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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 ...
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]
The normal curvature, k n, is the curvature of the curve projected onto the plane containing the curve's tangent T and the surface normal u; the geodesic curvature, k g, is the curvature of the curve projected onto the surface's tangent plane; and the geodesic torsion (or relative torsion), τ r, measures the rate of change of the surface ...
PC = point of curvature (point at which the curve begins) PT = point of tangent (point at which the curve ends) PI = point of intersection (point at which the two tangents intersect) T = tangent length; C = long chord length (straight line between PC and PT) L = curve length
In North America, equipment for unlimited interchange between railway companies is built to accommodate for a 288-foot (88 m) radius, but normally a 410-foot (125 m) radius is used as a minimum, as some freight carriages (freight cars) are handled by special agreement between railways that cannot take the sharper curvature.
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 ...
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This phenomenon permits a way of confirming that Earth's surface is locally convex: If the degree of curvature is determined to be the same everywhere on Earth's surface, and that surface was determined to be large enough, the constant curvature would show that Earth is spherical.