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Division Calculation of the mean plane

United Kingdom

Original text. Later amendments are not shown.

In the plane of the path, the topography (including terrain, mounds, embankments and other man-made obstacles, buildings, …) may be described by an ordered set of discrete points (xk, Hk ); k є {1,…,n}. This set of points defines a polyline, or equivalently, a sequence of straight segments Hk = akx + bk, x є [xk, xk + 1 ]; k є {1,…n}, where:

ak = (Hk + 1 – Hk )/(xk + 1 – xk )(2.5.2)
bk = (Hk · xk + 1 – Hk + 1 · xk )/(xk + 1 – xk )

The mean plane is represented by the straight line Z = ax + b; x є [x 1, xn ], which is adjusted to the polyline by means of a least-square approximation. The equation of the mean line can be worked out analytically.

Using:

A = 2 3 ∑ k = 1 n − 1 a k x 3 k + 1 − x 3 k + ∑ k = 1 n − 1 b k x 2 k + 1 − x 2 k (2.5.3)
B = ∑ k = 1 n − 1 a k x 2 k + 1 − x 2 k + 2 ∑ k = 1 n − 1 b k x k + 1 − x k

The coefficients of the straight line are given by:

a = 3 2 A − B x n + x 1 x n − x 1 3 (2.5.4)
b = 2 x 3 n − x 3 1 x n − x 1 4 B − 3 x n + x 1 x n − x 1 3 A

Where segments with xk + 1 = xk shall be ignored when evaluating eq. 2.5.3.