182:(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
4089:
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167:
4084:{\displaystyle {\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}}}
3269:
1703:
805:
20:
2644:
568:, to smoothen the abrupt change of curvature and suppress coupling to radiation modes, or in multimode waveguides, 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, with a hollow metal
487:
1425:
4345:{\displaystyle {\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}}}
2214:
3075:
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189:
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
105:
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
4618:
2434:
94:
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).
1886:
3566:
1009:
302:
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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.
2777:
4478:
2048:
2999:
257:(and later some civil engineers) also solved the calculus of the 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.
2883:
2309:
1698:{\displaystyle {\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\,ds\\&={\frac {1}{a}}\int _{0}^{L'}\sin \left({s}^{2}\right)\,ds\end{aligned}}}
4483:
3264:{\displaystyle {\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}}}
3274:
1732:
3430:
888:
1340:
2639:{\displaystyle {\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}}}
2043:
883:
4357:
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2053:
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893:
1965:
1182:
2807:
482:{\displaystyle \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 )}},}
1127:
1065:
4742:
Marie Alfred Cornu. M´ethode nouvelle pour la discussion des probl´emes de diffraction dans le cas d’une onde cylindrique. Journal de
Physique th´eoretique et appliqu´ee, pages 5–15, 1874.
611:
has released Spiro as a toolkit for curve design, especially font design, in 2007 under a free licence. This toolkit has been implemented quite quickly afterwards in the font design tool
2219:
1420:
3049:
572:
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.
1224:
2209:{\displaystyle {\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}}}
500:
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
1374:
124:
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,
675:
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. The two parameters of the
106:
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").
1229:
2788:
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
811:
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
78:. 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:
147:
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.
4613:{\displaystyle {\begin{aligned}\theta &=\theta _{s}\cdot {\frac {L^{2}}{L_{s}^{2}}}=L^{2}\\{\frac {1}{R}}&={\frac {d\theta }{dL}}=2L\end{aligned}}}
39:
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
3402:{\displaystyle \theta _{s}={\frac {L'_{s}}{2R'_{c}}}={\frac {\frac {1}{\sqrt {6}}}{2\times {\frac {3}{\sqrt {6}}}}}={\frac {1}{6}}\ \mathrm {radian} }
143:
The third independent discovery occurred in the 1800's when various railway engineers sought a formula for gradual curvature in track shape. By 1880
1970:
1881:{\displaystyle {\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}}}
832:
190:
is necessary, so that the centripetal acceleration increases smoothly with the traveled distance. Given the expression of centripetal acceleration
3561:{\displaystyle {\begin{aligned}x&=\int _{0}^{L}\cos \left(s^{2}\right)\,ds\\y&=\int _{0}^{L}\sin \left(s^{2}\right)\,ds\end{aligned}}}
4830:
4805:
4705:
170:
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
4093:
The normalized Euler spiral will converge to a single point in the limit as the parameter L approaches infinity, which can be expressed as:
5752:
1898:
1132:
1004:{\displaystyle {\begin{aligned}Rs={\text{constant}}&=R_{c}s_{o}\\{\frac {d\theta }{ds}}&={\frac {s}{R_{c}s_{o}}}\end{aligned}}}
588:
phase arrows for each time step between the two points. The arrows spiral around each endpoint forming what is termed a Cornu spiral.
5458:
5059:
5010:
4752:
Ziatdinov, R. (2012), "Family of superspirals with completely monotonic curvature given in terms of Gauss hypergeometric function",
1729:, which is the case for normalized Euler curve, then the Cartesian coordinates are given by Fresnel integrals (or Euler integrals):
1070:
1014:
4929:
Cherchi, M.; et al. (18 July 2013). "Dramatic size reduction of waveguide bends on a micron-scale silicon photonic platform".
247:(a polynomial curve of degree 3), which is an approximation of the Euler spiral for small angular changes in the same way that a
51:
can be illustrated by an Euler spiral, a connection first made by Alfred Marie Cornu in 1874. Euler's spiral is a type of
240:
120:
that defines the same spiral. He was unaware of Euler's integrals or the connection to the theory of elasticity. In 1874,
3004:
1379:
569:
5190:"The Morphology of the Rat Vibrissal Array: A Model for Quantifying Spatiotemporal Patterns of Whisker-Object Contact"
1187:
561:
75:
5544:
4473:{\displaystyle {\begin{aligned}2R_{c}L_{s}&=1\\\theta _{s}&={\frac {L_{s}}{2R_{c}}}=L_{s}^{2}\end{aligned}}}
584:
of quantum mechanics, the probability amplitude for propagation between two points can be visualized by connecting
232:, increases linearly with the traveled distance. This geometry is a "clothoid", another name for the Euler spiral.
82:
Its curvature begins with zero at the straight section (the tangent) and increases linearly with its curve length.
5593:
5052:
The
Perfect Corner: A Driver's Step-By-Step Guide to Finding Their Own Optimal Line Through the Physics of Racing
581:
560:
Bends with continuously varying radius of curvature following the Euler spiral are also used to reduce losses in
4730:
2994:{\displaystyle \theta _{s}={\frac {L_{s}}{2R_{c}}}={\frac {100}{2\times 300}}={\frac {1}{6}}\ \mathrm {radian} }
4847:
2772:{\displaystyle 2R'_{c}L'_{s}=2{\sqrt {\frac {R_{c}}{2L_{s}}}}{\sqrt {\frac {L_{s}}{2R_{c}}}}={\frac {2}{2}}=1}
5089:
5716:
179:
128:
praised the advantages of the "spiral of Cornu" over the "unpleasant multitude of hairy integral formulas".
1345:
735:
This can also be measured as the angle between the initial tangent and the tangent at the concerned point.
161:
5762:
5711:
5509:
5451:
5431:
5372:
2878:{\displaystyle {\begin{aligned}R_{c}&=300\,\mathrm {m} \\L_{s}&=100\,\mathrm {m} \end{aligned}}}
536:. This facilitates a rough computation of the attenuation of the plane wave by the knife edge of height
5660:
5244:
4948:
4893:
144:
5721:
2789:
85:
Where the Euler spiral meets the circular curve, its curvature becomes equal to that of the latter.
67:
63:
5357:
4798:
Principles of optics: electromagnetic theory of propagation, interference and diffraction of light
2304:{\displaystyle {\begin{aligned}L'&=aL\\a&={\frac {1}{\sqrt {2R_{c}L_{s}}}}.\end{aligned}}}
74:
between straight and curved sections of railways or roads. A similar application is also found in
5692:
5640:
5174:
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565:
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121:
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171:
125:
48:
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1717:
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117:
109:
71:
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826:
The spiral is a small segment of the above double-end Euler spiral in the first quadrant.
102:
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4952:
4897:
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5504:
5318:
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5232:
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5189:
4995:
4882:"Ultralow-loss, high-density SOI optical waveguide routing for macrochip interconnects"
656:
236:
183:
129:
98:
5347:
265:
In optics the term "Cornu spiral" is used. The Cornu spiral can be used to describe a
5746:
5697:
5009:
Taylor, Edwin F.; Vokos, Stamatis; O’Meara, John M.; Thornber, Nora S. (1998-03-01).
596:
Motorsport author Adam
Brouillard has shown the Euler spiral's use in optimizing the
497:
is the
Fresnel integral function, which forms the Cornu spiral on the complex plane.
23:
A double-end Euler spiral. The curve continues to converge to the points marked, as
5150:
166:
5667:
5613:
5559:
5111:
Bartholdi, Laurent; Henriques, André (2012). "Orange Peels and
Fresnel Integrals".
3569:
5412:
5386:
5379:
5308:
5206:
5731:
5583:
5534:
4772:
608:
597:
266:
113:
59:
4982:
Unger, H.G. (September 1957). "Normal Mode Bends for
Circular Electric Waves".
5705:
5608:
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5436:
5426:
5164:
5134:
820:
244:
5142:
5036:
5726:
5650:
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5549:
5494:
5417:
804:
612:
36:
5327:
5276:
5257:
5225:
4968:
4915:
1335:{\displaystyle x=\int _{0}^{L}\cos \theta \,ds=\int _{0}^{L}\cos \left\,ds}
19:
5682:
5598:
4960:
4906:
4881:
4800:(6. ed., reprinted (with corrections) ed.). Oxford: Pergamon Press.
672:
616:
248:
212:, the obvious solution is to provide an easement curve whose curvature,
5169:
680:
5097:
5467:
5027:
815:
axis) and a circle. The spiral starts at the origin in the positive
133:
178:
To travel along a circular path, an object needs to be subject to a
5050:
Development, Paradigm Shift Driver; Brouillard, Adam (2016-03-18).
647:
and flattening out the resulting shape yields an Euler spiral when
5475:
5471:
5125:
4943:
652:
165:
137:
765:
Length measured along the spiral curve from its initial position
5440:
4848:"New Waveguide Fabrication Techniques for Next-generation PLCs"
2038:{\displaystyle {\frac {1}{R}}={\frac {L}{R_{c}L_{s}}}=2a^{2}L}
878:{\displaystyle {\frac {1}{R}}={\frac {d\theta }{ds}}\propto s}
4228:
4109:
803:
512:
as the physical distances between the points represented by
5075:
3838:
3595:
299:
plane. Then the diffracted wave field can be expressed as
55:
that has the property of a monotonic curvature function.
4354:
Normalized Euler spirals have the following properties:
2333:
of the original Euler spiral by multiplying with factor
679:
for an Euler spiral segment might give insight into the
5165:"A Strange Map Projection (Euler Spiral) - Numberphile"
3231:
3214:
3148:
3104:
4667:, in agreement with the last mathematical statement.
4486:
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4099:
3578:
3433:
3277:
3078:
3007:
2891:
2810:
2652:
2437:
2222:
2051:
1973:
1901:
1735:
1428:
1382:
1348:
1232:
1190:
1135:
1073:
1017:
891:
835:
305:
269:
pattern. Consider a plane wave with phasor amplitude
5633:
5568:
5527:
5482:
5011:"Teaching Feynman's sum-over-paths quantum theory"
4612:
4472:
4344:
4083:
3560:
3401:
3263:
3043:
2993:
2877:
2771:
2638:
2303:
2208:
2037:
1960:{\displaystyle 2RL=2R_{c}L_{s}={\frac {1}{a^{2}}}}
1959:
1880:
1697:
1414:
1368:
1334:
1218:
1177:{\displaystyle a={\frac {1}{\sqrt {2R_{c}s_{o}}}}}
1176:
1121:
1059:
1003:
877:
729:Angle of curve from beginning of spiral (infinite
719:Radius of circular curve at the end of the spiral
481:
4698:Practical handbook of curve design and generation
5231:Starostin, E.L.; et al. (15 January 2020).
4241:
4122:
281:which is diffracted by a "knife edge" of height
140:who spin the thread of life in Greek mythology.
5375:with Formulas, Graphs, and Mathematical Tables.
5292:"On the intrinsic curvature of animal whiskers"
4725:
4723:
4721:
4719:
4717:
819:direction and gradually turns anticlockwise to
3427:Normalized Euler spirals can be expressed as:
1122:{\displaystyle 2a^{2}={\frac {1}{R_{c}s_{o}}}}
1060:{\displaystyle {\frac {d\theta }{ds}}=2a^{2}s}
132:chose to name the same curve "clothoid" after
5452:
4825:(3rd ed.). Addison-Wesley. p. 491.
2325:of an Euler spiral can thus be described as:
8:
5370:Milton Abramowitz and Irene A. Stegun, eds.
3423:Other properties of normalized Euler spirals
651:tends to the infinity. If the sphere is the
5349:The Transition Curve or Curve of Adjustment
4731:"The Euler spiral: a mathematical history."
627:Cutting a sphere along a spiral with width
600:during the corner entry portion of a turn.
235:Unaware of the solution of the geometry by
62:computations. They are also widely used in
5565:
5459:
5445:
5437:
2792:of the calculation, especially for bigger
794:
5427:Euler's spiral at 2-D Mathematical Curves
5317:
5307:
5266:
5256:
5215:
5205:
5188:Towal, R.B.; et al. (7 April 2011).
5124:
5026:
4942:
4905:
4577:
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3213:
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2811:
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2740:
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2709:
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2651:
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2601:
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2504:
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2110:
2085:
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2066:
2052:
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2007:
1997:
1987:
1974:
1972:
1949:
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1931:
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1857:
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1798:
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1016:
988:
978:
968:
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892:
890:
849:
836:
834:
453:
433:
394:
381:
361:
358:
343:
333:
306:
304:
251:is an approximation to a circular curve.
4846:Kohtoku, M.; et al. (7 July 2005).
2386:by scaling up (denormalize) with factor
18:
4688:
733:) to a particular point on the spiral.
27:tends to positive or negative infinity.
829:From the definition of the curvature,
16:Curve whose curvature changes linearly
3072:to normalized Euler spiral that has:
2313:The process of obtaining solution of
604:Typography and digital vector drawing
58:The Euler spiral has applications to
7:
5290:Luo, Y.; Hartmann, M.J. (Jan 2023).
2781:Generally the normalization reduces
5054:. Paradigm Shift Motorsport Books.
4880:Li, G.; et al. (11 May 2012).
3044:{\displaystyle 2R_{c}L_{s}=60\,000}
1415:{\displaystyle ds={\frac {ds'}{a}}}
5373:Handbook of Mathematical Functions
5233:"The Euler spiral of rat whiskers"
5090:"| Spiro 0.01 release | Typophile"
4996:10.1002/j.1538-7305.1957.tb01509.x
4251:
4132:
3987:
3857:
3738:
3614:
3395:
3392:
3389:
3386:
3383:
3380:
3163:
3119:
3053:We scale down the Euler spiral by
2987:
2984:
2981:
2978:
2975:
2972:
2867:
2837:
659:whose distortion tends to zero as
467:
457:
454:
444:
437:
434:
385:
382:
372:
365:
362:
14:
5432:Interactive example with JSXGraph
5352:(3rd ed.). New York: McGraw.
5346:Kellogg, Norman Benjamin (1907).
4984:The Bell System Technical Journal
101:'s work on the spiral came after
5518:
5177:from the original on 2021-12-21.
1219:{\displaystyle \theta =(as)^{2}}
307:
4754:Computer Aided Geometric Design
4700:. Boca Raton, Fla.: CRC Press.
2359:from the Fresnel integrals; and
2344:of the normalized Euler spiral;
615:and the digital vector drawing
5113:The Mathematical Intelligencer
4796:Born, Max; Wolf, Emil (1993).
4696:Von Seggern, David H. (1994).
4248:
4129:
4031:
4016:
4005:
3995:
3901:
3886:
3875:
3865:
3776:
3767:
3756:
3746:
3652:
3643:
3632:
3622:
2431:In the normalization process,
1895:For a given Euler curve with:
1818:
1812:
1749:
1743:
1207:
1197:
470:
461:
447:
441:
423:
411:
375:
369:
323:
311:
1:
1712:Expansion of Fresnel integral
683:mechanism of whisker growth.
575:
5387:"Roller Coaster Loop Shapes"
5309:10.1371/journal.pone.0269210
5207:10.1371/journal.pcbi.1001120
562:photonic integrated circuits
76:photonic integrated circuits
5366:(Uses πt²/2 instead of t².)
751:Angle of full spiral curve
5784:
5753:Transportation engineering
5194:PLoS Computational Biology
1715:
159:
5516:
5135:10.1007/s00283-012-9304-1
4771:Constantin (2016-03-07).
582:path integral formulation
112:, working in 1818 on the
1011:We write in the format,
671:Natural shapes of rats'
180:centripetal acceleration
116:of light, developed the
5377:New York: Dover, 1972.
789:Length of spiral curve
663:tends to the infinity.
576:Feynman's path integral
564:, either in singlemode
552:beyond the knife edge.
5258:10.1126/sciadv.aax5145
4614:
4474:
4346:
4085:
3991:
3861:
3742:
3618:
3562:
3403:
3265:
3045:
2995:
2879:
2773:
2640:
2305:
2210:
2039:
1961:
1882:
1699:
1416:
1370:
1336:
1220:
1178:
1123:
1061:
1005:
879:
808:
483:
175:
162:Track transition curve
156:Track transition curve
28:
4821:Eugene Hecht (1998).
4615:
4475:
4347:
4086:
3971:
3841:
3722:
3598:
3563:
3404:
3266:
3046:
2996:
2880:
2790:numerical instability
2774:
2641:
2306:
2211:
2040:
1962:
1883:
1700:
1417:
1371:
1369:{\displaystyle s'=as}
1337:
1221:
1179:
1124:
1062:
1006:
880:
807:
484:
169:
22:
5015:Computers in Physics
4961:10.1364/OE.21.017814
4907:10.1364/OE.20.012035
4855:NTT Technical Review
4484:
4358:
4097:
3576:
3431:
3275:
3076:
3005:
2889:
2808:
2650:
2435:
2220:
2049:
1971:
1899:
1733:
1426:
1380:
1346:
1230:
1188:
1133:
1071:
1015:
889:
833:
704:Radius of curvature
303:
145:Arthur Newell Talbot
5249:2020SciA....6.5145S
4953:2013OExpr..2117814C
4937:(15): 17814–17823.
4898:2012OExpr..2012035L
4892:(11): 12035–12039.
4540:
4465:
4270:
4151:
3962:
3713:
3522:
3462:
3322:
3305:
3199:
3186:
3139:
3095:
2681:
2668:
2551:
2454:
2170:
2095:
1842:
1773:
1657:
1573:
1532:
1472:
1287:
1253:
136:, one of the three
68:highway engineering
5410:Weisstein, Eric W.
4733:Rapp. tech (2008).
4610:
4608:
4526:
4470:
4468:
4451:
4342:
4340:
4256:
4255:
4137:
4136:
4081:
4079:
3836:
3593:
3558:
3556:
3508:
3448:
3399:
3310:
3293:
3261:
3259:
3242:
3225:
3187:
3174:
3159:
3127:
3115:
3083:
3041:
2991:
2875:
2873:
2769:
2669:
2656:
2636:
2634:
2539:
2442:
2301:
2299:
2206:
2204:
2151:
2076:
2035:
1957:
1878:
1876:
1828:
1759:
1695:
1693:
1638:
1559:
1518:
1453:
1412:
1366:
1332:
1273:
1239:
1216:
1174:
1119:
1057:
1001:
999:
875:
809:
655:, this produces a
479:
255:Marie Alfred Cornu
176:
122:Alfred Marie Cornu
47:. The behavior of
29:
5740:
5739:
5629:
5628:
5094:www.typophile.com
4832:978-0-201-30425-1
4807:978-0-08-026481-3
4707:978-0-8493-8916-0
4595:
4568:
4541:
4446:
4330:
4329:
4318:
4240:
4211:
4210:
4199:
4121:
4075:
4038:
3945:
3908:
3820:
3783:
3696:
3659:
3378:
3374:
3361:
3358:
3357:
3340:
3339:
3324:
3241:
3240:
3224:
3223:
3158:
3157:
3114:
3113:
2970:
2966:
2953:
2932:
2761:
2748:
2747:
2717:
2716:
2630:
2629:
2596:
2595:
2533:
2532:
2499:
2498:
2292:
2291:
2149:
2074:
2014:
1982:
1955:
1709:
1708:
1636:
1451:
1410:
1172:
1171:
1117:
1036:
995:
959:
908:
867:
844:
793:
792:
556:Integrated optics
474:
409:
408:
172:osculating circle
72:transition curves
49:Fresnel integrals
35:is a curve whose
5775:
5566:
5545:Boerdijk–Coxeter
5522:
5521:
5461:
5454:
5447:
5438:
5423:
5422:
5396:
5394:
5393:
5358:The Cornu spiral
5353:
5332:
5331:
5321:
5311:
5287:
5281:
5280:
5270:
5260:
5237:Science Advances
5229:
5219:
5209:
5185:
5179:
5178:
5161:
5155:
5154:
5128:
5108:
5102:
5101:
5096:. Archived from
5086:
5080:
5079:
5072:
5066:
5065:
5047:
5041:
5040:
5030:
5028:10.1063/1.168652
5006:
5000:
4999:
4990:(5): 1292–1307.
4979:
4973:
4972:
4946:
4926:
4920:
4919:
4909:
4877:
4871:
4870:
4868:
4866:
4852:
4843:
4837:
4836:
4818:
4812:
4811:
4793:
4787:
4786:
4784:
4783:
4768:
4762:
4761:
4749:
4743:
4740:
4734:
4727:
4712:
4711:
4693:
4666:
4658:
4656:
4655:
4647:
4644:
4636:
4619:
4617:
4616:
4611:
4609:
4596:
4594:
4586:
4578:
4569:
4561:
4555:
4554:
4542:
4539:
4534:
4525:
4524:
4515:
4510:
4509:
4479:
4477:
4476:
4471:
4469:
4464:
4459:
4447:
4445:
4444:
4443:
4430:
4429:
4420:
4411:
4410:
4387:
4386:
4377:
4376:
4351:
4349:
4348:
4343:
4341:
4331:
4322:
4321:
4319:
4311:
4303:
4294:
4290:
4289:
4269:
4264:
4254:
4232:
4231:
4212:
4203:
4202:
4200:
4192:
4184:
4175:
4171:
4170:
4150:
4145:
4135:
4113:
4112:
4090:
4088:
4087:
4082:
4080:
4076:
4074:
4060:
4059:
4041:
4039:
4037:
4014:
4013:
4012:
3993:
3990:
3985:
3964:
3961:
3956:
3951:
3947:
3946:
3944:
3930:
3929:
3911:
3909:
3907:
3884:
3883:
3882:
3863:
3860:
3855:
3821:
3819:
3805:
3804:
3786:
3784:
3782:
3765:
3764:
3763:
3744:
3741:
3736:
3715:
3712:
3707:
3702:
3698:
3697:
3695:
3681:
3680:
3662:
3660:
3658:
3641:
3640:
3639:
3620:
3617:
3612:
3568:or expressed as
3567:
3565:
3564:
3559:
3557:
3546:
3542:
3541:
3521:
3516:
3486:
3482:
3481:
3461:
3456:
3417:
3408:
3406:
3405:
3400:
3398:
3376:
3375:
3367:
3362:
3360:
3359:
3353:
3349:
3335:
3331:
3330:
3325:
3323:
3318:
3301:
3292:
3287:
3286:
3270:
3268:
3267:
3262:
3260:
3247:
3243:
3236:
3232:
3226:
3219:
3215:
3195:
3182:
3166:
3160:
3153:
3149:
3135:
3122:
3116:
3109:
3105:
3091:
3071:
3070:
3064:
3063:
3062:
3061:
3050:
3048:
3047:
3042:
3030:
3029:
3020:
3019:
3000:
2998:
2997:
2992:
2990:
2968:
2967:
2959:
2954:
2952:
2938:
2933:
2931:
2930:
2929:
2916:
2915:
2906:
2901:
2900:
2884:
2882:
2881:
2876:
2874:
2870:
2854:
2853:
2840:
2824:
2823:
2795:
2787:
2778:
2776:
2775:
2770:
2762:
2754:
2749:
2746:
2745:
2744:
2731:
2730:
2721:
2720:
2718:
2715:
2714:
2713:
2700:
2699:
2690:
2689:
2677:
2664:
2645:
2643:
2642:
2637:
2635:
2631:
2628:
2627:
2626:
2613:
2612:
2603:
2602:
2597:
2594:
2593:
2584:
2583:
2571:
2570:
2569:
2560:
2547:
2534:
2531:
2530:
2529:
2516:
2515:
2506:
2505:
2500:
2497:
2496:
2487:
2486:
2474:
2473:
2472:
2463:
2450:
2426:
2424:
2422:
2421:
2416:
2413:
2405:
2404:
2402:
2401:
2396:
2393:
2385:
2373:
2358:
2343:
2336:
2332:
2324:
2310:
2308:
2307:
2302:
2300:
2293:
2290:
2289:
2280:
2279:
2267:
2263:
2234:
2215:
2213:
2212:
2207:
2205:
2194:
2190:
2189:
2169:
2168:
2159:
2150:
2142:
2119:
2115:
2114:
2094:
2093:
2084:
2075:
2067:
2044:
2042:
2041:
2036:
2031:
2030:
2015:
2013:
2012:
2011:
2002:
2001:
1988:
1983:
1975:
1966:
1964:
1963:
1958:
1956:
1954:
1953:
1941:
1936:
1935:
1926:
1925:
1887:
1885:
1884:
1879:
1877:
1866:
1862:
1861:
1841:
1836:
1797:
1793:
1792:
1772:
1767:
1728:
1718:Fresnel integral
1704:
1702:
1701:
1696:
1694:
1683:
1679:
1678:
1673:
1656:
1655:
1646:
1637:
1629:
1621:
1610:
1606:
1605:
1600:
1596:
1572:
1567:
1552:
1531:
1526:
1496:
1492:
1491:
1471:
1470:
1461:
1452:
1444:
1421:
1419:
1418:
1413:
1411:
1406:
1405:
1393:
1375:
1373:
1372:
1367:
1356:
1341:
1339:
1338:
1333:
1324:
1320:
1319:
1314:
1310:
1286:
1281:
1252:
1247:
1225:
1223:
1222:
1217:
1215:
1214:
1183:
1181:
1180:
1175:
1173:
1170:
1169:
1160:
1159:
1147:
1143:
1128:
1126:
1125:
1120:
1118:
1116:
1115:
1114:
1105:
1104:
1091:
1086:
1085:
1066:
1064:
1063:
1058:
1053:
1052:
1037:
1035:
1027:
1019:
1010:
1008:
1007:
1002:
1000:
996:
994:
993:
992:
983:
982:
969:
960:
958:
950:
942:
936:
935:
926:
925:
909:
906:
884:
882:
881:
876:
868:
866:
858:
850:
845:
837:
818:
814:
795:
786:
777:
762:
758:
748:
732:
726:
716:
701:
696:
695:
662:
650:
646:
645:
643:
642:
637:
634:
551:
539:
535:
531:
528:for appropriate
527:
519:
511:
496:
488:
486:
485:
480:
475:
473:
460:
440:
431:
430:
426:
410:
407:
396:
395:
388:
368:
359:
357:
356:
338:
337:
310:
298:
291:
284:
280:
231:
230:
228:
227:
222:
219:
211:
210:
208:
207:
202:
199:
118:Fresnel integral
110:Augustin Fresnel
26:
5783:
5782:
5778:
5777:
5776:
5774:
5773:
5772:
5743:
5742:
5741:
5736:
5625:
5579:
5564:
5523:
5519:
5514:
5478:
5465:
5408:
5407:
5404:
5399:
5391:
5389:
5385:
5380:(See Chapter 7)
5345:
5341:
5339:Further reading
5336:
5335:
5302:(1): e0269210.
5289:
5288:
5284:
5243:(3): eaax5145.
5230:
5200:(4): e1001120.
5187:
5186:
5182:
5163:
5162:
5158:
5110:
5109:
5105:
5088:
5087:
5083:
5074:
5073:
5069:
5062:
5049:
5048:
5044:
5008:
5007:
5003:
4981:
4980:
4976:
4928:
4927:
4923:
4879:
4878:
4874:
4864:
4862:
4850:
4845:
4844:
4840:
4833:
4820:
4819:
4815:
4808:
4795:
4794:
4790:
4781:
4779:
4770:
4769:
4765:
4751:
4750:
4746:
4741:
4737:
4728:
4715:
4708:
4695:
4694:
4690:
4685:
4677:List of spirals
4673:
4664:
4653:
4648:
4645:
4642:
4641:
4639:
4638:
4633:
4629:
4623:
4607:
4606:
4587:
4579:
4570:
4557:
4556:
4546:
4516:
4501:
4494:
4482:
4481:
4467:
4466:
4435:
4431:
4421:
4412:
4402:
4399:
4398:
4388:
4378:
4368:
4356:
4355:
4339:
4338:
4302:
4281:
4277:
4233:
4223:
4220:
4219:
4183:
4162:
4158:
4114:
4104:
4095:
4094:
4078:
4077:
4061:
4042:
4015:
4004:
3994:
3963:
3931:
3912:
3885:
3874:
3864:
3840:
3837:
3829:
3823:
3822:
3806:
3787:
3766:
3755:
3745:
3714:
3682:
3663:
3642:
3631:
3621:
3597:
3594:
3586:
3574:
3573:
3555:
3554:
3533:
3529:
3501:
3495:
3494:
3473:
3469:
3441:
3429:
3428:
3425:
3416:
3412:
3411:The two angles
3341:
3306:
3278:
3273:
3272:
3258:
3257:
3245:
3244:
3200:
3168:
3167:
3140:
3124:
3123:
3096:
3074:
3073:
3068:
3066:
3059:
3057:
3056:
3054:
3021:
3011:
3003:
3002:
2942:
2921:
2917:
2907:
2892:
2887:
2886:
2872:
2871:
2855:
2845:
2842:
2841:
2825:
2815:
2806:
2805:
2802:
2793:
2782:
2736:
2732:
2722:
2705:
2701:
2691:
2648:
2647:
2633:
2632:
2618:
2614:
2604:
2585:
2575:
2561:
2552:
2536:
2535:
2521:
2517:
2507:
2488:
2478:
2464:
2455:
2433:
2432:
2417:
2414:
2411:
2410:
2408:
2407:
2397:
2394:
2391:
2390:
2388:
2387:
2375:
2363:
2348:
2338:
2334:
2330:
2314:
2298:
2297:
2281:
2271:
2255:
2249:
2248:
2235:
2227:
2218:
2217:
2203:
2202:
2181:
2177:
2161:
2134:
2128:
2127:
2106:
2102:
2086:
2059:
2047:
2046:
2022:
2003:
1993:
1992:
1969:
1968:
1945:
1927:
1917:
1897:
1896:
1893:
1875:
1874:
1853:
1849:
1821:
1806:
1805:
1784:
1780:
1752:
1731:
1730:
1723:
1720:
1714:
1692:
1691:
1668:
1664:
1648:
1619:
1618:
1589:
1585:
1584:
1580:
1550:
1549:
1511:
1505:
1504:
1483:
1479:
1463:
1436:
1424:
1423:
1398:
1394:
1378:
1377:
1349:
1344:
1343:
1303:
1299:
1298:
1294:
1228:
1227:
1206:
1186:
1185:
1161:
1151:
1131:
1130:
1106:
1096:
1095:
1077:
1069:
1068:
1044:
1028:
1020:
1013:
1012:
998:
997:
984:
974:
973:
961:
951:
943:
938:
937:
927:
917:
910:
887:
886:
859:
851:
831:
830:
816:
812:
784:
779:
775:
770:
760:
756:
746:
741:
730:
724:
714:
709:
699:
694:
689:
677:Cesàro equation
669:
660:
648:
638:
635:
632:
631:
629:
628:
625:
606:
594:
578:
558:
541:
537:
533:
529:
521:
513:
501:
490:
432:
400:
393:
389:
360:
339:
329:
301:
300:
293:
286:
282:
276:
270:
263:
223:
220:
217:
216:
214:
213:
203:
200:
195:
194:
192:
191:
164:
158:
153:
103:James Bernoulli
92:
24:
17:
12:
11:
5:
5781:
5779:
5771:
5770:
5765:
5760:
5755:
5745:
5744:
5738:
5737:
5735:
5734:
5729:
5724:
5719:
5714:
5709:
5702:
5701:
5700:
5690:
5685:
5680:
5675:
5670:
5665:
5664:
5663:
5658:
5653:
5643:
5637:
5635:
5631:
5630:
5627:
5626:
5624:
5623:
5622:
5621:
5611:
5606:
5601:
5596:
5591:
5586:
5581:
5577:
5572:
5570:
5563:
5562:
5557:
5552:
5547:
5542:
5537:
5531:
5529:
5525:
5524:
5517:
5515:
5513:
5512:
5507:
5502:
5497:
5492:
5486:
5484:
5480:
5479:
5466:
5464:
5463:
5456:
5449:
5441:
5435:
5434:
5429:
5424:
5413:"Cornu Spiral"
5403:
5402:External links
5400:
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130:Ernesto Cesaro
126:Henri Poincaré
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2191:
2186:
2182:
2178:
2174:
2171:
2165:
2162:
2156:
2152:
2146:
2143:
2138:
2136:
2131:
2124:
2121:
2116:
2111:
2107:
2103:
2099:
2096:
2090:
2087:
2081:
2077:
2071:
2068:
2063:
2061:
2056:
2032:
2027:
2023:
2019:
2016:
2008:
2004:
1998:
1994:
1989:
1984:
1979:
1976:
1950:
1946:
1942:
1937:
1932:
1928:
1922:
1918:
1914:
1911:
1908:
1905:
1902:
1891:Normalization
1890:
1888:
1871:
1868:
1863:
1858:
1854:
1850:
1846:
1843:
1838:
1833:
1829:
1825:
1823:
1815:
1809:
1802:
1799:
1794:
1789:
1785:
1781:
1777:
1774:
1769:
1764:
1760:
1756:
1754:
1746:
1740:
1726:
1719:
1711:
1705:
1688:
1685:
1680:
1675:
1670:
1665:
1661:
1658:
1652:
1649:
1643:
1639:
1633:
1630:
1625:
1623:
1615:
1612:
1607:
1602:
1597:
1593:
1590:
1586:
1581:
1577:
1574:
1569:
1564:
1560:
1556:
1554:
1546:
1543:
1539:
1536:
1533:
1528:
1523:
1519:
1515:
1513:
1508:
1501:
1498:
1493:
1488:
1484:
1480:
1476:
1473:
1467:
1464:
1458:
1454:
1448:
1445:
1440:
1438:
1433:
1407:
1402:
1399:
1395:
1389:
1386:
1383:
1363:
1360:
1357:
1353:
1350:
1329:
1326:
1321:
1316:
1311:
1307:
1304:
1300:
1295:
1291:
1288:
1283:
1278:
1274:
1270:
1267:
1264:
1260:
1257:
1254:
1249:
1244:
1240:
1236:
1233:
1211:
1203:
1200:
1194:
1191:
1166:
1162:
1156:
1152:
1148:
1144:
1139:
1136:
1111:
1107:
1101:
1097:
1092:
1087:
1082:
1078:
1074:
1054:
1049:
1045:
1041:
1038:
1032:
1029:
1024:
1021:
989:
985:
979:
975:
970:
965:
963:
955:
952:
947:
944:
932:
928:
922:
918:
914:
912:
902:
899:
896:
872:
869:
863:
860:
855:
852:
846:
841:
838:
827:
824:
822:
806:
802:
801:
797:
796:
788:
785:
776:
769:
768:
764:
755:
754:
750:
747:
740:
739:
736:
728:
723:
722:
718:
715:
708:
707:
703:
698:
697:
691:
686:
684:
682:
678:
674:
666:
664:
658:
654:
641:
622:
620:
618:
614:
610:
603:
601:
599:
591:
589:
587:
583:
573:
571:
567:
563:
555:
553:
549:
545:
525:
517:
509:
505:
498:
494:
476:
464:
450:
427:
420:
417:
414:
404:
401:
397:
390:
378:
353:
350:
347:
344:
340:
334:
330:
326:
320:
317:
314:
296:
289:
279:
273:
268:
260:
258:
256:
252:
250:
246:
242:
238:
233:
226:
206:
198:
187:
185:
181:
173:
168:
163:
155:
150:
148:
146:
141:
139:
135:
131:
127:
123:
119:
115:
111:
107:
104:
100:
96:
89:
84:
81:
80:
79:
77:
73:
69:
65:
61:
56:
54:
50:
46:
42:
38:
34:
21:
5763:Plane curves
5704:
5672:
5569:Biochemistry
5416:
5390:. Retrieved
5378:
5371:
5365:
5362:Hyperphysics
5361:
5348:
5299:
5295:
5285:
5240:
5236:
5197:
5193:
5183:
5168:
5159:
5116:
5112:
5106:
5098:the original
5093:
5084:
5070:
5051:
5045:
5018:
5014:
5004:
4987:
4983:
4977:
4934:
4930:
4924:
4889:
4885:
4875:
4863:. Retrieved
4858:
4854:
4841:
4822:
4816:
4797:
4791:
4780:. Retrieved
4776:
4766:
4760:(7): 510–518
4757:
4753:
4747:
4738:
4697:
4691:
4660:
4649:
4625:
4621:
4353:
4092:
3570:power series
3426:
3410:
3052:
2803:
2800:Illustration
2783:
2780:
2430:
2418:
2406:. Note that
2398:
2381:
2377:
2369:
2365:
2354:
2350:
2339:
2320:
2316:
2312:
1894:
1724:
1721:
828:
825:
823:the circle.
810:
780:
771:
742:
734:
710:
670:
639:
626:
607:
595:
579:
559:
547:
543:
523:
515:
507:
503:
499:
492:
294:
287:
277:
271:
264:
253:
234:
224:
204:
196:
188:
177:
151:Applications
142:
108:
97:
93:
57:
52:
45:Cornu spiral
44:
40:
33:Euler spiral
32:
30:
5717:Pitch angle
5693:Logarithmic
5641:Archimedean
5604:Polyproline
4637:also means
798:Derivation
687:Formulation
609:Raph Levien
598:racing line
592:Auto racing
267:diffraction
245:cubic curve
114:diffraction
60:diffraction
53:superspiral
5747:Categories
5706:On Spirals
5656:Hyperbolic
5392:2010-11-12
5119:(3): 1–3.
4865:24 January
4861:(7): 37–41
4782:2023-06-07
4683:References
4622:Note that
3065:, i.e. 100
2796:values.).
566:waveguides
243:cited the
70:to design
5727:Spirangle
5722:Theodorus
5661:Poinsot's
5651:Epispiral
5495:Curvature
5490:Algebraic
5418:MathWorld
5356:R. Nave,
5143:0343-6993
5126:1202.3033
5037:0894-1866
4944:1301.2197
4584:θ
4512:⋅
4503:θ
4492:θ
4404:θ
4333:≈
4324:π
4275:
4258:∫
4252:∞
4249:→
4229:′
4214:≈
4205:π
4156:
4139:∫
4133:∞
4130:→
4110:′
3999:−
3988:∞
3973:∑
3869:−
3858:∞
3843:∑
3750:−
3739:∞
3724:∑
3626:−
3615:∞
3600:∑
3527:
3510:∫
3467:
3450:∫
3346:×
3280:θ
3228:×
3211:×
2947:×
2894:θ
2175:
2153:∫
2100:
2078:∫
1847:
1830:∫
1778:
1761:∫
1662:
1640:∫
1578:
1561:∫
1540:θ
1537:
1520:∫
1477:
1455:∫
1292:
1275:∫
1261:θ
1258:
1241:∫
1192:θ
1025:θ
948:θ
870:∝
856:θ
613:Fontforge
570:waveguide
468:∞
465:−
451:−
445:∞
418:−
402:λ
379:−
373:∞
345:−
37:curvature
5758:Calculus
5683:Involute
5678:Fermat's
5619:Collagen
5555:Symmetry
5328:36607960
5296:PLoS One
5277:31998835
5226:21490724
5175:Archived
5151:52592272
4969:23938654
4916:22714189
4777:Pwayblog
4671:See also
3320:′
3303:′
3197:′
3184:′
3137:′
3093:′
2679:′
2666:′
2549:′
2452:′
2232:′
2166:′
2091:′
1653:′
1468:′
1403:′
1354:′
907:constant
821:osculate
673:whiskers
617:Inkscape
249:parabola
41:clothoid
5768:Spirals
5712:Padovan
5646:Cotes's
5634:Spirals
5540:Antenna
5528:Helices
5500:Gallery
5476:helices
5468:Spirals
5364:(2002)
5319:9821693
5268:6962041
5245:Bibcode
5217:3072363
5170:YouTube
5076:"Spiro"
4949:Bibcode
4894:Bibcode
4657:
4640:
3067:√
3055:√
2804:Given:
2423:
2409:
2403:
2389:
692:Symbols
644:
630:
580:In the
506:) − Fr(
292:on the
241:Rankine
229:
215:
209:
193:
90:History
64:railway
5698:Golden
5614:Triple
5594:Double
5560:Triple
5510:Topics
5483:Curves
5472:curves
5326:
5316:
5275:
5265:
5224:
5214:
5149:
5141:
5058:
5035:
4967:
4914:
4829:
4823:Optics
4804:
4704:
4336:0.6267
4217:0.6267
3377:
2969:
2425:> 1
2216:where
1067:where
885:i.e.,
586:action
489:where
285:above
261:Optics
134:Clotho
5673:Euler
5668:Doyle
5609:Super
5584:Alpha
5535:Angle
5147:S2CID
5121:arXiv
4939:arXiv
4851:(PDF)
2885:Then
2646:Then
2347:Find
2045:then
1422:Thus
1376:Then
1184:thus
653:globe
138:Fates
5732:Ulam
5688:List
5589:Beta
5550:Hemi
5505:List
5474:and
5324:PMID
5273:PMID
5222:PMID
5139:ISSN
5056:ISBN
5033:ISSN
4965:PMID
4912:PMID
4867:2017
4827:ISBN
4802:ISBN
4702:ISBN
4480:and
3271:and
3001:and
2362:Map
2329:Map
1226:Now
532:and
520:and
184:jerk
66:and
5314:PMC
5304:doi
5263:PMC
5253:doi
5212:PMC
5202:doi
5131:doi
5023:doi
4992:doi
4957:doi
4902:doi
4659:= 2
4635:= 1
4272:sin
4242:lim
4153:cos
4123:lim
3524:sin
3464:cos
3060:000
3039:000
2950:300
2940:100
2863:100
2833:300
2374:to
2368:′,
2353:′,
2337:to
2172:sin
2097:cos
1967:or
1844:sin
1775:cos
1727:= 1
1722:If
1659:sin
1575:sin
1534:sin
1474:cos
1342:If
1289:cos
1255:cos
1129:or
522:Fr(
514:Fr(
502:Fr(
491:Fr(
297:= 0
290:= 0
186:).
43:or
31:An
5749::
5599:Pi
5578:10
5470:,
5415:.
5360:,
5322:.
5312:.
5300:18
5298:.
5294:.
5271:.
5261:.
5251:.
5239:.
5235:.
5220:.
5210:.
5196:.
5192:.
5173:.
5167:.
5145:.
5137:.
5129:.
5117:34
5115:.
5092:.
5031:.
5019:12
5017:.
5013:.
4988:36
4986:.
4963:.
4955:.
4947:.
4935:21
4933:.
4910:.
4900:.
4890:20
4888:.
4884:.
4857:.
4853:.
4775:.
4758:29
4756:,
4716:^
3572::
3058:60
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2372:′)
2357:′)
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546:,
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5421:.
5395:.
5330:.
5306::
5279:.
5255::
5247::
5241:6
5228:.
5204::
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5133::
5123::
5078:.
5064:.
5039:.
5025::
4998:.
4994::
4971:.
4959::
4951::
4941::
4918:.
4904::
4896::
4869:.
4859:3
4835:.
4810:.
4785:.
4710:.
4663:s
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4646:/
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4626:R
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4601:2
4598:=
4592:L
4589:d
4581:d
4575:=
4566:R
4563:1
4552:2
4548:L
4544:=
4537:2
4532:s
4528:L
4522:2
4518:L
4507:s
4499:=
4462:2
4457:s
4453:L
4449:=
4441:c
4437:R
4433:2
4427:s
4423:L
4417:=
4408:s
4396:1
4393:=
4384:s
4380:L
4374:c
4370:R
4366:2
4327:2
4316:2
4313:1
4308:=
4300:s
4297:d
4292:)
4287:2
4283:s
4279:(
4267:L
4262:0
4246:L
4238:=
4225:y
4208:2
4197:2
4194:1
4189:=
4181:s
4178:d
4173:)
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4164:s
4160:(
4148:L
4143:0
4127:L
4119:=
4106:x
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4066:i
4063:4
4057:3
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4035:!
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4029:1
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3969:=
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3902:)
3899:1
3896:+
3893:i
3890:2
3887:(
3880:i
3876:)
3872:1
3866:(
3853:0
3850:=
3847:i
3834:=
3827:y
3817:1
3814:+
3811:i
3808:4
3802:1
3799:+
3796:i
3793:4
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3780:!
3777:)
3774:i
3771:2
3768:(
3761:i
3757:)
3753:1
3747:(
3734:0
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3728:i
3720:=
3710:L
3705:0
3700:|
3693:1
3690:+
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3684:4
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3675:+
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3656:!
3653:)
3650:i
3647:2
3644:(
3637:i
3633:)
3629:1
3623:(
3610:0
3607:=
3604:i
3591:=
3584:x
3552:s
3549:d
3544:)
3539:2
3535:s
3531:(
3519:L
3514:0
3506:=
3499:y
3492:s
3489:d
3484:)
3479:2
3475:s
3471:(
3459:L
3454:0
3446:=
3439:x
3415:s
3413:θ
3396:n
3393:a
3390:i
3387:d
3384:a
3381:r
3372:6
3369:1
3364:=
3355:6
3351:3
3343:2
3337:6
3333:1
3327:=
3316:c
3312:R
3308:2
3299:s
3295:L
3289:=
3284:s
3255:1
3252:=
3238:6
3234:1
3221:6
3217:3
3208:2
3205:=
3193:s
3189:L
3180:c
3176:R
3172:2
3164:m
3155:6
3151:1
3145:=
3133:s
3129:L
3120:m
3111:6
3107:3
3101:=
3089:c
3085:R
3069:6
3032:=
3027:s
3023:L
3017:c
3013:R
3009:2
2988:n
2985:a
2982:i
2979:d
2976:a
2973:r
2964:6
2961:1
2956:=
2944:2
2935:=
2927:c
2923:R
2919:2
2913:s
2909:L
2903:=
2898:s
2868:m
2860:=
2851:s
2847:L
2838:m
2830:=
2821:c
2817:R
2794:θ
2786:′
2784:L
2767:1
2764:=
2759:2
2756:2
2751:=
2742:c
2738:R
2734:2
2728:s
2724:L
2711:s
2707:L
2703:2
2697:c
2693:R
2686:2
2683:=
2675:s
2671:L
2662:c
2658:R
2654:2
2624:c
2620:R
2616:2
2610:s
2606:L
2599:=
2591:s
2587:L
2581:c
2577:R
2573:2
2567:s
2563:L
2557:=
2545:s
2541:L
2527:s
2523:L
2519:2
2513:c
2509:R
2502:=
2494:s
2490:L
2484:c
2480:R
2476:2
2470:c
2466:R
2460:=
2448:c
2444:R
2427:.
2419:a
2415:/
2412:1
2399:a
2395:/
2392:1
2384:)
2382:y
2378:x
2376:(
2370:y
2366:x
2364:(
2355:y
2351:x
2349:(
2342:′
2340:L
2335:a
2331:L
2323:)
2321:y
2317:x
2315:(
2295:.
2287:s
2283:L
2277:c
2273:R
2269:2
2265:1
2260:=
2253:a
2246:L
2243:a
2240:=
2229:L
2200:s
2197:d
2192:)
2187:2
2183:s
2179:(
2163:L
2157:0
2147:a
2144:1
2139:=
2132:y
2125:s
2122:d
2117:)
2112:2
2108:s
2104:(
2088:L
2082:0
2072:a
2069:1
2064:=
2057:x
2033:L
2028:2
2024:a
2020:2
2017:=
2009:s
2005:L
1999:c
1995:R
1990:L
1985:=
1980:R
1977:1
1951:2
1947:a
1943:1
1938:=
1933:s
1929:L
1923:c
1919:R
1915:2
1912:=
1909:L
1906:R
1903:2
1872:s
1869:d
1864:)
1859:2
1855:s
1851:(
1839:L
1834:0
1826:=
1819:)
1816:L
1813:(
1810:S
1803:s
1800:d
1795:)
1790:2
1786:s
1782:(
1770:L
1765:0
1757:=
1750:)
1747:L
1744:(
1741:C
1725:a
1689:s
1686:d
1681:)
1676:2
1671:s
1666:(
1650:L
1644:0
1634:a
1631:1
1626:=
1616:s
1613:d
1608:]
1603:2
1598:)
1594:s
1591:a
1587:(
1582:[
1570:L
1565:0
1557:=
1547:s
1544:d
1529:L
1524:0
1516:=
1509:y
1502:s
1499:d
1494:)
1489:2
1485:s
1481:(
1465:L
1459:0
1449:a
1446:1
1441:=
1434:x
1408:a
1400:s
1396:d
1390:=
1387:s
1384:d
1364:s
1361:a
1358:=
1351:s
1330:s
1327:d
1322:]
1317:2
1312:)
1308:s
1305:a
1301:(
1296:[
1284:L
1279:0
1271:=
1268:s
1265:d
1250:L
1245:0
1237:=
1234:x
1212:2
1208:)
1204:s
1201:a
1198:(
1195:=
1167:o
1163:s
1157:c
1153:R
1149:2
1145:1
1140:=
1137:a
1112:o
1108:s
1102:c
1098:R
1093:1
1088:=
1083:2
1079:a
1075:2
1055:s
1050:2
1046:a
1042:2
1039:=
1033:s
1030:d
1022:d
990:o
986:s
980:c
976:R
971:s
966:=
956:s
953:d
945:d
933:o
929:s
923:c
919:R
915:=
903:=
900:s
897:R
873:s
864:s
861:d
853:d
847:=
842:R
839:1
817:x
813:x
783:o
781:s
774:s
772:L
761:s
757:L
745:s
743:θ
731:R
725:θ
713:c
711:R
700:R
661:n
649:n
640:N
636:/
633:1
550:)
548:z
544:x
542:(
538:h
534:b
530:a
526:)
524:b
518:)
516:a
510:)
508:b
504:a
495:)
493:x
477:,
471:)
462:(
458:r
455:F
448:)
442:(
438:r
435:F
428:)
424:)
421:x
415:h
412:(
405:z
398:2
391:(
386:r
383:F
376:)
370:(
366:r
363:F
354:z
351:k
348:j
341:e
335:0
331:E
327:=
324:)
321:z
318:,
315:x
312:(
308:E
295:z
288:x
283:h
278:e
275:0
272:E
225:R
221:/
218:1
205:r
201:/
197:v
174:.
25:t
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