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A standard chord length is used: in the UK this is conventionally 30 metres, or sometimes 20 metres. Half chords, i.e. 15 metre or 10 metre intervals, are marked on the datum rail using chalk. The string, which is one full chord long, is then held taut with one end on two marks at each end of a chord, and the offset at the half chord mark measured.
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 method or tangential angle method is an angular technique for laying out circular curves by a combination of chaining and angles at circumference, fully exploiting the theodolite and making a substantial improvement in accuracy and productivity over existing methods. This method requires access to only one road/path of communication ...
The actual equation given in Rankine is that of a cubic curve, which is a polynomial curve of degree 3, at the time also known as a cubic parabola. In the UK, only from 1845, when legislation and land costs began to constrain the laying out of rail routes and tighter curves were necessary, were the principles beginning to be applied in practice.
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) = (+)
Important quantities in the Whewell equation. The Whewell equation of a plane curve is an equation that relates the tangential angle (φ) with arc length (s), where the tangential angle is the angle between the tangent to the curve at some point and the x-axis, and the arc length is the distance along the curve from a fixed point.
For a rectifiable curve these approximations don't get arbitrarily large (so the curve has a finite length). If a curve can be parameterized as an injective and continuously differentiable function (i.e., the derivative is a continuous function) f : [ a , b ] → R n {\displaystyle f\colon [a,b]\to \mathbb {R} ^{n}} , then the curve is ...
Free-space diagram of the red and the blue curve. In contrast to the definition in the text, which uses the parameter interval [0,1] for both curves, the curves are parameterized by arc length in this example. An important tool for calculating the Fréchet distance of two curves is the free-space diagram, which was introduced by Alt and Godau. [4]