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2. Traverse with curves

a. Segments and sectors

Area by coordinates is constrained to following straight lines. Where curves are involved, Figure H-3(a), the method returns an area bounded by their chords, Figure H-3(b).

(a) Traverse

(b) Area by coordinates

Figure H-3
Traverse with curves

To obtain the correct area, sector or segment areas are computed and added or subtracted accordingly. For the traverse in Figure H-3(a), the two segment areas are computed, Figure H-4(a),  and one subtracted from, the other added to, the area by coordinates, Figure H-4(b).

(a) Segments

(b) ATotal=A1-A2+A3

Figure H-4
Accounting for segments

b. Example area computation

Given the traverse in Figure H-5, determine its area.

Figure H-5
Traverse with curves

In the table below, the points and their coordinates are in the first three columns; their cross-products are in the last two.

Point North (ft) East (ft)
1 500.00 1200.00   737,634
2 614.70 1249.70 624,850. 843,243
3 674.76 1347.43 828,261 892,620.
4 662.46 1552.76 1,047,730 725,778
5 467.41 1516.87 1,004,860 633,075
6 417.36 1317.39 615,766 658,696
1 500.00 1200.00 500,828  
    sums: 4,491,046 4,622,300.

The area is

This is the area bounded by the chords. We must account for the segments.

Recall that the area of a segment, bounded by an arc and its chord, is

Equation H-2

Compute the area of each segment

Arc 2-3

Arc 4-5

Arc 6-1

Add or subtract the segments:

Total traverse area is 68,790 sq ft.

c. Segment or sector?

Consider the curvinlinear traverse in Figure H-5:

Figure H-5
Curvilinear traverse

Its area can be computed using the arc sector, Figure H-6,  

(a) Area to radial lines, AR

(b) Sector area, ASec

Figure H-6 Area = AR + ASec

or using the arc segment, Figure H-7

(a) Area to chord, AC

(b) Segment area, ASeg

Figure H-7
Area = AC + ASeg

Either method is fine. The only difference is that sectors require computing arc radius points, Figure H-6(a); segments do not, Figure H-7(a).

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