To enable distance data to be used on plans it is necessary to ensure that all measurements on a variety of slopes are converted into true horizontal distance. This data can then be used to calculate levels and co-ordinates at given points.

Table 5 shows the slope distances recorded during the survey

Table 5

True Distance = Cosine of Mean Vertical Angle x Slope Distance

Example @ AD/DA : Cos 4 degrees 30 mins x 39.522 = 39.400

Levels

If we assume that Station A has a height above Ordnance Datum of 100.000m then the levels at each of the other stations can be determined using the following data:-

Station A to Station B : True distance = 59.600

Mean Vertical Angle = Up 0 05 40

Station B to Station C : True distance =36.001

Mean Vertical Angle = Down 4 36 20

Station C to Station D : True distance = 43.601

Mean Vertical Angle = Down 0 23 35

Station D to Station A : True distance = 39.400

Mean Vertical Angle = Up 4 30 00

Note the check slope measurement AC was 67.858m

It is now possible to calculate the difference in level between each of these points using :-

Tan (Mean Vertical Angle) x True Distance

Therefore differences are:-

A to B Tan 0 05 40 x 59.600 = Plus 0.098m

B to C Tan 4 36 20 x 36.001 = Minus 2.900m

C to D Tan 0 23 35 x 43.601 = Minus 0.299m

D to A Tan 4 30 00 x 39.400 = Plus 3.101m

The levels at each Station can now be calculated:-

Station A (100.000) Station B 100.000 + 0.098 = 100.098

Back to Station A = 96.899 + 3.101= 100.000

**Co-ordinates**

Pegs have been placed at Stations A to D. For these to be used for setting out, the co-ordinates of each must be determined. We are assuming that Station A has the co-ordinates of 100.000 East and 100.000 North. (see Figure 6)

The remaining coordinates are calculated using the following procedure:

- Calculate the whole circle bearing (WCB) from Station A
- The start point for WCBs is the true north bearing
- Using calculation we determine clockwise angle from north
- We also calculate the true distance from Station A to each of the other points
- From this data it is then possible to determine eastings and northings for each main station

Figure 6

Whole Circle Bearings (WCB)

Table 6

Calculations (using Fig 6 and Table 6)

Data known

Line AB is 86 30 00 clockwise from north

True Distance AB = 59.600

True Distance BC = 36.001

True Distance AC = 67.800

Angle ABC = 86 38 27.5

Refer to Figure 7

Calculate Angle BAC using sine rule (a/ Sin A = b/ Sin B c/ Sin C)

Therefore Sin Angle BAC = (Sin 86 38 27.5 x 36.001) / 67.800 = 0.530075953

Using Shift Sin on your calculator - Angle BAC = 32 ( to nearest second)

Figure 7

Therefore:

WCB to Station C is 86 30 00 + 32 00 38 = 118 30 38

WCB to Station D is 86 30 00 + 69 20 17.5 = 155 50 17.5

We now have all the data to calculate the co-ordinates

Co-ordinates at A is 100.000 East and 100.000 North

To navigate to Station B we calculate distances east and north from A to B (see Figure 8)

Figure 8

Distance East from A = Sin 86 30 00 x 59.600 = 59 489m

added East and North co-ordinates at Station A to give Station B co-ordinates (159.489 East , 103.638 North)

Figure 9

118 30 38 Distance AC = 67.800

WCB minus 90 gives the angle shown above (28 30 38)

Distance East from A = Cos 28 30 38 x 67.800 = 59.578

Distance South from A = Sin 28 30 38 x 67.800 = 32.362

Co-ordinates at Station C are 159.578 East (100 + 59.578), 67.638 North (100 – 32.362). Remember that values are deducted when moving South or West.