Theodolite — Types, Parts, Uses, Traversing, and Angular Measurement in Surveying

The theodolite is the most versatile angular measurement instrument in surveying. It measures both horizontal and vertical angles with precision, making it essential for triangulation, traversing, setting out, and any survey where angular control is needed. Modern total stations have replaced the traditional theodolite in most applications, but understanding the theodolite remains fundamental for civil engineering examinations and for understanding total station principles.

Types of Theodolites

Parts of a Transit Theodolite

• Telescope: Magnification 20–40×; cross-hairs for target sighting; objective and eyepiece lenses
• Horizontal Circle: Graduated plate (0°–360°); lower plate (fixed) and upper plate (rotatable)
• Vertical Circle: Attached to trunnion axis; measures vertical angles
• Verniers: A and B verniers at 180° for accurate angle reading and eliminating eccentricity
• Standards (A-Frame): Support the trunnion axis (horizontal axis of telescope rotation)
• Levelling Head: Three foot screws on trivet stage; levelling the instrument
• Upper Clamp and Tangent Screw: Fine angular adjustment for upper plate
• Lower Clamp and Tangent Screw: Fix and fine-adjust lower plate (graduated circle)
• Plate Bubble: Two plate levels for horizontal levelling
• Altitude Bubble: For accurate vertical circle reading

Axes and Lines of a Theodolite

• Vertical Axis (VV'): Axis of rotation of instrument; must be truly vertical
• Trunnion / Horizontal Axis (HH'): Axis about which telescope rotates vertically; must be horizontal and perpendicular to VV'
• Line of Collimation: Line through the optical centre and intersection of cross-hairs; must be perpendicular to HH'
• Axis of Plate Bubble: Must lie in horizontal plane when bubble is centred

Temporary Adjustments

Done before every observation at a new instrument station:

1. Setting up over station: Optical plummet or plumb bob to centre over ground mark
2. Levelling up: Using foot screws; bring plate bubble to centre in two directions at right angles
3. Focusing the eyepiece: Sharpen cross-hairs (parallax elimination)
4. Focusing the objective: Sharpen image of target

Measurement of Horizontal Angles

Method of Repetition

1. Set horizontal circle to 0°0'0" by rotating upper plate and clamping lower plate to target A
2. Release lower clamp; rotate entire instrument to target B; read angle θ₁
3. Release upper clamp; rotate back to A (without changing lower plate position)
4. Clamp upper plate; swing to B — new reading = 2θ
5. Repeat n times → angle = Final reading / n (random and graduation errors reduced by √n)

This eliminates graduation errors and personal errors more effectively.

Method of Reiteration

Used when many angles are measured from one station (e.g., angular measurements for triangulation):

• Measure angle to each target in succession around the horizon
• Return to first target; check if reading = 0° (closing error in face left)
• Repeat in face right; average face left and face right readings

Face Left and Face Right — Elimination of Systematic Errors

Observations in FL (face left) and FR (face right) with mean taken eliminates:

• Error of non-perpendicularity of trunnion axis to vertical axis
• Error of non-perpendicularity of line of collimation to trunnion axis
• Eccentricity of vertical circle

This is why all precise observations use both faces: Mean angle = (FL + FR) / 2

Traversing

A traverse is a series of connected survey lines (legs) with measured lengths and angles, forming a network for determining positions of points.

Types of Traverse

• Closed Traverse: Starts and ends at the same known point (or two known points); allows closure check and adjustment — used for boundary surveys, property surveys
• Open Traverse: Starts at known point, ends at unknown or different point; no closure check — used for road centreline surveys, pipeline routes

Traverse Computation

1. Compute included angles from field readings
2. Check: Σ(included angles) = (2n−4) × 90° for closed traverse (n sides)
3. Apply angular correction: Distribute angular error equally to all angles
4. Compute bearings of each side: WCB or Reduced Bearing
5. Compute latitudes (Northing): L = length × cos(bearing)
6. Compute departures (Easting): D = length × sin(bearing)
7. Closure check: Σ Latitudes = 0, Σ Departures = 0 (for closed traverse)
8. Apply Bowditch (Compass) Rule for adjustment

Bowditch Rule (Compass Rule) for Traverse Adjustment

Distributes linear error proportional to the length of each traverse leg:

Correction to latitude of a leg = −(Error in latitude) × (Length of leg / Perimeter)

Correction to departure = −(Error in departure) × (Length of leg / Perimeter)

Transit Rule: Correction proportional to the magnitude of latitude and departure of each leg (preferred when angular accuracy > linear accuracy).

Relative Precision of Traverse

Linear closing error = √[(ΣL)² + (ΣD)²]

Relative precision = Linear error / Perimeter

Acceptable limits: 1/2000 for rough surveys; 1/5000 for third-order; 1/20000 for second-order.

Theodolite vs Total Station

Frequently Asked Questions

What is the difference between included angle and deflection angle in traversing?

Included angle is the interior or exterior angle at each traverse station between two adjacent legs — measured directly with a theodolite. Deflection angle is the angle between the extension of the preceding leg and the next leg — it is the deviation from the straight-ahead direction (right or left). Deflection angles are used in highway geometry (tangent deflection angle) and open traverses. Both can be used for traverse computation; the angular sum check differs for each.

What is the significance of face left and face right observations?

The mean of face left and face right observations cancels all systematic instrumental errors (collimation error, trunnion axis error, vertical circle index error). For GATE and precision surveys, all angles must be observed in both faces and averaged. A single-face observation has uncorrected systematic errors that can be significant.