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Where degree of curvature is based on 100 units of arc length, the conversion between degree of curvature and radius is Dr = 18000/π ≈ 5729.57795, where D is degree and r is radius. Since rail routes have very large radii, they are laid out in chords, as the difference to the arc is inconsequential; this made work easier before electronic ...
Radius of curvature and center of curvature. In differential geometry, the radius of curvature, R, is the reciprocal of the curvature. For a curve, it equals the radius of the circular arc which best approximates the curve at that point. For surfaces, the radius of curvature is the radius of a circle that best fits a normal section or ...
The relationship between speed and tilt can be calculated mathematically. We start with the formula for a balancing centripetal force: θ is the angle by which the train is tilted due to the cant, r is the curve radius in meters, v is the speed in meters per second, and g is the standard gravity, approximately equal to 9.81 m/s²:
The curvature is the reciprocal of radius of curvature. That is, the curvature is =, where R is the radius of curvature [5] (the whole circle has this curvature, it can be read as turn 2π over the length 2π R). This definition is difficult to manipulate and to express in formulas.
In North America, the measurement of curvature is expressed in degree of curvature. This is done by having a chord of 100 feet (30.48 m) connecting to two points on an arc of the reference rail, then drawing radii from the center to each of the chord's end points. The angle between the radii lines is the degree of curvature. [10]
When calculating the length of a short north-south line at the equator, the circle that best approximates that line has a radius of (which equals the meridian's semi-latus rectum), or 6335.439 km, while the spheroid at the poles is best approximated by a sphere of radius , or 6399.594 km, a 1% difference. So long as a spherical Earth is assumed ...
Geographical distance or geodetic distance is the distance measured along the surface of the Earth, or the shortest arch length.. The formulae in this article calculate distances between points which are defined by geographical coordinates in terms of latitude and longitude.
The second animation shows the increasing curvature of the transition curve which is able to connect to a circular arc of progressively smaller radius. In the horizontal plane, the radius of a transition curve varies continually over its length between the disparate radii of the sections that it joins—for example, from infinite radius at a ...