1. Theodolite — Parts & Temporary Adjustments

A transit theodolite is the standard instrument for measuring horizontal and vertical angles in civil engineering. It has two main circles: a horizontal circle (graduated 0°–360°) for measuring horizontal angles, and a vertical circle (graduated 0°–360° or 0°–90° each quadrant) for measuring vertical angles.

Key parts: Lower plate (horizontal circle, fixed during measurement), upper plate (vernier, rotates for reading), altitude bubble (levelling the vertical circle), plate bubble (levelling the horizontal plate), diaphragm (cross-hairs for sighting), focusing screw, clamp and tangent screws for both plates.

Temporary adjustments (done at each setup): (1) Centering over the station — using a plumb bob or optical plummet. (2) Levelling — using plate level bubble with foot screws (three-screw levelling). (3) Elimination of parallax — focus the eyepiece on the cross-hairs, then focus the objective on the target.

Permanent adjustments (in workshop): adjustment of plate level, cross-hair ring, collimation in azimuth, spire test (trunnion axis), altitude level, vertical arc index.

2. Angle Measurement Methods

Face Left (FL) + Face Right (FR): The mean of FL and FR readings eliminates collimation error, eccentricity of centres, and graduation errors. For a horizontal angle: Mean = (FL + FR)/2, but account for the 180° difference — FR reading = FL reading ± 180° theoretically. Discrepancy indicates collimation error.

3. Traverse — Types & Introduction

A traverse is a series of connected survey lines whose lengths and directions are measured. It is the most common method for establishing horizontal control.

• Closed traverse: Starts and ends at the same point (loop traverse) or at two known points (connecting traverse). Can be checked mathematically.
• Open traverse: Starts at one point and ends at a different, unknown point. No mathematical check possible — used only for preliminary surveys or where closure is impractical.
• Chain traverse: Bearings measured by compass; distances by chain. Lower accuracy.
• Theodolite traverse: Angles measured by theodolite; distances by tape or EDM. High accuracy. Gale’s traverse table is used for computation.

4. Latitude, Departure & Closing Error

5. Traverse Adjustment — Bowditch & Transit Rule

Least squares adjustment is the rigorous method — minimises the sum of squares of corrections. Used for high-precision control surveys. Not typically required for GATE but conceptually important.

6. Coordinate System & Gale’s Table

After adjusting the traverse, independent coordinates of each station are computed. Starting from the first station with assigned coordinates (often 0,0 or an arbitrary origin), successive coordinates are computed as:

Gale’s traverse table organises all traverse computations in one table: bearing, length, latitude, departure, corrections, corrected latitude/departure, and accumulated coordinates. The final row must return to the starting coordinates — providing the closure check.

Area of a closed traverse can be calculated from coordinates using the coordinate method (Gauss’s shoelace formula): Area = ½|Σ(E_n·N_n+1 − E_n+1·N_n)|

7. Worked Examples (GATE CE Level)

Example 1 — Traverse Closing Error & Accuracy

A closed traverse ABCDE has ΣLatitude = −0.082 m and ΣDeparture = +0.064 m. The perimeter is 850 m. Find the closing error and accuracy.

Solution:

e = √(0.082² + 0.064²) = √(0.006724 + 0.004096) = √0.01082 = 0.1040 m

Accuracy = 850 / 0.1040 = 8173 → 1 in 8173 (first-order work requires 1:10,000, so this is close)

Example 2 — Bowditch Rule Correction (GATE 2018 type)

Side AB has length 120 m in a traverse with perimeter 600 m. e_L = 0.15 m (excess northing), e_D = 0.10 m (excess easting). Find Bowditch corrections to L and D of AB.

Solution:

Corr to L_AB = −(0.15 × 120/600) = −0.030 m (reduce northing of AB by 0.030 m)

Corr to D_AB = −(0.10 × 120/600) = −0.020 m (reduce easting of AB by 0.020 m)

Example 3 — Angular Misclosure Distribution

A 5-sided closed traverse has measured interior angles: 95°12′, 128°30′, 97°45′, 115°22′, 103°14′. Find the angular misclosure and corrected angles.

Solution:

Sum of measured angles = 95°12′ + 128°30′ + 97°45′ + 115°22′ + 103°14′ = 540°03′

Theoretical sum = (5−2)×180° = 540°00′

Misclosure = +3′ (excess)

Correction per angle = −3’/5 = −0’36” per angle

Corrected angles: 95°11’24”, 128°29’24”, 97°44’24”, 115°21’24”, 103°13’24”. Sum = 540°00′ ✓

Common Mistakes

• Not taking FL and FR readings: A single face reading retains all instrumental errors — always observe both faces and average.
• Using WCB directly in latitude formula without quadrant check: With WCB, latitude = d·cosθ will give correct sign automatically (cos is positive for 0–90° and 270–360° range, negative for 90–270°). However, departure = d·sinθ: sin is positive for 0–180°, negative for 180–360°. No quadrant adjustment needed if you are careful with the calculator, but many students get the sign wrong.
• Choosing the wrong adjustment rule: Bowditch for chain traverses (equal precision); Transit for theodolite traverses (better angular precision).
• Forgetting to correct angles before computing latitudes/departures: First distribute the angular misclosure, recompute bearings, then calculate L and D.
• Applying the closing error in the same direction instead of opposite: The correction to each side’s L and D must be opposite in sign to e_L and e_D.

Frequently Asked Questions