Track transition curve

A track transition curve, or spiral easement, is a mathematically-calculated curve on a section of highway, or railroad track, 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 is reached. A track transition curve, or spiral easement, is a mathematically-calculated curve on a section of highway, or railroad track, 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 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 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 that by 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, at the time also known as a cubic parabola.