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

Euler spiral - Wikipedia

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2021-07-20 05:30:05

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.

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