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Measuring horizontal angles
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Horizontal angles between any two points or stations can measured by using Theodolite. While Surveying random errors occur quite frequently, so certain methods and corrections are applied to minimize these errors to adjust the values of angles. Here we will only discuss different methods which are employed to obtain the observed value of quantity. The three common methods used for calculating horizontal angles are numbered below:
- Ordinary Method
- Repetition Method
- Retrietion Method
Ordinary method can be understood by taking a simple example. Suppose we have to measure horizontal angle between points A and B from station C (both the points are visible from station C and point B is placed on right of A when viewed by the observer in the plane of paper), now the theodolite is set at C with all its required adjustments. We know that theodolite has two scales viz.
- Main Scale, with least count of 20 minutes
- Vernier Scale with least count of 20 seconds. Also vernier A and Vernier B is used for measuring Horizontal angles while C and D are used for measuring vertical angles.
Now set the vernier A at zero (coinciding with the zero of main scale) and vernier B at 180° by opening both tangent screws. Once the vernier A is set at zero close the upper tangent screw, and rotate the telescope so that the line of sight cuts A. Now close the lower tangent screw and open the upper tangent screw. since the line of sight is at A, rotate the telescope until the line of sight cuts B. Note that when upper tangent screw is open and lower tangent screw is closed, theodolite starts reading the angle but when upper tangent screw is closed and lower tangent screw is open reading of the theodolite does not change rather it can be rotated in any direction. Now take the reading directly from horizontal circle with the help of graduated main and vernier scales. The angle so obtained is horizontal angle by ordinary method. In ordinary method horizontal angle is measured only once (one with face left and other with face right).
In this method, the angle between the two points A and B is measured repeatedly commonly three times and the arthemetic mean of three readings gives the most probable value (MVP) of the angle. Here we proceed as follows :
- Set the theodolite at station C.
- Make temporary and permanent adjustments of theodolite.
- Firstly measure the angle between A and B same as in ordinary method. Suppose the horizontal angle between A and B is 35'.
- Now after measuring the angle, close the upper tangent screw and open the lower tangent screw. Rotate the theodolite from B to A in the clockwise direction. Closure of upper tangent screw will not allow reading of theodolite to change. The reading of theodolite will still be 35' at A. Now close the lower tangent screw and open the upper tangent screw so that theodolite starts reading the angle again.
- Rotate the theodolite until line of sight cuts B. The total angle measurad by theodolite will be some what double than previous reading. Let the total angle be 71'.
- Repeat the step 4 for taking another reading. The last reading of theodolite will be approximately thrice than the ist one. Let it be 107'. So by measuring same angle three times the total angle is 107'. Most probable value of the angle is equal to total sum divided by number of observations.
- Most probable value= Total sum/number of observations
- In this case MPV= 107/3
This method can be summarised by taking an example. Firstly it should be noted that this method is quite different from the first two methods. Let the station O be at the centre of circle and points A, B, C and D are somewhere on the circumference of this circle. Let the circle be in horizontal plane on the ground. We have to determine horizontal angles between A and B, B and C, C and D, D and A. Which implies that we have to determine angles AOB, BOC, COD and DOA. The sum of all these angles should be equal to 360' because they are closing the horizon. Here we proceed as follows :
- Set the theodolite at O and level it accurately.
- Set the Vernier A at zero coinciding with the zero of main scale and B at 180' same as in ordinary method.
- Now close the upper tangent screw and rotate the telescope until the line of sight cuts A. Now close the lower tangent screw and open the upper tangent screw.
- Assume we are rotating the theodolite in clockwise direction.
- Rotate the theodolite until line of sight cuts B. The reading of the theodolite will give angle AOB. Rotate the theodolite further until the line of sight cuts C. The angle so obtained from theodolite is AOC. Therefore angle BOC= angle AOC minus angle AOB.
- Similarly rotate the theodolite until line of sight cuts D and then A again.
- angle COD= angle AOD minus angle AOC and angle DOA=angle AOA' (observed value) minus angle AOD.
- Note if error is zero A will coincide with A'.
This method is known as Retrietion Method in which only one station is used and angles are obtained by closing the horizon. If some random errors occur, they can be eliminated by Normal Equation method or by Method of differences.