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| [[File:Euler spiral.svg|300px|thumb|A double-end Euler spiral]] | | [url=http://www.saintjeanmarc.com/images/Subscript.asp?tid=525]ミュウミュウ 長財布 激安[/url] Taylor Abduction and Murder Trial Day 3.txt |
| 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'''.
| | [url=http://www.saintjeanmarc.com/images/Subscript.asp?tid=72]ミュウミュウ 財布 リボン ラウンドファスナー ブ[/url] 12 restaurants with 1.txt |
| | | [url=http://www.saintjeanmarc.com/images/Subscript.asp?tid=700]miumiu バッグ 店舗 め[/url] Alabama's Nick Saban learned from Bill Belichick with the Browns before.txt |
| Euler spirals have applications to [[diffraction]] computations. They are also widely used as transition curve in [[Railway systems engineering|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:
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| *Its curvature begins with zero at the straight section (the tangent) and increases linearly with its curve length.
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| *Where the Euler spiral meets the circular curve, its curvature becomes equal to that of the latter.
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| ==Applications==
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| ===Track transition curve===
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| {{Main|Track transition curve}}
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| An object traveling on a circular path experiences a [[Uniform circular motion|centripetal acceleration]]. When a vehicle traveling on a straight path suddenly transitions to a tangential circular path, it experiences a sudden centripetal acceleration starting at the tangent point; and this centripetal force acts instantly causing much discomfort (causing [[Jerk (physics)|jerk]]).
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| 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 {{math| V² / R}}, the obvious solution is to provide an easement curve whose curvature, {{math| 1 / R}}, increases linearly with the traveled distance. This geometry is an Euler spiral.
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| Unaware of the solution of the geometry by [[Leonhard Euler]], [[William John Macquorn Rankine|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.
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| [[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.
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| ===Optics===
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| The Cornu <!--In optics the term Cornu Spiral is universal, please don't change to Euler spiral merely for consistency with the rest of the article without making this clear --> spiral can be used to describe a [[diffraction]] pattern.<ref name=Hecht>
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| {{cite book
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| | title = Optics (3rd edition)
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| | author = Eugene Hecht
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| | publisher = Addison-Wesley
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| | year = 1998
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| | isbn = 0-201-30425-2
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| | page = 491
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| }}</ref>
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| ==Formulation==
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| ===Symbols===
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| {|
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| | <math>R\,</math> || Radius of curvature
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| |-
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| | <math>R_c\,</math> || Radius of Circular curve at the end of the spiral
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| |-
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| | <math>\theta\,</math> || Angle of curve from beginning of spiral (infinite <math>R</math>) to a particular point on the spiral.
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| |-
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| | ||This can also be measured as the angle between the initial tangent and the tangent at the concerned point.
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| |-
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| | <math>\theta _s\,</math> || Angle of full spiral curve
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| |-
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| | <math>L , s\,</math> || Length measured along the spiral curve from its initial position
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| |-
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| | <math>L_s , s_o\,</math> || Length of spiral curve
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| |}
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| {{cot|Derivation}}
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| [[File:Easement curve.svg|360px|right]]
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| 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 [[osculation|osculate]] the circle.
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| The spiral is a small segment of the above double-end Euler spiral in the first quadrant.
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| : From the definition of the curvature,
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| :: <math>\frac {1}{R} = \frac {d\theta}{ds} \propto s </math>
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| :i.e.
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| :: <math>R s = \text{constant} = R_c s_o\,</math>
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| :: <math>\frac {d\theta}{ds} = \frac {s}{R_c s_o}</math>
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| : We write in the format,
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| ::<math>\frac {d\theta}{ds} = 2a^2 s</math>
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| :Where
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| ::<math>2a^2= \frac {1}{R_c s_o}</math>
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| :Or
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| ::<math>a = \frac {1}{\sqrt {2R_c s_o} }</math>
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| :Thus
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| ::<math>\theta = (a s)^2\,</math>
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| :Now
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| :: <math>
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| \begin{align}
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| x & = \int_0^L \cos\theta \, ds \\
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| & = \int_0^L \cos \left[ (a s)^2 \right] ds
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| \end{align}
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| </math>
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| :If
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| ::<math>s' = a s \,</math>
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| :Then
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| ::<math>ds = \frac{ds'}{a}\,</math>
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| :Thus
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| :: <math>x = \frac{1}{a} \int_0^{L'} \cos {s}^2 ds</math>
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| :: <math>
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| \begin{align}
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| y & = \int_0^L \sin\theta \, ds \\
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| & = \int_0^L \sin \left[ (a s)^2 \right] ds \\
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| & = \frac{1}{a} \int_0^{L'} \sin {s}^2 \, ds
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| \end{align}
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| </math>
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| {{cob}}
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| ===Expansion of Fresnel integral===
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| {{Main|Fresnel integral}}
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| If a = 1, which is the case for normalized Euler curve, then the Cartesian coordinates are given by Fresnel integrals (or Euler integrals):
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| :<math>\begin{align} C(L) &=\int_0^L\cos s^2 \, ds\\
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| S(L) &= \int_0^L\sin s^2 \, ds\end{align}</math>
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| Expand C(L) according to [[Trigonometric functions#Series definitions|power series]] expansion of cosine:
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| :<math> \cos \theta = 1 - \frac{\theta^2}{2!} + \frac{\theta^4}{4!} - \frac{\theta^6}{6!} + \cdots</math>
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| :<math>\begin{align} C(L) &= \int_0^L \cos s^2 \, ds \\
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| &= \int_0^L (1 - \frac{s^4}{2!} + \frac{s^8}{4!} - \frac{s^{12}}{6!} + \cdots) \, ds\\
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| &= L - \frac{L^5}{5 \times 2!} + \frac{L^9}{9 \times 4!} - \frac{L^{13}}{13 \times 6!} +\cdots\end{align}</math>
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| Expand S(L) according to power series expansion of sine:
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| :<math> \sin \theta = \theta - \frac{\theta^3}{3!} + \frac{\theta^5}{5!} - \frac{\theta^7}{7!} + \cdots</math>
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| :<math>\begin{align} S(L) &= \int_0^L \sin s^2 \, ds\\
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| &= \int_0^L (s^2 - \frac{s^6}{3!} + \frac{s^{10}}{5!} - \frac{s^{14}}{7!} + \cdots) \, ds\\
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| &= \frac{L^3}{3} - \frac{L^7}{7 \times 3!} + \frac{L^{11}}{11 \times 5!} - \frac{L^{15}}{15 \times 7!} +\cdots\end{align}</math>
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| === Normalization and conclusion ===
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| For a given Euler curve with:
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| :<math>2RL = 2R_c L_s = \frac{1}{a^2} \, </math>
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| or
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| :<math>\frac{1}{R} = \frac{L}{R_c L_s} = 2a^2L \, </math>
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| then
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| :<math>x=\frac{1}{a} \int_0^{L'} \cos s^2 \, ds</math>
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| :<math>y=\frac{1}{a} \int_0^{L'} \sin s^2 \, ds \, </math>
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| where <math>L' = aL \,</math> and <math>a = \frac{1}{\sqrt{2R_c L_s}}</math>.
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| The process of obtaining solution of {{math|(''x'', ''y'')}} of an Euler spiral can thus be described as:
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| * Map L of the original Euler spiral by multiplying with factor <math>a</math> to <math>L'</math> of the normalized Euler spiral;
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| * Find {{math|(''x''′, ''y''′)}} from the Fresnel integrals; and
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| * Map {{math|(''x''′, ''y''′)}} to {{math|(''x'', ''y'')}} by scaling up (denormalize) with factor <math> 1/a </math>. Note that <math> 1/a > 1 </math>.
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| In the normalization process,
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| :<math>
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| \begin{align}
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| R'_c & = \frac{R_c}{\sqrt{2 R_c L_s}} \\
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| & = \sqrt{\frac{R_c}{2L_s}} \\
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| \end{align}
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| </math>
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| :<math>
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| \begin{align}
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| L'_s & = \frac{L_s}{\sqrt{2R_c L_s}} \\
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| & = \sqrt{\frac{L_s}{2R_c}}
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| \end{align}
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| </math>
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| Then
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| :<math>
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| \begin{align}
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| 2R'_c L'_s & = 2 \sqrt{\frac{R_c}{2L_s} } \sqrt{\frac{L_s}{2 R_c}} \\
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| & = \tfrac{2}{2} \\
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| & = 1
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| \end{align}
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| </math>
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| 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 stability|numerical instability]] of the calculation, esp. for bigger ''<math>\theta\,</math>'' values.).
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| <!-- Original Research?
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| ==== Proposed distinction between Euler spiral and Cornu spiral ==== | |
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| It is proposed that the term '''Euler spiral''' applies generally where as the term '''Cornu spiral''' shall only apply to the scaled down version of Euler spiral that has 2Rc'.Ls' = 1.
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| -->
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| ==== Illustration ====
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| Given:
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| :<math> | |
| \begin{align}
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| R_c & = 300\mbox{m} \\
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| L_s &= 100\mbox{m}
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| \end{align}
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| </math>
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| Then
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| :<math>
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| \begin{align}
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| \theta_s & = \frac{L_s} {2R_c} \\
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| & = \frac{100} {2 \times 300} \\
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| & = 0.1667 \ \mbox{radian} \\
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| \end{align}
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| </math>
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| And
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| :<math> 2R_c L_s = 60,000 \,</math>
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| We scale down the Euler spiral by √60,000, i.e.100√6 to normalized Euler spiral that has:
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| :<math>
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| \begin{align}
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| R'_c = \tfrac{3}{\sqrt{6}}\mbox{m} \\
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| L'_s = \tfrac{1}{\sqrt{6}}\mbox{m} \\
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| \\
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| \end{align}
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| </math>
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| :<math>
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| \begin{align}
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| 2R'_c L'_s & = 2 \times \tfrac{3}{\sqrt{6}} \times \tfrac{1}{\sqrt{6}} \\
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| & = 1
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| \end{align}
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| </math>
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| And
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| :<math>
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| \begin{align}
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| \theta_s & = \frac{L'_s}{2R'_c} \\
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| & = \frac{\tfrac{1}{\sqrt{6}}} {2 \times \tfrac{3}{\sqrt{6}}} \\
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| & = 0.1667 \ \mbox{radian} \\
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| \end{align}
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| </math>
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| The two angles <math>\theta_s\,</math> 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.
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| ===Other properties of normalized Euler spiral=== | |
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| Normalized Euler spiral can be expressed as:
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| :: <math>x = \int_0^L \cos s^2 ds</math>
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| :: <math>y = \int_0^L \sin s^2 ds</math>
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| Normalized Euler spiral has the following properties:
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| :<math>2 R_c L_s = 1 \,\!</math>
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| :<math>\theta_s = \frac{L_s}{2 R_c} = L_s ^2</math>
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| And
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| :<math>\theta = \theta _s\cdot\frac{L^2}{L_s^2} = L^2</math>
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| :<math>\frac{1}{R} = \frac{d\theta}{dL} = 2L</math>
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| Note that <math>2R_c L_s = 1</math> also means <math>1/R_c = 2L_s</math>, in agreement with the last mathematical statement.
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| == Code for producing an Euler spiral ==
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| The following [[Sage (mathematics software)|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.
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| var('L')
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| p = integral(taylor(cos(L^2), L, 0, 12), L)
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| q = integral(taylor(sin(L^2), L, 0, 12), L)
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| r1 = parametric_plot([p, q], (L, 0, 1), color = 'red')
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|
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| r2 = line([(-1.0, 0), (0,0)], rgbcolor = 'blue')
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|
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| x1 = p.subs(L = 1)
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| y1 = q.subs(L = 1)
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| R = 0.5
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| x2 = x1 - R*sin(1.0)
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| y2 = y1 + R*cos(1.0)
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| r3 = circle((x2, y2), R, rgbcolor = 'green')
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| show(r1 + r2 + r3, aspect_ratio = 1, axes=false)
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| The following is [[Mathematica]] code for the Euler spiral component (it works directly in wolframalpha.com):
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| <code>
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| ParametricPlot[
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| {FresnelC[Sqrt[2/\[Pi]] t]/Sqrt[2/\[Pi]],
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| FresnelS[Sqrt[2/\[Pi]] t]/Sqrt[2/\[Pi]]},
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| {t, -10, 10}]
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| </code>
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| == See also ==
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| *[[Fresnel integral]]
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| * [[Geometric design of roads]]
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| * [[Track transition curve]]
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| ==References==
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| {{reflist}}
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| {{refbegin}}
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| *{{cite book
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| | last = Kellogg
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| | first = Norman Benjamin
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| | title = The Transition Curve or Curve of Adjustment
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| | publisher = McGraw
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| | location = New York
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| | year = 1907
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| | edition = 3rd edition
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| | url = http://books.google.com/books?id=ZVZCzW2codgC}}
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| {{refend}}
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| *{{mathworld|title=Cornu Spiral|urlname=CornuSpiral}}
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| *R. Nave, [http://hyperphysics.phy-astr.gsu.edu/hbase/phyopt/cornu.html#c1 The Cornu spiral], ''Hyperphysics'' (2002) ''(Uses πt²/2 instead of t².)''
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| * 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)]''
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| *{{cite web
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| |url=http://physics.gu.se/LISEBERG/eng/loop_pe.html
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| |title=Roller Coaster Loop Shapes
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| |accessdate=2010-11-12}}
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| == External links ==
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| * [http://www.2dcurves.com/spiral/spirale.html Euler's spiral at 2-D Mathematical Curves]
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| * [http://jsxgraph.uni-bayreuth.de/wiki/index.php/Euler%27s_spiral_(Clothoid) Interactive example with JSXGraph]
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| {{DEFAULTSORT:Euler Spiral}}
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| [[Category:Transport engineering]]
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| [[Category:Calculus]]
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| [[Category:Curves]]
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| [[Category:Spirals]]
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