Surveying I– coordinate account

Page takes part in the competition "Notes on the Internet" organized by the Rector of Academy of Science and Technology in Cracow.
V o t e !

Back>>

Analytical method of calculating the surface area of the Gauss's formulae

The analytical method is based on points with known coordinates or measures of angular and linear measured directly in the field. To determine the surface area of the Gauss's formulae we need to know the coordinates of the collapse of the outline.

Figure 1. The siting of points 1-2-3-4-5 in the collapse of a contour in coordinate system

Search:
P


Data:
1 (X₁, Y₁)
2 (X₂, Y₂)
3 (X₃, Y₃)
4 (X₄, Y₄)
5 (X₅, Y₅)

Field P polygon 1-2-3-4-5 we calculate one of the Gauss's formula (second formula provides control):

–2P = ∑₁ⁿ (X_i+1 – X_i-1) Y_i
2P = ∑₁ⁿ (Y_i+1 – Y_i-1) X_i

where:
n – number of contour points of collapse,
i – the number of point.

An additional control is to determine whether:
∑₁ⁿ (X_i+1 – X_i-1) = 0
∑₁ⁿ (Y_i+1 – Y_i-1) = 0

Field measurements are given in hectares, remembering that 1ha = 100a = 10 000m².

The following is a table that facilitates the calculation of surface area.

Example

Calculate the surface area of a polygon 1-2-3-4 with the collapse of a contour point coordinates.

Data:
1 (2m; 1m)
2 (5m; 2m)
3 (7m; 5m)
4 (4m; 7m)

SOLUTION

Point number	Xi	Yi	Xi+1 – Xi-1	Yi+1 – Yi-1
1	2	1	1	–5
2	5	2	5	4
3	7	5	–1	5
4	4	7	–5	–4
1	2	1	∑ = 0	∑ = 0

–2P = ∑₁ⁿ (X_i+1 – X_i-1)Y_i = 1•1 + 5•2 + (–1)•5 + (–5)•7 = -29m²
P = 14,5m² = 0,00145ha ≈ 0,0014ha


Control
2P = ∑₁ⁿ (Y_i+1 – Y_i-1)X_i = (–5)•2 + 4•5 + 5•7 + (–4)•4 = 29m²
P = 14,5m² = 0,00145ha ≈ 0,0014ha