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.
Euler spirals have applications to diffraction computations. They are also widely used as transition curves in railroad engineering/highway engineering for connecting and transitioning the geometry between a tangent and a circular curve. A similar application is also found in photonic 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:
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.
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. 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 asConsider a plane wave with phasor amplitude
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,to smoothen the abrupt change of curvature and 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 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.
Raph Levien has released Spiro as a toolkit for curve design, especially font design, in 2007under 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. If the sphere is the globe, this produces a map projection whose distortion tends to zero as n tends to the infinity.
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.
|R||Radius of curvature|
|Rc||Radius of circular curve at the end of the spiral|
|θ||Angle of curve from beginning of spiral (infinite R) to a particular point on the spiral. |
This can also be measured as the angle between the initial tangent and the tangent at the concerned point.
|θs||Angle of full spiral curve|
|L, s||Length measured along the spiral curve from its initial position|
|Ls, so||Length of spiral curve|
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 osculate the circle.
The spiral is a small segment of the above double-end Euler spiral in the first quadrant.
If a = 1, which is the case for normalized Euler curve, then the Cartesian coordinates are given by Fresnel integrals (or Euler integrals):
For a given Euler curve with:
The process of obtaining solution of (x, y) of an Euler spiral can thus be described as:
In the normalization process,
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.).
We scale down the Euler spiral by √, i.e. 100√ to normalized Euler spiral that has:
The two angles θs 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:
or expressed as power series:
The normalized Euler spiral will converge to a single point in the limit as the parameter L approaches infinity, which can be expressed as:
Normalized Euler spirals have the following properties:
Note that 2RcLs = 1 also means 1/Rc = 2Ls, in agreement with the last mathematical statement.
In mathematics, the arithmetic–geometric mean (AGM) of two positive real numbers x and y is defined as follows:
Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that for any real number x:
In mathematical physics and mathematics, the Pauli matrices are a set of three 2 × 2 complex matrices which are Hermitian and unitary. Usually indicated by the Greek letter sigma, they are occasionally denoted by tau when used in connection with isospin symmetries.
In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point is called the pole, and the ray from the pole in the reference direction is the polar axis. The distance from the pole is called the radial coordinate, radial distance or simply radius, and the angle is called the angular coordinate, polar angle, or azimuth. The radial coordinate is often denoted by r or ρ, and the angular coordinate by φ, θ, or t. Angles in polar notation are generally expressed in either degrees or radians.
The Archimedean spiral is a spiral named after the 3rd-century BC Greek mathematician Archimedes. It is the locus corresponding to the locations over time of a point moving away from a fixed point with a constant speed along a line that rotates with constant angular velocity. Equivalently, in polar coordinates (r, θ) it can be described by the equation
In mathematics, a spiral is a curve which emanates from a point, moving farther away as it revolves around the point.
A helix, plural helixes or helices, is a shape like a corkscrew or spiral staircase. It is a type of smooth space curve with tangent lines at a constant angle to a fixed axis. Helices are important in biology, as the DNA molecule is formed as two intertwined helices, and many proteins have helical substructures, known as alpha helices. The word helix comes from the Greek word ἕλιξ, "twisted, curved". A "filled-in" helix – for example, a "spiral" (helical) ramp – is called a helicoid.
A hyperbolic spiral is a plane curve, which can be described in polar coordinates by the equation
A Fermat's spiral or parabolic spiral is a plane curve named after Pierre de Fermat. Its polar coordinate representation is given by
In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function of the form
The Fresnel integralsS(x) and C(x) are two transcendental functions named after Augustin-Jean Fresnel that are used in optics and are closely related to the error function. They arise in the description of near-field Fresnel diffraction phenomena and are defined through the following integral representations:
In mathematics, theta functions are special functions of several complex variables. They are important in many areas, including the theories of Abelian varieties and moduli spaces, and of quadratic forms. They have also been applied to soliton theory. When generalized to a Grassmann algebra, they also appear in quantum field theory.
In mathematics, physics and engineering, the sinc function, denoted by sinc(x), has two forms, normalized and unnormalized.
In geometry, the area enclosed by a circle of radius r is πr2. Here the Greek letter π represents the constant ratio of the circumference of any circle to its diameter, approximately equal to 3.1416.
Arc length is the distance between two points along a section of a curve.
In optics, the Fresnel diffraction equation for near-field diffraction is an approximation of the Kirchhoff–Fresnel diffraction that can be applied to the propagation of waves in the near field. It is used to calculate the diffraction pattern created by waves passing through an aperture or around an object, when viewed from relatively close to the object. In contrast the diffraction pattern in the far field region is given by the Fraunhofer diffraction equation.
In mathematics, the sine is a trigonometric function of an angle. The sine of an acute angle is defined in the context of a right triangle: for the specified angle, it is the ratio of the length of the side that is opposite that angle, to the length of the longest side of the triangle. For an angle , the sine function is denoted simply as .
In geometry, various formalisms exist to express a rotation in three dimensions as a mathematical transformation. In physics, this concept is applied to classical mechanics where rotational kinematics is the science of quantitative description of a purely rotational motion. The orientation of an object at a given instant is described with the same tools, as it is defined as an imaginary rotation from a reference placement in space, rather than an actually observed rotation from a previous placement in space.
In integral calculus, the Weierstrass substitution or tangent half-angle substitution is a method for evaluating integrals, which converts a rational function of trigonometric functions of into an ordinary rational function of by setting . No generality is lost by taking these to be rational functions of the sine and cosine. The general transformation formula is