**6. Coordinate Calculations**

**6.1. Co-ordinate calculations**

Calculations involving rectangular co-ordinates are:

a) Calculation of Whole Circle Bearing (WCB) and distance between two co-ordinated points.

i) Calculate the difference in easting( E) and difference in northings ( N) between the two points.

ii) Determine the Reduced Bearing from:

E

Reduced bearing= tan ------

N

Iii) Convert the reduced bearing to a Whole Circle Bearing.

1st quadrant same

2nd quadrant 180 – rb

3rd quadrant 180 + rb

4th quadrant 360 – rb

The correct quadrant can be found by inspection of the relative positions of the co-ordinated points.

iii)Calculate the horizontal distance from:

E N

Distance = -------------- and --------------

Sin WCB Cos WCB

Using both these formulae separately gives a check on the calculation of the WCB

A further formulae could be used to determine distance but this does no check WCB calculation.

Distance = E + N

Example calculation

Given: Stn A 179.724 mE 414.132 mN

Stn B 142.171 mE 372.916 mN

A 179.724 414.132

B 142.171 372.916

------------ -----------

E 37.553 N 41.216

Reduced bearing = tan –1

__37.553__

41.216

= 42.2015

WCB= 180 + 42 20 15

= 222 20 15

Distance

__= 37.553__= -55.758 m

Sin 222 20 15

=

__41,216__= -55.758 m

cos 222 20 15

The two distances should be the same and of course, the negative sign is ignored.

a) Calculation of rectangular co-ordinates of a point (B) knowing the co-ordinates of another point (A), the WCB from that point to B (A to B) and the horizontal distance between them.

The procedure is as follows:

i) Calculate the difference in eastings and northings using the following formulae:

E = distance x sin WCB

N = distance x cos WCB

ii) Add E and N to the eastings and northings respectively of the known point.

Example- calculation

Given: Co-ords A 137.629 mE 473.126 mN

WCB A to B 136 27 19

Distance A to B 53.249

E = 53.249 x sin 136 27 19

= 36.684 m

N = 53.249 x cos 136 27 19

= 38.597 m

Co-ords A 137.629 473.126

Difference 36.684 -38.597

------------ --------------

Co-ords B 174.313E 434.529N

-------------- --------------

**6.2. Traverse calculation**

The commonest form of traverse is the ring traverse, which starts and finishes at the same point, and is calculated as follows:

a)The following data is required to compute the traverse.

i) All internal horizontal angles

ii) All horizontal distances

iii) The WCB of one leg.

iv) The rectangular co-ordinates of one point.

i)and (ii) measured in the field

(iii) Usually an assumed bearing but could be magnetic, grid or true.

(vi) Usually an assumed value unless the traverse is to be tied to the Ordnance survey horizontal control or similar.

b)Distribute angular misclose equally to each measured angle.

c)Calculate the forward bearing of each line, working anticlockwise around the traverse, using:

Forward bearing = back bearing of previous line + included angle

Ensure you finish back on the original bearing as a check against error.

d)Calculate E and N for each leg using the forward bearing and measured distance.

e) Add up all E’s and N’s to check for large errors in observation or calculation. The sums should be close to zero.

f)Starting at the origin station, compute the co-ordinates of each station around the traverse (working anticlockwise) finishing back at the origin. Any variation in the co-ordinates of this station is the eastings and northings misclose.

g)Distribute the misclosures using either the Bowditch or Transit methods.

Bowditch correction =

__Distance from origin x E (or N) misclose__

Traverse length

Transit correction (for eastings) =

__E from origin x easting misclose__

E

Note that all E are treated as positive when calculating corrections.

h) Calculate corrected co-ordinates.

An example traverse calculation is tabulated in Figure 4.10.

**6.3 Co-ordinate setting out**

Co-ordinated points can be set out from a horizontal control by either:

a) Using bearing and distance.

b) By intersecting theodolite rays

The following notes detail the general approach.

a) Bearing and distance method

i) Two control stations are required. Calculate the WCB and distance from each station to the point to be set out and the WCB between the two stations.

ii) Set a theodolite over one control station and sight the other station (RO) with the computed WCB between the two stations.

iii) Release the theodolite upper plate and rotate the instrument until the computed WCB to the point is obtained.

iv) Mark the direction with a nail in a peg at approximately he correct distance away.

v) Change face and repeat to check for instrument error.

vi) Measure the distance to the peg at the trial position applying whatever corrections are deemed necessary (see paragraph-4). Using a pocket tape, position a new peg at the correct position from the trial position.

vii) Check by using a WCB or distance from another station.

**a) Intersection by theodolite**

i) Calculate WCB and distance as in (a) (i)

ii) Two theodolites are required one set up over each control station.

iii) Each theodolite is sighted on to the opposite station with the computed bearing on the lower plate.

iv) Each theodolite is then turned until the required WCB to the point is found. Establish a peg on line and change face to check.

v) The required point is at the intersection of the two theoidolite lines

vi) The set out position must be checked by either a distance measured from a station or a WCB from a third station.

vii) This method should only be used if the resultant triangle formed by the two control stations and the set out point is well onditioned.

**(b) Other checks**

Once all the points are set out from the control, it is essential that they are checked relative to each other. Some slight adjustment to peg position may be required.