the centripetal acceleration increases linearly with the traveled distance. Given the expression of centripetal acceleration V² / R, the obvious solution is to provide an easement curve whose curvature, 1 / R, increases linearly with the traveled distance. This geometry is an Euler spiral.

Unaware of the solution of the geometry by Leonhard Euler, 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.

Marie Alfred Cornu (and later some civil engineers) also solved the calculus of Euler spiral independently. Euler spirals are now widely used in rail and highway engineering for providing a transition or an easement between a tangent and a horizontal circular curve.


The Cornu spiral can be used to describe a diffraction pattern.[1]



R\, Radius of curvature
R_c\, Radius of Circular curve at the end of the spiral
\theta\, Angle of curve from beginning of spiral (infinite R) to a particular point on the spiral.
This can also be measured as the angle between the initial tangent and the tangent at the concerned point.
\theta _s\, Angle of full spiral curve
L , s\, Length measured along the spiral curve from its initial position
L_s , s_o\, Length of spiral curve


Easement Curve.png

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.

The spiral is a small segment of the above double-end Euler spiral in the first quadrant.

From the definition of the curvature,
\frac {1}{R} = \frac {d\theta}{ds} \propto s
R s = \text{constant} = R_c s_o\,
\frac {d\theta}{ds} = \frac {s}{R_c s_o}
We write in the format,
\frac {d\theta}{ds} = 2a^2 s
2a^2= \frac {1}{R_c s_o}
a = \frac {1}{\sqrt {2R_c s_o} }
\theta = (a s)^2\,

x & = \int_0^L \cos\theta \, ds \\
  & = \int_0^L \cos \left[ (a s)^2 \right] ds
s' = a s \,
ds = \frac{ds'}{a}\,
x = \frac{1}{a} \int_0^{L'} \cos {s}^2 ds

y & = \int_0^L \sin\theta \, ds \\
  & = \int_0^L \sin \left[ (a s)^2 \right] ds \\
  & = \frac{1}{a} \int_0^{L'} \sin {s}^2 \, ds 

Expansion of Fresnel integral

If a = 1, which is the case for normalized Euler curve, then the Cartesian coordinates are given by Fresnel integrals (or Euler integrals):

      C(L) =\int_0^L\cos s^2 \, ds
      S(L) = \int_0^L\sin s^2 \, ds

Expand C(L) according to power series expansion of cosine:

      \cos \theta = 1 - \frac{\theta^2}{2!} + \frac{\theta^4}{4!} - \frac{\theta^6}{6!} + \cdots
      C(L) = \int_0^L \cos s^2 \, ds
           = \int_0^L (1 - \frac{s^4}{2!} + \frac{s^8}{4!} - \frac{s^{12}}{6!} + \cdots) \,  ds
           = L - \frac{L^5}{5 \times 2!} + \frac{L^9}{9 \times 4!} - \frac{L^{13}}{13 \times 6!} +\cdots

Expand S(L) according to power series expansion of sine:

      \sin \theta = \theta - \frac{\theta^3}{3!} + \frac{\theta^5}{5!} - \frac{\theta^7}{7!} + \cdots
      S(L) = \int_0^L \sin s^2 \, ds
           = \int_0^L (s^2 - \frac{s^6}{3!} + \frac{s^{10}}{5!} - \frac{s^{14}}{7!} + \cdots) \,  ds
           = \frac{L^3}{3} - \frac{L^7}{7 \times 3!} + \frac{L^{11}}{11 \times 5!} - \frac{L^{15}}{15 \times 7!} +\cdots

Normalization and conclusion

For a given Euler curve with:

2RL = 2R_c L_s = \frac{1}{a^2} \,


\frac{1}{R} = \frac{L}{R_c L_s} = 2a^2L \,


x=\frac{1}{a} \int_0^{L'} \cos s^2 \, ds
y=\frac{1}{a} \int_0^{L'} \sin s^2 \, ds \,

where L' = aL \, and a = \frac{1}{\sqrt{2R_c L_s}}.

The process of obtaining solution of (x, y) of an Euler spiral can thus be described as:

  • Map L of the original Euler spiral by multiplying with factor a to L' of the normalized Euler spiral;
  • Find (x′, y′) from the Fresnel integrals; and
  • Map (x′, y′) to (x, y) by scaling up (denormalize) with factor 1 / a. Note that 1 / a > 1.

In the normalization process,

R'_c & = \frac{R_c}{\sqrt{2 R_c L_s}} \\
     & = \sqrt{\frac{R_c}{2L_s}} \\

L'_s & = \frac{L_s}{\sqrt{2R_c L_s}} \\
     & = \sqrt{\frac{L_s}{2R_c}}


2R'_c L'_s & = 2 \sqrt{\frac{R_c}{2L_s} } \sqrt{\frac{L_s}{2 R_c}} \\
           & = \tfrac{2}{2} \\
           & = 1

Generally the normalization reduces L' to a small value (<1) and results in good converging characteristics of the Fresnel integral manageable with only a few terms (at a price of increased numerical instability of the calculation, esp. for bigger \theta\, values.).



   R_c & = 300\mbox{m} \\
   L_s &= 100\mbox{m}


   \theta_s & = \frac{L_s} {2R_c} \\
       & = \frac{100} {2 \times 300} \\
       & = 0.1667 \ \mbox{radian} \\


 2R_c L_s = 60,000 \,

We scale down the Euler spiral by √60,000, i.e.100√6 to normalized Euler spiral that has:

      R'_c = \tfrac{3}{\sqrt{6}}\mbox{m} \\
      L'_s = \tfrac{1}{\sqrt{6}}\mbox{m} \\

   2R'_c L'_s & = 2 \times \tfrac{3}{\sqrt{6}} \times \tfrac{1}{\sqrt{6}} \\
              & = 1 


    \theta_s & = \frac{L'_s}{2R'_c} \\
       & = \frac{\tfrac{1}{\sqrt{6}}} {2 \times \tfrac{3}{\sqrt{6}}} \\
       & = 0.1667 \ \mbox{radian} \\

The two angles \theta_s\, are the same. This thus confirm that the original and normalized Euler spirals are having geometric similarity. The locus of the normalized curve can be determined from Fresnel Integral, while the locus of the original Euler spiral can be obtained by scaling back / up or denormalizing.

Other properties of normalized Euler spiral

Normalized Euler spiral can be expressed as:

x = \int_0^L \cos s^2 ds
y = \int_0^L \sin s^2 ds

Normalized Euler spiral has the following properties:

2 R_c L_s = 1 \,\!
\theta_s = \frac{L_s}{2 R_c} = L_s ^2


\theta = \theta _s\cdot\frac{L^2}{L_s^2} = L^2
\frac{1}{R} = \frac{d\theta}{dL} = 2L

Note that 2RcLs = 1 also means 1 / Rc = 2Ls, in agreement with the last mathematical statement.

Code for producing an Euler spiral

The following Sage code produces the second graph above. The first four lines express the Euler spiral component. Fresnel functions could not be found. Instead, the integrals of two expanded Taylor series are adopted. The remaining code expresses respectively the tangent and the circle, including the computation for the center coordinates.

  p = integral(taylor(cos(L^2), L, 0, 12), L)
  q = integral(taylor(sin(L^2), L, 0, 12), L)
  r1 = parametric_plot([p, q], (L, 0, 1), color = 'red')
  r2 = line([(-1.0, 0), (0,0)], rgbcolor = 'blue')
  x1 = p.subs(L = 1)
  y1 = q.subs(L = 1)
  R = 0.5
  x2 = x1 - R*sin(1.0)
  y2 = y1 + R*cos(1.0)
  r3 = circle((x2, y2), R, rgbcolor = 'green')
  show(r1 + r2 + r3, aspect_ratio = 1, axes=false)

The following is Mathematica code for the Euler spiral component (it works directly in

   {FresnelC[Sqrt[2/\[Pi]] t]/Sqrt[2/\[Pi]],
    FresnelS[Sqrt[2/\[Pi]] t]/Sqrt[2/\[Pi]]},
   {t, -10, 10}]

See also


  1. ^ Eugene Hecht (1998). Optics (3rd edition). Addison-Wesley. p. 491. ISBN 0201304252. 

External links

Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Ulam spiral — The Ulam spiral to 150 iterations. Red dots represent prime numbers; blue dots represent composite numbers, with the size of the dot indicating the degree of compositeness. The Ulam spiral, or prime spiral (in other languages also called the Ulam …   Wikipedia

  • Sacks spiral — Robert Sacks devised the Sacks spiral, a variant of the Ulam spiral, in 1994. It differs from Ulam s in three ways: it places points on an Archimedean spiral rather than the square spiral used by Ulam, it places zero in the center of the spiral,… …   Wikipedia

  • Fresnel integral — S(x) and C(x) The maximum of C(x) is about 0.977451424. If πt²/2 were used instead of t², then the image would be scaled vertically and horizontally (see below). Fresnel integrals, S(x) and C(x), are two transcendental functions named aft …   Wikipedia

  • Track transition curve — The red Euler spiral is an example of an easement curve between a blue straight line and a circular arc, shown in green …   Wikipedia

  • Trigonometric integral — Si(x) (blue) and Ci(x) (green) plotted on the same plot. In mathematics, the trigonometric integrals are a family of integrals which involve trigonometric functions. A number of the basic trigonometric integrals are discussed at the list of… …   Wikipedia

  • Fresnel diffraction — In optics, the Fresnel diffraction equation for near field diffraction, is an approximation of Kirchhoff Fresnel diffraction that can be applied to the propagation of waves in the near field.[1] The near field can be specified by the Fresnel… …   Wikipedia

  • Cycle Messenger World Championships — The Cycle Messenger World Championships, or CMWCs, are an annual urban cycling competition whereby cycle messengers and cycling enthusiasts showcase their skills in an array of events, many of which simulate everyday tasks for a cycle messenger.… …   Wikipedia

  • Polar coordinate system — Points in the polar coordinate system with pole O and polar axis L. In green, the point with radial coordinate 3 and angular coordinate 60 …   Wikipedia

  • mathematics — /math euh mat iks/, n. 1. (used with a sing. v.) the systematic treatment of magnitude, relationships between figures and forms, and relations between quantities expressed symbolically. 2. (used with a sing. or pl. v.) mathematical procedures,… …   Universalium

  • Fictitious force — Classical mechanics Newton s Second Law History of classical mechanics  …   Wikipedia