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A track transition curve, or spiral easement, is a mathematically-calculated curve on a section of highway, or
railroad track A railway track (British English and UIC terminology) or railroad track (American English), also known as permanent way or simply track, is the structure on a railway or railroad consisting of the rails, fasteners, railroad ties (sleepers, ...
, in which a straight section changes into a curve. It is designed to prevent sudden changes in lateral (or centripetal) acceleration. In plane (viewed from above), the start of the transition of the horizontal curve is at infinite radius, and at the end of the transition, it has the same radius as the curve itself and so forms a very broad spiral. At the same time, in the vertical plane, the outside of the curve is gradually raised until the correct degree of
bank A bank is a financial institution that accepts Deposit account, deposits from the public and creates a demand deposit while simultaneously making loans. Lending activities can be directly performed by the bank or indirectly through capital m ...
is reached. If such an easement were not applied, the lateral acceleration of a rail vehicle would change abruptly at one point (the
tangent point In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More ...
where the straight track meets the curve) with undesirable results. With a road vehicle, a transition curve allows the driver to alter the steering in a gradual manner.


History

On early railroads, because of the low speeds and wide-radius curves employed, the surveyors were able to ignore any form of easement, but during the 19th century, as speeds increased, the need for a track curve with gradually increasing curvature became apparent. Rankine's 1862 "Civil Engineering" cites several such curves, including an 1828 or 1829 proposal based on the " curve of sines" by
William Gravatt William Gravatt FRS (14 July 1806 – 30 May 1866), was a noted English civil engineer and scientific instrument maker. Apprenticed as a mechanical engineer in London from aged 15, after interview he worked with Sir Marc Isambard Brunel on t ...
, and the ''curve of adjustment'' by
William Froude William Froude (; 28 November 1810 in Devon – 4 May 1879 in Simonstown, South Africa) was an English engineer, hydrodynamicist and naval architect. He was the first to formulate reliable laws for the resistance that water offers to ships (su ...
around 1842 approximating the elastic curve. The actual equation given in Rankine is that of a cubic curve, which is a polynomial curve of degree 3, at the time also known as a cubic parabola. In the UK, only from 1845, when legislation and land costs began to constrain the laying out of rail routes and tighter curves were necessary, were the principles beginning to be applied in practice. The 'true spiral', whose curvature is exactly linear in arclength, requires more sophisticated mathematics (in particular, the ability to integrate its
intrinsic equation In geometry, an intrinsic equation of a curve is an equation that defines the curve using a relation between the curve's intrinsic properties, that is, properties that do not depend on the location and possibly the orientation of the curve. The ...
) to compute than the proposals that were cited by Rankine. Several late-19th century civil engineers seem to have derived the equation for this curve independently (all unaware of the original characterization of the curve by
Leonhard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries ...
in 1744). Charles Crandall gives credit to one Ellis Holbrook, in the Railroad Gazette, Dec. 3, 1880, for the first accurate description of the curve. Another early publication was ''The Railway Transition Spiral'' by
Arthur N. Talbot Arthur Newell Talbot (October 21, 1857 – April 3, 1942) was an American civil engineer. He made many contributions to several engineering fields including structures, sewage management, and education. He is considered to be a pioneer in ...
, originally published in 1890. Some early 20th century authors call the curve "Glover's spiral" and attribute it to James Glover's 1900 publication. The equivalence of the railroad transition spiral and the
clothoid 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. Eu ...
seems to have been first published in 1922 by Arthur Lovat Higgins. Since then, "clothoid" is the most common name given the curve, but the correct name (following standards of academic attribution) is 'the
Euler spiral 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. E ...
'.


Geometry

While railroad
track geometry Track geometry is concerned with the properties and relations of points, lines, curves, and surfaces in the three-dimensional positioning of railroad track. The term is also applied to measurements used in design, construction and maintenance of t ...
is intrinsically
three-dimensional Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called '' parameters'') are required to determine the position of an element (i.e., point). This is the inform ...
, for practical purposes the vertical and horizontal components of track geometry are usually treated separately. The overall design pattern for the vertical geometry is typically a sequence of constant grade segments connected by vertical transition curves in which the local grade varies linearly with distance and in which the elevation therefore varies quadratically with distance. Here grade refers to the tangent of the angle of rise of the track. The design pattern for horizontal geometry is typically a sequence of straight line (i.e., a
tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...
) and curve (i.e. a circular arc) segments connected by transition curves. The degree of banking in railroad track is typically expressed as the difference in elevation of the two rails, commonly quantified and referred to as the superelevation. Such difference in the elevation of the rails is intended to compensate for the centripetal acceleration needed for an object to move along a curved path, so that the lateral acceleration experienced by passengers/the cargo load will be minimized, which enhances passenger comfort/reduces the chance of load shifting (movement of cargo during transit, causing accidents and damage). It is important to note that superelevation is not the same as the roll angle of the rail which is used to describe the "tilting" of the individual rails instead of the banking of the entire track structure as reflected by the elevation difference at the "top of rail". Regardless of the horizontal alignment and the superelevation of the track, the individual rails are almost always designed to "roll"/"cant" towards gage side (the side where the wheel is in contact with the rail) to compensate for the horizontal forces exerted by wheels under normal rail traffic. The change of superelevation from zero in a tangent segment to the value selected for the body of a following curve occurs over the length of a transition curve that connects the tangent and the curve proper. Over the length of the transition the curvature of the track will also vary from zero at the end abutting the tangent segment to the value of curvature of the curve body, which is numerically equal to one over the radius of the curve body. The simplest and most commonly used form of transition curve is that in which the superelevation and horizontal curvature both vary linearly with distance along the track.
Cartesian coordinates A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
of points along this spiral are given by the Fresnel integrals. The resulting shape matches a portion of an
Euler spiral 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. E ...
, which is also commonly referred to as a "clothoid", and sometimes "Cornu spiral". A transition curve can connect a track segment of constant non-zero curvature to another segment with constant curvature that is zero or non-zero of either sign. Successive curves in the same direction are sometimes called progressive curves and successive curves in opposite directions are called reverse curves. The Euler spiral provides the shortest transition subject to a given limit on the rate of change of the track superelevation (i.e. the twist of the track). However, as has been recognized for a long time, it has undesirable dynamic characteristics due to the large (conceptually infinite) roll acceleration and rate of change of centripetal acceleration at each end. Because of the capabilities of personal computers it is now practical to employ spirals that have dynamics better than those of the Euler spiral.


See also

* Degree of curvature *
Minimum railway curve radius The minimum railway curve radius is the shortest allowable design radius for the centerline of railway tracks under a particular set of conditions. It has an important bearing on construction costs and operating costs and, in combination with ...
*
Railway systems engineering Railway engineering is a multi-faceted engineering discipline dealing with the design, construction and operation of all types of rail transport systems. It encompasses a wide range of engineering disciplines, including civil engineering, comput ...


References


Sources

* * * *
''Railway Track Design''
pdf from The American Railway Engineering and Maintenance of Way Association, accessed 4 December 2006. * {{Rail tracks Transportation engineering Spirals Track geometry