Euler spiral

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 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).

On early railroads this instant application of lateral force was not an issue since low speeds and wide-radius curves were employed (lateral forces on the passengers and the lateral sway was small and tolerable). As speeds of rail vehicles increased over the years, it became obvious that an easement is necessary, so that the centripetal acceleration increases linearly with the traveled distance. Given the expression of centripetal acceleration v²/r, the obvious solution is to provide an easement curve whose curvature, 1/R, increases linearly with the traveled distance. This geometry is an Euler spiral.[3]

Unaware of the solution of the geometry by Leonhard Euler, Rankine cited the cubic curve (a polynomial curve of degree 3), which is an approximation of the Euler spiral for small angular changes in the same way that a parabola is an approximation to a circular curve.

Marie Alfred Cornu (and later some civil engineers) also solved the calculus of 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.

The Cornu spiral can be used to describe a diffraction pattern.[4] Consider a plane wave with phasor amplitude E₀e^−jkz which is diffracted by a "knife edge" of height h above x = 0 on the z = 0 plane. Then the diffracted wave field can be expressed as

$${\displaystyle \mathbf {E} (x,z)=E_{0}e^{-jkz}{\frac {\mathrm {Fr} (\infty )-\mathrm {Fr} \left({\sqrt {\frac {2}{\lambda z}}}(h-x)\right)}{\mathrm {Fr} (\infty )-\mathrm {Fr} (-\infty )}},}$$

where 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 Fr(a) − Fr(b) as the physical distances between the points represented by Fr(a) and Fr(b) for appropriate a and b. This facilitates a rough computation of the attenuation of the plane wave by the knife edge of height h at a location (x, z) beyond the knife edge.

Bends with continuously varying radius of curvature following the Euler spiral are also used to reduce losses in photonic integrated circuits, either in singlemode waveguides,[5][6] to smoothen the abrupt change of curvature and suppress coupling to radiation modes, or in multimode waveguides,[7] 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,[8] with a hollow metal 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.

Motorsport author Adam Brouillard has shown the Euler spiral's use in optimizing the racing line during the corner entry portion of a turn.[9]

Raph Levien has released Spiro as a toolkit for curve design, especially font design, in 2007[10][11] under a free licence. This toolkit has been implemented quite quickly afterwards in the font design tool Fontforge and the digital vector drawing Inkscape.

Cutting a sphere along a spiral with width 1/N and flattening out the resulting shape yields an Euler spiral when n tends to the infinity.[12] If the sphere is the globe, this produces a map projection whose distortion tends to zero as n tends to the infinity.[13]

Natural shapes of rat's mystacial pad vibrissae (whiskers) are well approximated by pieces of the Euler spiral. When all these pieces for a single rat are assembled together, they span an interval extending from one coiled domain of the Euler spiral to the other.[14]

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

$${\displaystyle {\begin{aligned}C(L)&=\int _{0}^{L}\cos \left(s^{2}\right)\,ds\\S(L)&=\int _{0}^{L}\sin \left(s^{2}\right)\,ds\end{aligned}}}$$

For a given Euler curve with:

$${\displaystyle 2RL=2R_{c}L_{s}={\frac {1}{a^{2}}}}$$

or

$${\displaystyle {\frac {1}{R}}={\frac {L}{R_{c}L_{s}}}=2a^{2}L}$$

then

$${\displaystyle {\begin{aligned}x&={\frac {1}{a}}\int _{0}^{L'}\cos \left(s^{2}\right)\,ds\\y&={\frac {1}{a}}\int _{0}^{L'}\sin \left(s^{2}\right)\,ds\end{aligned}}}$$

where

$${\displaystyle {\begin{aligned}L'&=aL\\a&={\frac {1}{\sqrt {2R_{c}L_{s}}}}.\end{aligned}}}$$

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

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

In the normalization process,

$${\displaystyle {\begin{aligned}R'_{c}&={\frac {R_{c}}{\sqrt {2R_{c}L_{s}}}}={\sqrt {\frac {R_{c}}{2L_{s}}}}\\L'_{s}&={\frac {L_{s}}{\sqrt {2R_{c}L_{s}}}}={\sqrt {\frac {L_{s}}{2R_{c}}}}\end{aligned}}}$$

Then

$${\displaystyle 2R'_{c}L'_{s}=2{\sqrt {\frac {R_{c}}{2L_{s}}}}{\sqrt {\frac {L_{s}}{2R_{c}}}}={\frac {2}{2}}=1}$$

Generally the normalization reduces 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 instability of the calculation, especially for bigger θ values.).

Given:

$${\displaystyle {\begin{aligned}R_{c}&=300\,\mathrm {m} \\L_{s}&=100\,\mathrm {m} \end{aligned}}}$$

Then

$${\displaystyle \theta _{s}={\frac {L_{s}}{2R_{c}}}={\frac {100}{2\times 300}}={\frac {1}{6}}\ \mathrm {radian} }$$

and

$${\displaystyle 2R_{c}L_{s}=60\,000}$$

We scale down the Euler spiral by √60000, i.e. 100√6 to normalized Euler spiral that has:

$${\displaystyle {\begin{aligned}R'_{c}&={\tfrac {3}{\sqrt {6}}}\,\mathrm {m} \\L'_{s}&={\tfrac {1}{\sqrt {6}}}\,\mathrm {m} \\2R'_{c}L'_{s}&=2\times {\tfrac {3}{\sqrt {6}}}\times {\tfrac {1}{\sqrt {6}}}\\&=1\end{aligned}}}$$

and

$${\displaystyle \theta _{s}={\frac {L'_{s}}{2R'_{c}}}={\frac {\frac {1}{\sqrt {6}}}{2\times {\frac {3}{\sqrt {6}}}}}={\frac {1}{6}}\ \mathrm {radian} }$$

The two angles θ_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.

Normalized Euler spirals can be expressed as:

$${\displaystyle {\begin{aligned}x&=\int _{0}^{L}\cos \left(s^{2}\right)\,ds\\y&=\int _{0}^{L}\sin \left(s^{2}\right)\,ds\end{aligned}}}$$

or expressed as power series:

$${\displaystyle {\begin{aligned}x&=\left.\sum _{i=0}^{\infty }{\frac {(-1)^{i}}{(2i)!}}{\frac {s^{4i+1}}{4i+1}}\right|_{0}^{L}&&=\sum _{i=0}^{\infty }{\frac {(-1)^{i}}{(2i)!}}{\frac {L^{4i+1}}{4i+1}}\\y&=\left.\sum _{i=0}^{\infty }{\frac {(-1)^{i}}{(2i+1)!}}{\frac {s^{4i+3}}{4i+3}}\right|_{0}^{L}&&=\sum _{i=0}^{\infty }{\frac {(-1)^{i}}{(2i+1)!}}{\frac {L^{4i+3}}{4i+3}}\end{aligned}}}$$

The normalized Euler spiral will converge to a single point in the limit as the parameter L approaches infinity, which can be expressed as:

$${\displaystyle {\begin{aligned}x^{\prime }&=\lim _{L\to \infty }\int _{0}^{L}\cos \left(s^{2}\right)\,ds&&={\frac {1}{2}}{\sqrt {\frac {\pi }{2}}}\approx 0.6267\\y^{\prime }&=\lim _{L\to \infty }\int _{0}^{L}\sin \left(s^{2}\right)\,ds&&={\frac {1}{2}}{\sqrt {\frac {\pi }{2}}}\approx 0.6267\end{aligned}}}$$

Normalized Euler spirals have the following properties:

$${\displaystyle {\begin{aligned}2R_{c}L_{s}&=1\\\theta _{s}&={\frac {L_{s}}{2R_{c}}}=L_{s}^{2}\end{aligned}}}$$

and

$${\displaystyle {\begin{aligned}\theta &=\theta _{s}\cdot {\frac {L^{2}}{L_{s}^{2}}}=L^{2}\\{\frac {1}{R}}&={\frac {d\theta }{dL}}=2L\end{aligned}}}$$

Note that 2R_cL_s = 1 also means 1/R_c = 2L_s, in agreement with the last mathematical statement.