Track transition curves

A track transition curve, or spiral easement, is a mathematically calculated curve on a section of highway, or railroad track, where a straight section changes into a curve. It is designed to prevent sudden changes in lateral (or centripetal) acceleration. In plan (i.e., 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, thus forming 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 is reached.

If such easement were not applied, the lateral acceleration of a rail vehicle would change abruptly at one point – the tangent point where the straight track meets the curve – with undesirable results. With a road vehicle the driver naturally applies the steering alteration in a gradual manner and the curve is designed to permit this, using the same principle.

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, and the curve of adjustment by William Froude 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. This was also known as cubic parabola at that time.

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, did the principles start to be applied in practice.

The "true spiral", where the curvature is exactly linear in arclength, requires more sophisticated mathematics (in particular, the ability to integrate its intrinsic equation) to compute than the proposals 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 this curve by Leonhard Euler 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, originally published in 1890. Some early 20th century authors call the curve "Glover's spiral", attributing it to James Glover's 1900 publication.

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