Euler spiral
[1][2] The behavior of Fresnel integrals can be illustrated by an Euler spiral, a connection first made by Marie Alfred Cornu in 1874.
[3] Euler's spiral is a type of superspiral that has the property of a monotonic curvature function.
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: The spiral has multiple names reflecting its discovery and application in multiple fields.
[2] Leonhard Euler's work on the spiral came after James 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?
Thirty-eight years later, in 1781, he reported his discovery of the formula for the limit (by "happy chance").
He was unaware of Euler's integrals or the connection to the theory of elasticity.
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" after Clotho, one of the three Fates who spin the thread of life in Greek mythology.
[2] The third independent discovery occurred in the 1800s when various railway engineers sought a formula for gradual curvature in track shape.
By 1880 Arthur Newell Talbot worked out the integral formulas and their solution, which he called the "railway transition spiral".
[2] Unaware of the solution of the geometry by Euler, William 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.
[citation needed] 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).
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 acceleration v2/r, the obvious solution is to provide an easement curve whose curvature, 1/R, increases linearly with the traveled distance.
where Fr(x) is the Fresnel integral function, which forms the Cornu spiral on the complex plane.
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,[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.
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.
In the path integral formulation of quantum mechanics, the probability amplitude for propagation between two points can be visualized by connecting action phase arrows for each time step between the two points.
[11] Motorsport author Adam Brouillard has shown the Euler spiral's use in optimizing the racing line during the corner entry portion of a turn.
This toolkit has been implemented quite quickly afterwards in the font design tool Fontforge and the digital vector drawing Inkscape.
[15] If the sphere is the globe, this produces a map projection whose distortion tends to zero as n tends to the infinity.
[16] 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 the Cesàro equation for an Euler spiral segment might give insight into the keratinization mechanism of whisker growth.
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.
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.).
This thus confirms that the original and normalized Euler spirals are geometrically similar.
The normalized Euler spiral will converge to a single point in the limit as the parameter L approaches infinity, which can be expressed as:
