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A curve should not become a straight all at once, but should gradually increase in radius over time (a distance of around 40–80 m (130–260 ft) for a line with a maximum speed of about 100 km/h (62 mph)). Even sharper than curves with no transition are reverse curves with no intervening straight track.
Vertical curve is the curve in vertical layout to connect two track gradients together whether it is for changing from an upgrade to a downgrade (summit), changing from a downgrade to an upgrade (sag or valley), changing in two levels of upgrades or changing in two levels of downgrades.
Sag vertical curves are those that have a tangent slope at the end of the curve that is higher than that of the beginning of the curve. When driving on a road, a sag curve would appear as a valley, with the vehicle first going downhill before reaching the bottom of the curve and continuing uphill or level.
Road curves are irregular bends in roads to bring a gradual change of direction. Similar curves are on railways and canals. Curves provided in the horizontal plane are known as horizontal curves and are generally circular or parabolic. Curves provided in the vertical plane are known as vertical curve.
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 combinations thereof. [1] [2] [3]
Curvature is usually measured in radius of curvature.A small circle can be easily laid out by just using radius of curvature, but degree of curvature is more convenient for calculating and laying out the curve if the radius is as large as a kilometer or mile, as is needed for large scale works like roads and railroads.
Similarly, on highways, transition curves allow drivers to change steering gradually when entering or exiting curves. Transition curves also serve as a transition in the vertical plane, whereby the elevation of the inside or outside of the curve is lowered or raised to reach the nominal amount of bank for the curve.
Slope illustrated for y = (3/2)x − 1.Click on to enlarge Slope of a line in coordinates system, from f(x) = −12x + 2 to f(x) = 12x + 2. The slope of a line in the plane containing the x and y axes is generally represented by the letter m, [5] and is defined as the change in the y coordinate divided by the corresponding change in the x coordinate, between two distinct points on the line.