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Crest vertical curves are those that have a tangent slope at the end of the curve that is lower than that of the beginning of the curve. When driving on a crest curve, the road appears as a hill, with the vehicle first going uphill before reaching the top of the curve and continuing downhill. The profile also affects road drainage.
Too tight a 'crest' curve could result in the train leaving the track as it drops away beneath it; too tight a 'trough' and the train will plough downwards into the rails and damage them. More precisely, the support force R exerted by the track on a train as a function of the curve radius r , the train mass m , and the speed v , is given by
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 ...
Curvature of general surfaces was first studied by Euler. In 1760 [4] he proved a formula for the curvature of a plane section of a surface and in 1771 [5] he considered surfaces represented in a parametric form. Monge laid down the foundations of their theory in his classical memoir L'application de l'analyse à la géometrie which appeared in ...
A plane curve with non-vanishing curvature has zero torsion at all points. Conversely, if the torsion of a regular curve with non-vanishing curvature is identically zero, then this curve belongs to a fixed plane. The curvature and the torsion of a helix are constant. Conversely, any space curve whose curvature and torsion are both constant and ...
In differential geometry, constant-mean-curvature (CMC) surfaces are surfaces with constant mean curvature. [1] [2] This includes minimal surfaces as a subset, but typically they are treated as special case. Note that these surfaces are generally different from constant Gaussian curvature surfaces, with the important exception of the sphere.
Ontario (Education), 2023 ONCA 788 is a Canadian constitutional law decision concerning s. 15(1) of the Canadian Charter of Rights and Freedoms, being Part I of the Constitution Act, 1982. It concerned the constitutionality of Ontario's requirement that teachers pass a proficiency test in mathematics.
A vertex of a smooth curve is a point where its curvature is a local maximum or minimum; for a circular arc, all points are vertices, but non-circular curves may have a finite discrete set of vertices. For a curve that is not smooth, the points where it is not smooth can also be considered as vertices, of infinite curvature.