View source for Euler spiral

{{short description|Curve whose curvature changes linearly}}
[[File:Euler spiral.svg|300px|thumb|A double-end Euler spiral.  The curve continues to converge to the points marked, as {{mvar|t}} tends to positive or negative infinity.]]
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'''.<ref>{{Cite book |last=Von Seggern |first=David H. |title=Practical handbook of curve design and generation |date=1994 |publisher=CRC Press |isbn=978-0-8493-8916-0 |location=Boca Raton, Fla.}}</ref><ref name=LevienHistory>Levien, Raph. [https://digitalassets.lib.berkeley.edu/techreports/ucb/text/EECS-2008-111.pdf "The Euler spiral: a mathematical history."] Rapp. tech (2008).</ref> The behavior of [[Fresnel integrals]] can be illustrated by an Euler spiral, a connection first made by [[Marie Alfred Cornu]] in 1874.<ref>Marie Alfred Cornu. Méthode nouvelle pour la discussion des problèmes de diffraction dans le cas d’une onde cylindrique. Journal de Physique théorique et appliquée, pages 5–15, 1874.</ref>  Euler's spiral is a type of '''superspiral''' that has the property of a monotonic curvature function.<ref>{{Citation
| last = Ziatdinov
| first = R.
| title = Family of superspirals with completely monotonic curvature given in terms of Gauss hypergeometric function
| journal = Computer Aided Geometric Design
| volume = 29
| issue = 7
| pages = 510–518
| year = 2012
| doi = 10.1016/j.cagd.2012.03.006
| language = English
}}</ref>

The Euler spiral has applications to [[diffraction]] computations. They are also widely used in [[railway engineering|railway]] and [[highway engineering]] to design [[transition curve]]s between straight and curved sections of railways or roads. A similar application is also found in [[photonic integrated circuit]]s. 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.

== History ==
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).<ref name=LevienHistory/>

[[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").<ref name=LevienHistory/>

[[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]].<ref name=LevienHistory/>

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.<ref name=LevienHistory/>

Unaware of the solution of the geometry by Euler, [[William John Macquorn Rankine|William Rankine]] cited the [[Polynomial#Graphs|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.{{Cn|date=September 2024}}

==Applications==

===Track transition curve===
{{Main article|Track transition curve}}

[[File:CornuSpiralAnimation.gif|thumb|Animation depicting evolution of a Cornu spiral with the tangential circle with the same radius of curvature as at its tip, also known as an [[osculating circle]].]]

To travel along a circular path, an object needs to be subject to a [[Uniform circular motion|centripetal acceleration]] (for example: the Moon circles around the Earth because of gravity; a car turns its front wheels inward to generate a centripetal force). If a vehicle traveling on a straight path were to suddenly transition to a tangential circular path, it would require centripetal acceleration suddenly switching at the tangent point from zero to the required value; this would be difficult to achieve (think of a driver instantly moving the steering wheel from straight line to turning position, and the car actually doing it), putting mechanical stress on the vehicle's parts, and causing much discomfort (due to lateral [[Jerk (physics)|jerk]]).

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 smoothly with the traveled distance. Given the expression of centripetal acceleration {{math|{{sfrac|''v''<sup>2</sup>|''r''}}}}, the obvious solution is to provide an easement curve whose curvature, {{math|{{sfrac|1|''R''}}}}, increases linearly with the traveled distance.

===Optics===

In [[optics]], the term Cornu spiral is used.<ref>{{Cite book |last1=Born |first1=Max |title=Principles of optics: electromagnetic theory of propagation, interference and diffraction of light |last2=Wolf |first2=Emil |date=1993 |publisher=Pergamon Press |isbn=978-0-08-026481-3 |edition=6. ed., reprinted (with corrections) |location=Oxford}}</ref>{{rp|432}}
The Cornu spiral can be used to describe a [[diffraction]] pattern.<ref name=Hecht>{{cite book | title = Optics | edition = 3rd | author = Eugene Hecht | publisher = Addison-Wesley | year = 1998 | isbn = 978-0-201-30425-1 | page = 491 }}</ref>
Consider a plane wave with phasor amplitude {{math|''E''<sub>0</sub>''e''<sup>−''jkz''</sup>}} which is diffracted by a "knife edge" of height {{mvar|h}} above {{math|1=''x'' = 0}} on the {{math|1=''z'' = 0}} plane. Then the diffracted wave field can be expressed as
<math display="block"> \mathbf{E}(x, z) = E_0 e^{-jkz} \frac{\mathrm{Fr}(\infty) - \mathrm{Fr}\left(\sqrt{\frac{2}{\lambda z}}(h-x)\right)}{\mathrm{Fr}(\infty) - \mathrm{Fr}(-\infty)},</math>
where {{math|Fr(''x'')}} is the [[Fresnel integral]] function, which forms the Cornu spiral on the complex plane.

So, to simplify the calculation of plane wave attenuation as it is diffracted from the knife-edge, one can use the diagram of a Cornu spiral by representing the quantities {{math|Fr(''a'') − Fr(''b'')}} as the physical distances between the points represented by {{math|Fr(''a'')}} and {{math|Fr(''b'')}} for appropriate {{mvar|a}} and {{mvar|b}}. This facilitates a rough computation of the attenuation of the plane wave by the knife edge of height {{mvar|h}} at a location {{math|(''x'', ''z'')}} beyond the knife edge.

=== Integrated optics ===
Bends with continuously varying radius of curvature following the Euler spiral are also used to reduce losses in [[photonic integrated circuit]]s, either in singlemode [[waveguide (optics)|waveguides]],<ref>
{{cite journal
 | last = Kohtoku
 | first = M.
 | display-authors = etal
 | date = 7 July 2005
 | title = New Waveguide Fabrication Techniques for Next-generation PLCs
 | url = https://www.ntt-review.jp/archive/ntttechnical.php?contents=ntr200507037.pdf
 | journal    = NTT Technical Review
 | volume     = 3
 | issue      = 7
 | pages      = 37–41
 | access-date = 24 January 2017
}}</ref><ref>
{{cite journal
| last       = Li
| first      = G.
| display-authors      = etal
| date       = 11 May 2012
| title      = Ultralow-loss, high-density SOI optical waveguide routing for macrochip interconnects
| journal    = Optics Express
| volume     = 20
| issue      = 11
| pages      = 12035–12039
| doi        = 10.1364/OE.20.012035
| pmid      = 22714189
| bibcode = 2012OExpr..2012035L
| doi-access      = free
}}
</ref> to smoothen the abrupt change of curvature and suppress coupling to radiation modes, or in multimode waveguides,<ref>
{{cite journal
| last       = Cherchi
| first      = M.
| display-authors      = etal
| date       = 18 July 2013
| title      = Dramatic size reduction of waveguide bends on a micron-scale silicon photonic platform
| journal    = Optics Express
| volume     = 21
| issue      = 15
| pages      = 17814–17823
| doi        = 10.1364/OE.21.017814
| pmid      = 23938654
| arxiv      = 1301.2197
| bibcode = 2013OExpr..2117814C
}}
</ref> in order to suppress coupling to higher order modes and ensure effective singlemode operation.
A pioneering and very elegant application of the Euler spiral to waveguides had been made as early as 1957,<ref>
{{cite journal
| last       = Unger
| first      = H.G.
| date       = September 1957
| title      = Normal Mode Bends for Circular Electric Waves
| journal    = The Bell System Technical Journal
| volume     = 36
| issue      = 5
| pages      = 1292–1307
| doi        = 10.1002/j.1538-7305.1957.tb01509.x
}}</ref> with a hollow metal [[waveguide (electromagnetism)|waveguide]] for microwaves. There the idea was to exploit the fact that a straight metal waveguide can be physically bent to naturally take a gradual bend shape resembling an Euler spiral.

=== Feynman's path integral ===
In the [[path integral formulation]] of quantum mechanics, the probability amplitude for propagation between two points can be visualized by connecting [[action (physics)|action]] phase arrows for each time step between the two points. The arrows spiral around each endpoint forming what is termed a Cornu spiral.<ref>{{Cite journal |last1=Taylor |first1=Edwin F. |last2=Vokos |first2=Stamatis |last3=O’Meara |first3=John M. |last4=Thornber |first4=Nora S. |date=1998-03-01 |title=Teaching Feynman's sum-over-paths quantum theory |url=https://pubs.aip.org/cip/article/12/2/190/510288/Teaching-Feynman-s-sum-over-paths-quantum-theory |journal=Computers in Physics |language=en |volume=12 |issue=2 |pages=190–199 |doi=10.1063/1.168652 |issn=0894-1866|doi-access=free |bibcode=1998ComPh..12..190T }}</ref>

=== Auto racing ===
Motorsport author Adam Brouillard has shown the Euler spiral's use in optimizing the [[racing line]] during the corner entry portion of a turn.<ref>{{Cite book|title=The Perfect Corner: A Driver's Step-By-Step Guide to Finding Their Own Optimal Line Through the Physics of Racing|last1=Development|first1=Paradigm Shift Driver|last2=Brouillard|first2=Adam|date=2016-03-18| publisher=Paradigm Shift Motorsport Books| isbn=9780997382426|language=en}}</ref>

=== Typography and digital vector drawing ===
[[Raph Levien]] has released Spiro as a toolkit for curve design, especially font design, in 2007<ref>{{Cite web | title=Spiro | url=http://levien.com/spiro/}}</ref><ref>{{cite web |url=http://www.typophile.com/node/33531 |title={{!}} Spiro 0.01 release {{!}} Typophile |website=www.typophile.com |url-status=dead |archive-url=https://web.archive.org/web/20070510051459/http://typophile.com/node/33531 |archive-date=2007-05-10}}</ref> under a free licence. This toolkit has been implemented quite quickly afterwards in the font design tool [[Fontforge]] and the digital vector drawing [[Inkscape]].

=== Map projection ===
Cutting a sphere along a spiral with width {{math|{{sfrac|1|''N''}}}} and flattening out the resulting shape yields an Euler spiral when {{mvar|n}} tends to the infinity.<ref>{{Cite journal|last1=Bartholdi|first1=Laurent|last2=Henriques|first2=André |date=2012|title=Orange Peels and Fresnel Integrals|journal=The Mathematical Intelligencer|language=en|volume=34|issue=3| pages=1–3| doi=10.1007/s00283-012-9304-1|issn=0343-6993|arxiv=1202.3033|s2cid=52592272}}</ref> If the sphere is the [[globe]], this produces a [[map projection]] whose distortion tends to zero as {{mvar|n}} tends to the infinity.<ref name=":0">{{Cite web| url=https://www.youtube.com/watch?v=D3tdW9l1690 |archive-url=https://ghostarchive.org/varchive/youtube/20211221/D3tdW9l1690 |archive-date=2021-12-21 |url-status=live|title=A Strange Map Projection (Euler Spiral) - Numberphile|website=[[YouTube]] |date=13 November 2018 }}{{cbignore}}</ref>

=== Whisker shapes ===
Natural shapes of rats' [[whiskers]] are well approximated by segments of Euler spirals; for a single rat all of the whiskers can be approximated as segments of the same spiral.<ref>{{cite journal
| last = Towal | first = R.B. | display-authors = etal
| date = 7 April 2011
| title = The Morphology of the Rat Vibrissal Array: A Model for Quantifying Spatiotemporal Patterns of Whisker-Object Contact 
| journal = PLOS Computational Biology
| volume = 7 | issue = 4 | pages = e1001120
| doi = 10.1371/journal.pcbi.1001120 | doi-access= free | pmid = 21490724 | pmc = 3072363
| bibcode = 2011PLSCB...7E1120T }} {{pb}} {{cite journal
| last = Starostin | first = E.L. | display-authors = etal
| date = 15 January 2020
| title = The Euler spiral of rat whiskers
| journal = Science Advances
| volume = 6 | issue = 3 | pages = eaax5145
| doi = 10.1126/sciadv.aax5145 | doi-access= free  | pmid = 31998835 | pmc = 6962041 | bibcode = 2020SciA....6.5145S
}}</ref> The two parameters of the [[Cesàro equation]] for an Euler spiral segment might give insight into the [[keratin]]ization mechanism of whisker growth.<ref>{{cite journal
| last1 = Luo |first1=Y. |last2=Hartmann |first2=M.J. 
| date = Jan 2023
| title = On the intrinsic curvature of animal whiskers
| journal = PLOS ONE 
| volume = 18  | issue = 1  | pages = e0269210 
| doi = 10.1371/journal.pone.0269210 | doi-access = free | pmid = 36607960  | pmc = 9821693
|bibcode=2023PLoSO..1869210L }}</ref>

==Formulation==

===Symbols===
{|
| {{mvar|R}} || Radius of curvature
|-
| {{math|''R<sub>c</sub>''}} || Radius of circular curve at the end of the spiral
|-
| {{mvar|θ}} || Angle of curve from beginning of spiral (infinite {{mvar|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.
|-
| {{math|''θ<sub>s</sub>''}} || Angle of full spiral curve
|-
| {{mvar|L}}, {{mvar|s}} || Length measured along the spiral curve from its initial position
|-
| {{math|''L<sub>s</sub>''}}, {{math|''s<sub>o</sub>''}} || Length of spiral curve
|}

{{math proof | title = Derivation | proof =
[[File:Easement curve.svg|360px|right]]
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 {{mvar|x}} axis) and a circle.  The spiral starts at the origin in the positive {{mvar|x}} direction and gradually turns anticlockwise to [[osculation|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, <math display="block">\frac {1}{R} = \frac {d\theta}{ds} \propto s </math> i.e., <math display="block">\begin{align}
R s = \text{constant} &= R_c s_o \\
\frac {d\theta}{ds} &= \frac {s}{R_c s_o}
\end{align}</math>
We write in the format, <math display="block">\frac {d\theta}{ds} = 2a^2 s</math> where <math display="block">2a^2= \frac {1}{R_c s_o}</math> or <math display="block">a = \frac {1}{\sqrt {2R_c s_o} }</math>
thus <math display="block">\theta = (a s)^2</math>
Now <math display="block">x = \int_0^L \cos\theta \, ds = \int_0^L \cos \left[ \left(a s\right)^2 \right] \,ds</math>
If <math display="block">s' = a s </math>
Then <math display="block">ds = \frac{ds'}{a}</math>
Thus <math display="block">\begin{align}
x &= \frac{1}{a} \int_0^{L'} \cos \left(s^2\right) \,ds \\
y & = \int_0^L \sin\theta \, ds \\
  & = \int_0^L \sin \left[ \left(a s\right)^2 \right] \,ds \\
  & = \frac{1}{a} \int_0^{L'} \sin \left({s}^2\right) \, ds 
\end{align}
</math>
}}

===Expansion of Fresnel integral===
{{Main article|Fresnel integral}}

If {{math|1=''a'' = 1}}, which is the case for normalized Euler curve, then the Cartesian coordinates are given by Fresnel integrals (or Euler integrals):
<math display="block">\begin{align}
C(L) &=\int_0^L\cos\left(s^2\right) \, ds\\
S(L) &= \int_0^L\sin\left(s^2\right) \, ds
\end{align}</math>

=== Normalization ===

For a given Euler curve with:
<math display="block">2RL = 2R_c L_s = \frac{1}{a^2} </math>
or
<math display="block">\frac{1}{R} = \frac{L}{R_c L_s} = 2a^2L </math>
then
<math display="block">\begin{align}
x&=\frac{1}{a} \int_0^{L'} \cos \left(s^2\right) \, ds \\
y&=\frac{1}{a} \int_0^{L'} \sin \left(s^2\right) \, ds
\end{align}</math>
where
<math display="block">\begin{align}
L' &= aL \\
a &= \frac{1}{\sqrt{2R_c L_s}}.
\end{align}</math>

The process of obtaining solution of {{math|(''x'', ''y'')}} of an Euler spiral can thus be described as:
* Map {{mvar|L}} of the original Euler spiral by multiplying with factor {{mvar|a}} to {{math|''L''′}} of the normalized Euler spiral;
* Find {{math|(''x''′, ''y''′)}} from the Fresnel integrals; and
* Map {{math|(''x''′, ''y''′)}} to {{math|(''x'', ''y'')}} by scaling up (denormalize) with factor {{math|{{sfrac|1|''a''}}}}. Note that {{math|{{sfrac|1|''a''}} > 1}}.

In the normalization process,
<math display="block">\begin{align}
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}}
\end{align}</math>
Then
<math display="block">2R'_c L'_s = 2 \sqrt{\frac{R_c}{2L_s} } \sqrt{\frac{L_s}{2 R_c}} = \frac{2}{2} = 1</math>

Generally the normalization reduces {{math|''L''′}} to a small value (less than 1) and results in good converging characteristics of the Fresnel integral manageable with only a few terms (at a price of increased [[Numerical stability|numerical instability]] of the calculation, especially for bigger {{mvar|θ}} values.).
==== Illustration ====
Given:
<math display="block">\begin{align}
R_c & = 300\,\mathrm{m} \\
L_s &= 100\,\mathrm{m}
\end{align}</math>
Then
<math display="block">\theta_s = \frac{L_s} {2R_c}  = \frac{100} {2 \times 300} = \frac{1}{6} \ \mathrm{radian}</math>
and
<math display="block"> 2R_c L_s = 60\,000 </math>

We scale down the Euler spiral by {{radic|{{val|60000}}}}, i.e. 100{{radic|6}} to normalized Euler spiral that has:
<math display="block">\begin{align}
      R'_c &= \tfrac{3}{\sqrt{6}}\,\mathrm{m} \\
      L'_s &= \tfrac{1}{\sqrt{6}}\,\mathrm{m} \\
2R'_c L'_s & = 2 \times \tfrac{3}{\sqrt{6}} \times \tfrac{1}{\sqrt{6}} \\
           & = 1 
\end{align}</math>
and
<math display="block">\theta_s = \frac{L'_s}{2R'_c} = \frac{\frac{1}{\sqrt{6}}} {2 \times \frac{3}{\sqrt{6}}} = \frac{1}{6} \ \mathrm{radian}</math>

The two angles {{mvar|θ<sub>s</sub>}} are the same. This thus confirms that the original and normalized Euler spirals are geometrically similar. 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 up or denormalizing.

===Other properties of normalized Euler spirals===

Normalized Euler spirals can be expressed as:
<math display="block">\begin{align}
x &= \int_0^L \cos \left(s^2\right) \,ds \\
y &= \int_0^L \sin \left(s^2\right) \,ds
\end{align}</math>
or expressed as [[power series]]:
<math display="block">\begin{align}
x &= \left . \sum_{i=0}^{\infty} \frac{(-1)^i}{(2i)!} \frac{s^{4i+1}}{4i+1} \right |_0^{L} &&=\sum_{i=0}^{\infty} \frac{(-1)^i}{(2i)!} \frac{L^{4i+1}}{4i+1} \\
y &= \left . \sum_{i=0}^{\infty} \frac{(-1)^i}{(2i+1)!} \frac{s^{4i+3}}{4i+3} \right |_0^{L} &&=\sum_{i=0}^{\infty} \frac{(-1)^i}{(2i+1)!} \frac{L^{4i+3}}{4i+3}
\end{align} </math>

The normalized Euler spiral will converge to a single point in the limit as the parameter L approaches infinity, which can be expressed as:
<math display="block">\begin{align}
x^\prime &= \lim_{L \to \infty} \int_0^{L} \cos \left(s^2\right) \,ds &&= \frac{1}{2} \sqrt{\frac{\pi}{2}} \approx 0.6267 \\
y^\prime &= \lim_{L \to \infty} \int_0^{L} \sin \left(s^2\right) \,ds &&= \frac{1}{2} \sqrt{\frac{\pi}{2}} \approx 0.6267
\end{align}</math>

Normalized Euler spirals have the following properties:
<math display="block">\begin{align}
2 R_c L_s &= 1 \\
\theta_s &= \frac{L_s}{2 R_c} = L_s ^2
\end{align}</math>
and
<math display="block">\begin{align}
\theta &= \theta _s\cdot\frac{L^2}{L_s^2} = L^2 \\
\frac{1}{R} &= \frac{d\theta}{dL} = 2L
\end{align}</math>

Note that {{math|1=2''R<sub>c</sub>L<sub>s</sub>'' = 1}} also means {{math|1={{sfrac|1|''R<sub>c</sub>''}} = 2''L<sub>s</sub>''}}, in agreement with the last mathematical statement.

== See also ==
* [[List of spirals]]

==References==
{{reflist}}

==Further reading==
{{refbegin}}
*{{cite book
 | last = Kellogg
 | first = Norman Benjamin
 | title = The Transition Curve or Curve of Adjustment
 | publisher = McGraw
 | location = New York
 | year = 1907
 | edition = 3rd
 | url = https://archive.org/details/transitioncurve01nrgoog}}
*R. Nave, [http://hyperphysics.phy-astr.gsu.edu/hbase/phyopt/cornu.html#c1 The Cornu spiral], ''Hyperphysics'' (2002) ''(Uses πt²/2 instead of t².)''
* Milton Abramowitz and Irene A. Stegun, eds. ''[[Abramowitz and Stegun|Handbook of Mathematical Functions]] with Formulas, Graphs, and Mathematical Tables.'' New York: Dover, 1972. ''[http://www.math.sfu.ca/~cbm/aands/page_297.htm (See Chapter 7)]''
*{{cite web
|url=http://physics.gu.se/LISEBERG/eng/loop_pe.html
|title=Roller Coaster Loop Shapes
|access-date=2010-11-12}}
{{refend}}

== External links ==
* {{mathworld|title=Cornu Spiral|urlname=CornuSpiral}}
* [http://www.2dcurves.com/spiral/spirale.html Euler's spiral at 2-D Mathematical Curves]
* [http://jsxgraph.uni-bayreuth.de/wiki/index.php/Euler%27s_spiral_(Clothoid) Interactive example with JSXGraph]

{{Spirals}}

{{DEFAULTSORT:Euler Spiral}}
[[Category:Transportation engineering]]
[[Category:Calculus]]
[[Category:Plane curves]]
[[Category:Spirals]]