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The proper equation depends on whether the vertical curve is shorter or longer than the available sight distance. Normally, both equations are solved, then the results are compared to the curve length. [4] [5] sight distance > curve length (S > L) = (+)
The distance from (x 0, y 0) to this line is measured along a vertical line segment of length |y 0 - (-c/b)| = |by 0 + c| / |b| in accordance with the formula. Similarly, for vertical lines (b = 0) the distance between the same point and the line is |ax 0 + c| / |a|, as measured along a horizontal line segment.
The length of the curve is given by the formula = | ′ | where | ′ | is the Euclidean norm of the tangent vector ′ to the curve. To justify this formula, define the arc length as limit of the sum of linear segment lengths for a regular partition of [ a , b ] {\displaystyle [a,b]} as the number of segments approaches infinity.
Some countries do not have specification on the exact geometry of vertical curves beyond general specification on vertical alignment. Australia has specification that the shape of vertical curves should be based on quadratic parabola but the length of a given vertical curve is calculated based on circular curve. [5]
However, if both ends of the curve (P 1 and P 2) are at the same level (y 1 = y 2), it can be shown that [58] = where L is the total length of the curve between P 1 and P 2 and h is the sag (vertical distance between P 1, P 2 and the vertex of the curve).
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.
Other lengths may be used—such as 100 metres (330 ft) where SI is favoured or a shorter length for sharper curves. 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.
Rankine's 1862 "Civil Engineering" [2] 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 ...