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Euler spiral

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(Redirected fromClothoid)

Curve whose curvature changes linearly

A double-end Euler spiral. The curve continues to converge to the points marked, ast tends to positive or negative infinity.

AnEuler spiral is a curve whosecurvature 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 aclothoid orCornu spiral.^[1]^[2] The behavior ofFresnel integrals can be illustrated by an Euler spiral, a connection first made byMarie Alfred Cornu in 1874.^[3]

The Euler spiral has applications todiffraction computations. They are also widely used inrailway andhighway engineering to designtransition curves between straight and curved sections of railways or roads. A similar application is also found inphotonic 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:

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).^[2]

Leonhard Euler's work on the spiral came afterJames 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").^[2]

Augustin Fresnel, working in 1818 on thediffraction of light, developed theFresnel 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" afterClotho, one of the threeFates who spin the thread of life inGreek mythology.^[2]

The third independent discovery occurred in the 1800s when various railway engineers sought a formula for gradual curvature in track shape. By 1880Arthur 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.^[2]

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

Formulation

Symbols

R	Radius of curvature
R_c	Radius of circular curve at the end of the spiral
θ	Angle of curve from beginning of spiral (infiniteR) 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.
θ_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

Derivation

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 negativex axis) and a circle. The spiral starts at the origin in the positivex direction and gradually turns anticlockwise toosculate 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, ${\frac {1}{R}}={\frac {d\theta }{ds}}\propto s$ i.e., ${\begin{aligned}Rs={\text{constant}}&=R_{c}s_{o}\\{\frac {d\theta }{ds}}&={\frac {s}{R_{c}s_{o}}}\end{aligned}}$ We write in the format, ${\frac {d\theta }{ds}}=2a^{2}s$ where $2a^{2}={\frac {1}{R_{c}s_{o}}}$ or $a={\frac {1}{\sqrt {2R_{c}s_{o}}}}$ thus $\theta =(as)^{2}$ Now $x=\int _{0}^{L}\cos \theta \,ds=\int _{0}^{L}\cos \left[\left(as\right)^{2}\right]\,ds$ If $s'=as$ Then $ds={\frac {ds'}{a}}$ Thus ${\begin{aligned}x&={\frac {1}{a}}\int _{0}^{L'}\cos \left(s^{2}\right)\,ds\\y&=\int _{0}^{L}\sin \theta \,ds\\&=\int _{0}^{L}\sin \left[\left(as\right)^{2}\right]\,ds\\&={\frac {1}{a}}\int _{0}^{L'}\sin \left({s}^{2}\right)\,ds\end{aligned}}$

Relation to Fresnel integral

Main article:Fresnel integral

Ifa = 1, which is the case for normalized Euler curve, then the Cartesian coordinates are given by Fresnel integrals (or Euler integrals): ${\begin{aligned}C(L)&=\int _{0}^{L}\cos \left(s^{2}\right)\,ds\\S(L)&=\int _{0}^{L}\sin \left(s^{2}\right)\,ds\end{aligned}}$

Relation to superspirals

Euler's spiral is a type ofsuperspiral that has the property of a monotonic curvature function.^[4]

Applications

Track transition curve

Main article:Track transition curve

Animation depicting evolution of a Cornu spiral with the tangential circle with the same radius of curvature as at its tip, also known as anosculating circle.

To travel along a circular path, an object needs to be subject to acentripetal 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 lateraljerk).

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 acceleration⁠v²/r⁠, the obvious solution is to provide an easement curve whose curvature,⁠1/R⁠, increases linearly with the traveled distance.

Optics

Inoptics, the term Cornu spiral is used.^[5]^: 432The Cornu spiral can be used to describe adiffraction pattern.^[6]Consider a plane wave with phasor amplitudeE₀e^−jkz which is diffracted by a "knife edge" of heighth abovex = 0 on thez = 0 plane. Then the diffracted wave field can be expressed as $\mathbf {E} (x,z)=E_{0}e^{-jkz}{\frac {\mathrm {Fr} (\infty )-\mathrm {Fr} \left({\sqrt {\frac {2}{\lambda z}}}(h-x)\right)}{\mathrm {Fr} (\infty )-\mathrm {Fr} (-\infty )}},$ where $Fr(x)=C(x)+iS(x)$ is theFresnel integral function on the complex plane.^{[citation needed]}

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 quantitiesFr(a) − Fr(b) as the physical distances between the points represented byFr(a) andFr(b) for appropriatea andb. This facilitates a rough computation of the attenuation of the plane wave by the knife edge of heighth at a location(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 inphotonic integrated circuits, either in singlemodewaveguides,^[7]^[8] to smoothen the abrupt change of curvature and suppress coupling to radiation modes, or in multimode waveguides,^[9] 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,^[10] with a hollow metalwaveguide 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 thepath integral formulation of quantum mechanics, the probability amplitude for propagation between two points can be visualized by connectingaction phase arrows for each time step between the two points. The arrows spiral around each endpoint forming what is termed a Cornu spiral.^[11]

Auto racing

Motorsport author Adam Brouillard has shown the Euler spiral's use in optimizing theracing line during the corner entry portion of a turn.^[12]

Typography and digital vector drawing

Raph Levien has released Spiro as a toolkit for curve design, especially font design, in 2007^[13]^[14] under a free licence. This toolkit has been implemented quite quickly afterwards in the font design toolFontforge and the digital vector drawingInkscape.

Map projection

Cutting a sphere along a spiral with width⁠1/N⁠ and flattening out the resulting shape yields an Euler spiral whenn tends to the infinity.^[15] If the sphere is theglobe, this produces amap projection whose distortion tends to zero asn tends to the infinity.^[16]

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.^[17] The two parameters of theCesàro equation for an Euler spiral segment might give insight into thekeratinization mechanism of whisker growth.^[18]

Normalization

For a given Euler curve with: $2RL=2R_{c}L_{s}={\frac {1}{a^{2}}}$ or ${\frac {1}{R}}={\frac {L}{R_{c}L_{s}}}=2a^{2}L$ then ${\begin{aligned}x&={\frac {1}{a}}\int _{0}^{L'}\cos \left(s^{2}\right)\,ds\\y&={\frac {1}{a}}\int _{0}^{L'}\sin \left(s^{2}\right)\,ds\end{aligned}}$ where ${\begin{aligned}L'&=aL\\a&={\frac {1}{\sqrt {2R_{c}L_{s}}}}.\end{aligned}}$

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

MapL of the original Euler spiral by multiplying with factora toL′ of the normalized Euler spiral;
Find(x′,y′) from the Fresnel integrals; and
Map(x′,y′) to(x,y) by scaling up (denormalize) with factor⁠1/a⁠. Note that⁠1/a⁠ > 1.

In the normalization process, ${\begin{aligned}R'_{c}&={\frac {R_{c}}{\sqrt {2R_{c}L_{s}}}}={\sqrt {\frac {R_{c}}{2L_{s}}}}\\L'_{s}&={\frac {L_{s}}{\sqrt {2R_{c}L_{s}}}}={\sqrt {\frac {L_{s}}{2R_{c}}}}\end{aligned}}$ Then $2R'_{c}L'_{s}=2{\sqrt {\frac {R_{c}}{2L_{s}}}}{\sqrt {\frac {L_{s}}{2R_{c}}}}={\frac {2}{2}}=1$

Generally the normalization reducesL′ 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 increasednumerical instability of the calculation, especially for biggerθ values.).

Illustration

Given: ${\begin{aligned}R_{c}&=300\,\mathrm {m} \\L_{s}&=100\,\mathrm {m} \end{aligned}}$ Then $\theta _{s}={\frac {L_{s}}{2R_{c}}}={\frac {100}{2\times 300}}={\frac {1}{6}}\ \mathrm {radian}$ and $2R_{c}L_{s}=60\,000$

We scale down the Euler spiral by√60000, i.e. 100√6 to normalized Euler spiral that has: ${\begin{aligned}R'_{c}&={\tfrac {3}{\sqrt {6}}}\,\mathrm {m} \\L'_{s}&={\tfrac {1}{\sqrt {6}}}\,\mathrm {m} \\2R'_{c}L'_{s}&=2\times {\tfrac {3}{\sqrt {6}}}\times {\tfrac {1}{\sqrt {6}}}\\&=1\end{aligned}}$ and $\theta _{s}={\frac {L'_{s}}{2R'_{c}}}={\frac {\frac {1}{\sqrt {6}}}{2\times {\frac {3}{\sqrt {6}}}}}={\frac {1}{6}}\ \mathrm {radian}$

The two anglesθ_s 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: ${\begin{aligned}x&=\int _{0}^{L}\cos \left(s^{2}\right)\,ds\\y&=\int _{0}^{L}\sin \left(s^{2}\right)\,ds\end{aligned}}$ or expressed aspower series: ${\begin{aligned}x&=\left.\sum _{i=0}^{\infty }{\frac {(-1)^{i}}{(2i)!}}{\frac {s^{4i+1}}{4i+1}}\right|_{0}^{L}&&=\sum _{i=0}^{\infty }{\frac {(-1)^{i}}{(2i)!}}{\frac {L^{4i+1}}{4i+1}}\\y&=\left.\sum _{i=0}^{\infty }{\frac {(-1)^{i}}{(2i+1)!}}{\frac {s^{4i+3}}{4i+3}}\right|_{0}^{L}&&=\sum _{i=0}^{\infty }{\frac {(-1)^{i}}{(2i+1)!}}{\frac {L^{4i+3}}{4i+3}}\end{aligned}}$

The normalized Euler spiral will converge to a single point in the limit as the parameter L approaches infinity, which can be expressed as: ${\begin{aligned}x^{\prime }&=\lim _{L\to \infty }\int _{0}^{L}\cos \left(s^{2}\right)\,ds&&={\frac {1}{2}}{\sqrt {\frac {\pi }{2}}}\approx 0.6267\\y^{\prime }&=\lim _{L\to \infty }\int _{0}^{L}\sin \left(s^{2}\right)\,ds&&={\frac {1}{2}}{\sqrt {\frac {\pi }{2}}}\approx 0.6267\end{aligned}}$

Normalized Euler spirals have the following properties: ${\begin{aligned}2R_{c}L_{s}&=1\\\theta _{s}&={\frac {L_{s}}{2R_{c}}}=L_{s}^{2}\end{aligned}}$ and ${\begin{aligned}\theta &=\theta _{s}\cdot {\frac {L^{2}}{L_{s}^{2}}}=L^{2}\\{\frac {1}{R}}&={\frac {d\theta }{dL}}=2L\end{aligned}}$

Note that2R_cL_s = 1 also means⁠1/R_c⁠ = 2L_s, in agreement with the last mathematical statement.

See also

List of spirals

References

^Von Seggern, David H. (1994).Practical handbook of curve design and generation. Boca Raton, Fla.: CRC Press.ISBN 978-0-8493-8916-0.
^^a ^b ^c ^d ^eLevien, Raph."The Euler spiral: a mathematical history." Rapp. tech (2008).
^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.
^Ziatdinov, R. (2012), "Family of superspirals with completely monotonic curvature given in terms of Gauss hypergeometric function",Computer Aided Geometric Design,29 (7):510–518,doi:10.1016/j.cagd.2012.03.006
^Born, Max; Wolf, Emil (1993).Principles of optics: electromagnetic theory of propagation, interference and diffraction of light (6. ed., reprinted (with corrections) ed.). Oxford: Pergamon Press.ISBN 978-0-08-026481-3.
^Eugene Hecht (1998).Optics (3rd ed.). Addison-Wesley. p. 491.ISBN 978-0-201-30425-1.
^Kohtoku, M.; et al. (7 July 2005)."New Waveguide Fabrication Techniques for Next-generation PLCs"(PDF).NTT Technical Review.3 (7):37–41. Retrieved24 January 2017.
^Li, G.; et al. (11 May 2012)."Ultralow-loss, high-density SOI optical waveguide routing for macrochip interconnects".Optics Express.20 (11):12035–12039.Bibcode:2012OExpr..2012035L.doi:10.1364/OE.20.012035.PMID 22714189.
^Cherchi, M.; et al. (18 July 2013). "Dramatic size reduction of waveguide bends on a micron-scale silicon photonic platform".Optics Express.21 (15):17814–17823.arXiv:1301.2197.Bibcode:2013OExpr..2117814C.doi:10.1364/OE.21.017814.PMID 23938654.
^Unger, H.G. (September 1957). "Normal Mode Bends for Circular Electric Waves".The Bell System Technical Journal.36 (5):1292–1307.doi:10.1002/j.1538-7305.1957.tb01509.x.
^Taylor, Edwin F.; Vokos, Stamatis; O'Meara, John M.; Thornber, Nora S. (1998-03-01)."Teaching Feynman's sum-over-paths quantum theory".Computers in Physics.12 (2):190–199.Bibcode:1998ComPh..12..190T.doi:10.1063/1.168652.ISSN 0894-1866.
^Development, Paradigm Shift Driver; Brouillard, Adam (2016-03-18).The Perfect Corner: A Driver's Step-By-Step Guide to Finding Their Own Optimal Line Through the Physics of Racing. Paradigm Shift Motorsport Books.ISBN 978-0-9973824-2-6.
^"Spiro".
^"| Spiro 0.01 release | Typophile".www.typophile.com. Archived fromthe original on 2007-05-10.
^Bartholdi, Laurent; Henriques, André (2012). "Orange Peels and Fresnel Integrals".The Mathematical Intelligencer.34 (3):1–3.arXiv:1202.3033.doi:10.1007/s00283-012-9304-1.ISSN 0343-6993.S2CID 52592272.
^"A Strange Map Projection (Euler Spiral) - Numberphile".YouTube. 13 November 2018.Archived from the original on 2021-12-21.
^Towal, R.B.; et al. (7 April 2011)."The Morphology of the Rat Vibrissal Array: A Model for Quantifying Spatiotemporal Patterns of Whisker-Object Contact".PLOS Computational Biology.7 (4) e1001120.Bibcode:2011PLSCB...7E1120T.doi:10.1371/journal.pcbi.1001120.PMC 3072363.PMID 21490724.
Starostin, E.L.; et al. (15 January 2020)."The Euler spiral of rat whiskers".Science Advances.6 (3) eaax5145.Bibcode:2020SciA....6.5145S.doi:10.1126/sciadv.aax5145.PMC 6962041.PMID 31998835.
^Luo, Y.; Hartmann, M.J. (Jan 2023)."On the intrinsic curvature of animal whiskers".PLOS ONE.18 (1) e0269210.Bibcode:2023PLoSO..1869210L.doi:10.1371/journal.pone.0269210.PMC 9821693.PMID 36607960.

Further reading

Kellogg, Norman Benjamin (1907).The Transition Curve or Curve of Adjustment (3rd ed.). New York: McGraw.
R. Nave,The Cornu spiral,Hyperphysics (2002)(Uses πt²/2 instead of t².)
Milton Abramowitz and Irene A. Stegun, eds.Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972.(See Chapter 7)
"Roller Coaster Loop Shapes". Retrieved2010-11-12.

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