Uses of Theodolite

Table of Contents

The use of theodolites is essential in various surveying tasks due to their precision in measuring angles and distances. These instruments are crucial for both horizontal and vertical measurements, providing accurate data for engineering, construction, and land surveying. The use of theodolites extends beyond simple angle measurement, allowing surveyors to determine magnetic bearings, lay off angles, prolong straight lines, and run direct and deflection angles with high precision. Their versatility makes them indispensable for many field operations requiring accuracy and reliability.

Theodolites serve a variety of purposes in surveying, such as:

Measurement of Horizontal Angles
Measurement of Vertical Angles
Determining the Magnetic Bearing of a Line
Direct Angle Measurements
Deflection Angle Measurements
Prolonging a Straight Line
Establishing a Straight Line Between Two Points
Laying Off Angles Using the Repetition Method

1. Measurement of Horizontal Angles

i. Direct Method of Measuring the Angle

To measure a horizontal angle, say ABC, formed by lines BA and BC, follow these steps:

Set up and Level the Theodolite: Position the theodolite over point B, centering and leveling it properly.
Initial Reading: Loosen the upper plate, set the vernier to zero, and clamp the upper plate. Loosen the lower plate and rotate the telescope until point A is aligned. Secure the lower clamp and fine-tune the bisection using the lower tangent screw. Record the readings from both verniers and take the average.
Sight the Second Point: Unclamp the upper plate and rotate the telescope clockwise to align with point C. Clamp the upper plate and accurately bisect point C using the upper tangent screw.
Record the Readings: Take the vernier readings for C and A, then calculate the difference, which gives the required angle ABC.
Repeat on the Opposite Face: Reverse the face of the instrument and repeat the procedure to get another measurement of angle ABC.
Calculate the Final Angle: The mean of the two measurements taken from both faces gives the final value of the angle ABC.

Notes

Observations on both faces of the instrument help eliminate errors due to imperfect adjustments.
Reading both verniers reduces errors caused by eccentricity in the circle or verniers.

ii. Measurement of Horizontal Angles by the Method of Repetition

When precise angular measurements are needed, especially for small angles, the method of repetition is often employed. This approach minimizes errors by accumulating the angle multiple times and then averaging the results to enhance accuracy. It is particularly useful when small errors in sine values could significantly affect the computed distances.

Procedure:

To measure a small horizontal angle, ABC, using the repetition method:

Setup and Initial Sighting:
- Position and level the theodolite over point B, with the instrument’s face left.
- Set the vernier to zero and swing the telescope to sight point A. Accurately bisect A using the lower tangent screw.
First Measurement:
- Record the readings from both verniers and calculate the mean.
- Rotate the telescope clockwise to sight point C, using the upper tangent screw for precise alignment.
- Record the mean vernier readings again. The difference between these readings provides an initial approximation of angle ABC.
Repetition Process:
- Unclamp the lower plate, swing the telescope back to point A, and bisect it without altering the vernier readings.
- Rotate clockwise once more to sight C, doubling the angle measurement. Repeat this process until the angle has been measured multiple times (e.g., 5 repetitions).
Final Calculation:
- After repeating the measurement, the accumulated angle should equal the number of repetitions multiplied by the approximate angle.
- Divide this accumulated value by the number of repetitions to find the correct value of angle ABC.
Face Right Observation:
- Change the instrument’s face to the right and repeat the entire procedure to ensure precision.
Final Angle:
- The mean of the values obtained on both face left and face right gives the required angle ABC.

Notes

Error Reduction: This method reduces several errors, such as those from eccentricity, imperfect adjustments, pointing inaccuracies, and circle graduations, making the measurement more precise.
Enhanced Accuracy: By measuring the angle repeatedly, it is possible to achieve greater accuracy than the least count of the vernier would otherwise allow.
Common Applications: The method of repetition is especially valuable in subtense bar measurements and triangulation base extensions, where small angles can significantly impact the accuracy of computed distances.

2. Measurement of Vertical Angles

A vertical angle is the angle formed between the horizontal line of sight and the inclined line of sight, observed in a vertical plane at a particular station. If the point being sighted lies above the horizontal axis of the theodolite, it is referred to as an angle of elevation. Conversely, if the point is below the horizontal axis, it is termed an angle of depression.

The measurement of vertical angles is crucial in various surveying applications, including determining elevation differences and calculating heights and slopes.

Procedure to Measure a Vertical Angle:

To measure the vertical angle subtended by station B at the instrument station A, follow these steps:

Set up the theodolite: Position and level the theodolite accurately over the ground station mark A using the altitude bubble.
Align the vertical scale:
- Set the zero of the vertical vernier to align exactly with the zero of the vertical scale.
- Use the vertical clamp and vertical tangent screw to make this adjustment.
- Ensure that the altitude level bubble is centered. If not, use the clip screw to adjust it. In this position, the line of collimation of the telescope is horizontal, and the verniers should read zero.
Sight the station:
- Loosen the vertical circle clamp and rotate the telescope vertically until the station B appears in the field of view.
- Use the vertical tangent screw for fine adjustments and accurate bisection of the point.
Record the angle:
- Read both verniers of the vertical circle, and take the mean of these readings. This provides the value of the vertical angle.
Change the face:
- Flip the instrument to face right and repeat the measurement process in a similar manner.
Average the measurements:
- The average of the vertical angle measurements from both faces (left and right) gives the final value of the vertical angle.

3. Determining the Magnetic Bearing of a Line

To determine the magnetic bearing of a line AB, a theodolite equipped with a circular or trough compass is required. The following steps outline the process:

Center and level the theodolite:
- Set up and level the theodolite accurately over station A.
Set the vernier:
- Adjust the vernier to read zero.
Release the magnetic needle:
- Loosen the lower plate and release the magnetic needle to allow it to move freely.
Align with magnetic north:
- Rotate the telescope around its vertical axis until the magnetic needle aligns with the N-S graduations on the compass box scale.
Clamp and adjust the needle:
- Tighten the lower plate, and use the lower tangent screw to bring the north end of the magnetic needle into exact alignment with the zero graduation of the scale.
Orient to the magnetic meridian:
- At this point, the line of collimation of the telescope is aligned with the magnetic meridian, while the vernier still reads zero. The theodolite is now oriented on the magnetic meridian.
Sight the target:
- Loosen the upper plate, rotate the instrument, and use the upper tangent screw to accurately sight station B.
Record the bearing:
- Read both verniers and take the mean of the two readings to obtain the magnetic bearing of line AB.
Reverse the instrument:
- Change the face of the theodolite (move to the opposite face) and repeat the process to observe the magnetic bearing from the other side.
Calculate the mean:
The mean of the magnetic bearings observed on both faces provides the accurate value of the magnetic bearing of the line AB.

4. Measurement of Direct Angles

A direct angle is the angle measured in a clockwise direction from the preceding line to the following line. This angle, sometimes referred to as the azimuth from the back line or the angle to the right, typically ranges between 0° and 360°. Below is the procedure for measuring a direct angle, such as BCD.

Procedure

Set up the theodolite:
- Position the theodolite over station C and ensure it is centered and leveled properly.
Set the vernier to zero:
- With the face of the instrument to the left, adjust the vernier to read 0° by turning the upper plate.
Bisect the preceding station:
- Loosen the lower clamp and use the lower tangent screw to accurately bisect station B. Tighten the lower plate.
Bisect the forward station:
- Unclamp the upper plate, swing the telescope clockwise, and bisect the forward station D. Read both verniers.
Reverse the instrument:
- Plunge the telescope, loosen the lower clamp, and once again bisect the preceding station B without altering the vernier readings.
Repeat the bisection:
- Unclamp the upper plate, swing the telescope clockwise again, and accurately bisect station D. Read both verniers once more.
Calculate the angle:
- Take the mean of the final vernier readings. The angle will now be doubled, so divide the value by two to obtain the direct angle BCD.

Notes

Direct angles should generally be measured on both faces of the instrument to eliminate instrumental errors.
Two measurements of the angle can be obtained by calculating the difference between:
- The first and second readings.
- The final and second readings.
The average of the two measurements gives the most accurate value of the direct angle.
Direct angles are commonly used in a theodolite traverse for precise surveying.

5. Measurement of Deflection Angles

A deflection angle is the angle formed between any survey line and the prolongation of the preceding line. These angles can range from 0° to 180° and are classified as right deflection angles when measured clockwise, and left deflection angles when measured anticlockwise. For instance, in the figure below, angles α and δ at stations B and E are left deflection angles, whereas angles β and γ at stations C and D are right deflection angles.

Procedure to Measure Deflection Angle (α at Station B):

Set up the instrument:
- Position the theodolite accurately over station B, ensuring it is centered and leveled.
Set the verniers to zero:
- Adjust the verniers to read 0° and take a back sight on station A. Tighten both the lower and upper plates.
Transit the telescope:
- Rotate the telescope so that the line of sight is in the direction of the prolonged line AB. The verniers should still read zero.
Swing to the forward station:
- Unclamp the upper plate and rotate the telescope anticlockwise to sight station C. Read both verniers.
Return to the back station:
- Loosen the lower plate and swing the telescope to sight station A again. Ensure that the vernier readings have not changed. Tighten the lower plate and transit the telescope.
Repeat the observation:
- Loosen the upper clamp and swing the telescope anticlockwise again to sight station C. Bisect the station mark C accurately using the upper tangent screw, and read both verniers.
Calculate the deflection angle:
- Since observations are taken on both faces, the deflection angle is doubled. Half of the final reading gives the required deflection angle at B.

Notes

- After plunging the telescope and sighting the forward station:
  - If the vernier reading is less than 180°, the deflection angle is left.
  - If the reading is more than 180°, the deflection angle is right.
- Deflection angles are commonly used in theodolite traverses conducted along linear paths, such as for the alignment of highways, railways, and cana

6. Prolongation of a Straight Line

The prolongation of a straight line refers to extending a line beyond a given point. In surveying, this can be done using a theodolite to ensure accuracy over long distances. Here is the procedure for prolonging a straight line AB to a point F using the first method:

Procedure:

Set up the theodolite:
- Position the theodolite accurately at point A, ensuring it is centered and leveled.
Bisect point B:
- Using the theodolite, sight and bisect an arrow that is centered over the mark at point B.
Establish point C:
- Sight through the telescope and establish point C in line with AB at a convenient distance from point B.
Shift the instrument to point B:
- Move the theodolite to point B, center and level it accurately.
Sight point C:
- Using the theodolite at point B, sight point C and establish another point D further along the line.
Repeat the process:
- Continue this process by shifting the instrument to each newly established point (C, D, etc.) until the desired point F is reached.

7. Laying Off Angles Using the Repetition Method

The repetition method is employed to lay off angles with greater precision, especially when high accuracy is required. Below is the procedure for laying off an angle ABC using this method with a theodolite having a 20″ least count, and the desired angle ABC = 30° 40′ 13″.

Procedure:

Set up the theodolite:
- Position and center the theodolite accurately at station B, and level it carefully.
Sight station A:
- Aim the telescope at point A and adjust the vernier to read zero.
Set the angle:
- Unclamp the upper plate, swing the telescope until the reading is approximately equal to the required angle (30° 40′ 13″). Fine-tune the angle using the upper tangent screw to exactly 30° 40′ 20″.
Fix point C':
- Mark point C' at a convenient distance in the line of sight, creating the approximate desired angle.
Measure the angle by repetition:
- Use the repetition method to measure the angle ABC'. After five repetitions, if the accumulated angle ABC' equals 153° 22′ 20″, calculate the average:
- Average angle ABC = (153° 22′ 20″) / 5 = 30° 40′ 28″
Determine the correction:
- Calculate the correction needed by comparing the measured angle to the desired angle:
- Correction = 30° 40′ 28″ − 30° 40′ 13″ = 15″
Apply the correction:
- Since the correction is small, apply it linearly by calculating the offset C'C using the formula:
- C'C = BC' × tan(angle correction) = 250 × tan(15″) = 0.018 m
Establish the final point C:
- Measure BC', which is 250 m, and then set a perpendicular from C' to line BC'. Measure the offset C'C = 0.018 m and mark the correct location of point C.
Verify accuracy:
- Re-measure the angle ABC using the repetition method to verify its accuracy. Apply further corrections if necessary.