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PRC Geodetic Engineering Licensure Examination

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6.2 Angular Measurements, Transit/Theodolite/Total Station Operations, Bearings, Azimuths, and Magnetic Declination

Key Takeaways

  • Horizontal angles include interior angles, exterior angles, angles to the right, and deflection angles, with geometric closure governed by (n - 2) * 180 deg.
  • Bearings use a quadrantal format (N/S 0-90 deg E/W), whereas Azimuths express continuous full-circle direction (0-360 deg) referenced to North.
  • Magnetic declination is the horizontal angle between True North and Magnetic North; True Azimuth = Magnetic Azimuth +/- Declination.
  • Local attraction is detected when forward and back magnetic azimuths of a line differ by an amount other than exactly 180 degrees.
  • Taking the mean of Direct (Face Left) and Reverse (Face Right) telescope observations completely eliminates horizontal collimation and horizontal axis tilt errors.
Last updated: July 2026

6.2 Angular Measurements, Total Station Operations, Bearings, Azimuths, and Magnetic Declination

Angular measurement forms the secondary framework of plane surveying, enabling the orientation of survey lines, determination of positions via triangulation and traversing, and calculation of boundary bearings. This section covers horizontal/vertical angle types, directional systems (bearings vs. azimuths), magnetic declination dynamics, local attraction correction, and instrument error elimination techniques.

1. Types of Horizontal and Vertical Angles

Surveying horizontal angles are measured in a horizontal plane between two intersecting lines of sight.

A. Horizontal Angle Classifications

  1. Interior Angles: Angles measured inside a closed polygon between adjacent sides. For a closed traverse with $n$ sides:

∑Interior Angles=(n−2)×180∘\sum \text{Interior Angles} = (n - 2) \times 180^\circ

  1. Exterior Angles: Angles measured outside a closed polygon between adjacent sides:

∑Exterior Angles=(n+2)×180∘\sum \text{Exterior Angles} = (n + 2) \times 180^\circ

  1. Angles to the Right (Clockwise Angles): Angles measured clockwise from the backsight line to the foresight line. Angle to the right ranges from $0^\circ$ to $360^\circ$.
  2. Deflection Angles: The angle between the prolongation of the preceding line (backsight extended) and the forward line (foresight).
    • Range: $0^\circ$ to $180^\circ$, designated as Right (R) (clockwise) or Left (L) (counterclockwise).
    • Geometric rule for closed polygon: Algebraic sum of deflection angles (treating R as positive and L as negative) equals $\pm 360^\circ$.

Deflection Angle Δ=Azimuthforward−Azimuthback prolongation\text{Deflection Angle } \Delta = \text{Azimuth}_{\text{forward}} - \text{Azimuth}_{\text{back prolongation}}

B. Vertical Angles and Zenith Angles

  • Vertical Angle ($\alpha$): Angle measured above (+) or below (-) the horizontal plane. Range: $-90^\circ$ to $+90^\circ$.
  • Zenith Angle ($Z$): Angle measured downward from the vertical zenith line ($0^\circ$ directly overhead, $90^\circ$ on horizontal, $180^\circ$ directly downward).

α=90∘−Z\alpha = 90^\circ - Z

2. Bearings versus Azimuths

Line orientations are expressed in two primary systems: Bearings and Azimuths.

FeatureBearing System (Quadrantal)Azimuth System (Full-Circle)
Reference MeridianNorth or SouthNorth (or South in some military contexts)
Angular Range$0^\circ$ to $90^\circ$$0^\circ$ to $360^\circ$
FormatQuadrant Letter, Angle, Direction Letter (e.g., N $42^\circ 15' \text{ E}$)Angle only (e.g., $42^\circ 15'$)
QuadrantsNE, SE, SW, NWI ($0^\circ\text{--}90^\circ$), II ($90^\circ\text{--}180^\circ$), III ($180^\circ\text{--}270^\circ$), IV ($270^\circ\text{--}360^\circ$)
Forward/BackReverse letters (N $\leftrightarrow$ S, E $\leftrightarrow$ W)Add or subtract $180^\circ$ ($\text{Back Az} = \text{Fwd Az} \pm 180^\circ$)

Conversion Rules Between Bearings and Azimuths (North Reference)

  • Quadrant I (NE): $\text{Azimuth} = \text{Bearing}$. Example: N $35^\circ \text{ E} \implies \text{Azimuth} = 35^\circ$.
  • Quadrant II (SE): $\text{Azimuth} = 180^\circ - \text{Bearing}$. Example: S $42^\circ \text{ E} \implies \text{Azimuth} = 180^\circ - 42^\circ = 138^\circ$.
  • Quadrant III (SW): $\text{Azimuth} = 180^\circ + \text{Bearing}$. Example: S $68^\circ \text{ W} \implies \text{Azimuth} = 180^\circ + 68^\circ = 248^\circ$.
  • Quadrant IV (NW): $\text{Azimuth} = 360^\circ - \text{Bearing}$. Example: N $25^\circ \text{ W} \implies \text{Azimuth} = 360^\circ - 25^\circ = 335^\circ$.

3. Meridians, Magnetic Declination, and Local Attraction

Types of Meridians

  1. True Meridian (Geographic Meridian): The true north-south line passing through the geographic poles of Earth. Constant direction; reference for official land surveys in the Philippines.
  2. Magnetic Meridian: Line parallel to magnetic lines of force defining Magnetic North. Subject to secular, annual, daily, and irregular variation.
  3. Grid Meridian: Line parallel to the central meridian of a cartographic map projection (e.g., Philippine Transverse Mercator / PTM).

Magnetic Declination ($\delta$)

Magnetic declination is the horizontal angle between True North and Magnetic North:

  • East Declination ($\delta_E$): Magnetic North lies East of True North.
  • West Declination ($\delta_W$): Magnetic North lies West of True North.

True Azimuth=Magnetic Azimuth±Declination\text{True Azimuth} = \text{Magnetic Azimuth} \pm \text{Declination}

Conversion Rules:

  • $\text{True Azimuth} = \text{Magnetic Azimuth} + \delta_E \quad (\text{for East declination})$
  • $\text{True Azimuth} = \text{Magnetic Azimuth} - \delta_W \quad (\text{for West declination})$

Local Attraction Analysis and Correction

Local attraction refers to magnetic disturbances caused by artificial structures (steel pipes, power cables, vehicles, reinforced concrete) affecting compass needles.

  • In an unaffected line $AB$: $|\text{Forward Magnetic Azimuth} - \text{Back Magnetic Azimuth}| = 180^\circ00'$.
  • Any deviation from $180^\circ00'$ indicates local attraction at station $A$, station $B$, or both.

Worked Field Calculation Table: Local Attraction Correction

Consider a 4-station closed magnetic traverse with observed forward and back bearings:

LineObserved Forward BearingForward AzimuthObserved Back BearingBack AzimuthDiscrepancy ($\Delta$)Corrected Forward AzimuthCorrected Bearing
ABN $30^\circ00'\text{ E}$$30^\circ00'$S $30^\circ00'\text{ W}$$210^\circ00'$$0^\circ00'$ (Free of Local Att.)$30^\circ00'$N $30^\circ00'\text{ E}$
BCS $45^\circ00'\text{ E}$$135^\circ00'$N $42^\circ30'\text{ W}$$317^\circ30'$$+2^\circ30'$ (Error at C)$135^\circ00'$S $45^\circ00'\text{ E}$
CDS $50^\circ00'\text{ W}$$230^\circ00'$ (Obs)N $50^\circ00'\text{ E}$$70^\circ00'$ (Obs)$+2^\circ30'$ at C$227^\circ30'$ (Corr)S $47^\circ30'\text{ W}$
DAN $20^\circ00'\text{ W}$$340^\circ00'$S $20^\circ00'\text{ E}$$160^\circ00'$$0^\circ00'$$340^\circ00'$N $20^\circ00'\text{ W}$

Analysis: Stations $A$ and $B$ are free from local attraction because the difference between forward azimuth $AB$ ($30^\circ00'$) and back azimuth $BA$ ($210^\circ00'$) is exactly $180^\circ00'$. Therefore, the observed forward bearing of $BC$ ($135^\circ00'$) is correct, but back bearing $CB$ has a $+2^\circ30'$ local attraction error at station $C$. Applying $-2^\circ30'$ correction to all readings taken at station $C$ yields the true magnetic orientation for line $CD$.

4. Total Station / Transit Instrumental Errors and Elimination

Modern Total Stations integrate electronic digital theodolites and EDMs. Precise angular measurement requires understanding systematic instrumental errors:

Instrumental ErrorDescriptionElimination Method
Collimation Error (Sight Line Error)Line of sight is not perpendicular to horizontal axisDouble-centering: Average Face Left (Direct) and Face Right (Reverse) readings
Horizontal Axis Tilt ErrorHorizontal rotation axis is not perpendicular to vertical axisDouble-centering / dual-axis liquid electronic compensator
Index Error (Vertical Circle)Vertical index zero mark is offset from true zenith/horizontalTake direct and reverse vertical angle readings; $\alpha = \frac{\alpha_L - \alpha_R}{2}$
Eccentricity of CirclesGeometric center of graduated glass circle does not coincide with rotation axisTake readings using diametrically opposed optical sensors or verniers
Plumb / Level Bubble ErrorPlate bubble axis not perpendicular to vertical axisRe-level instrument; rotate $180^\circ$ and eliminate half-parallax using leveling screws

Double-Centering (Face Left / Face Right) Technique

When turning horizontal angles, systematically taking the mean of Face Left (Direct) and Face Right (Reverse) readings completely eliminates:

  1. Horizontal collimation error
  2. Horizontal axis tilt error
  3. Circle eccentricity error

θtrue=θFace Left+(θFace Right±180∘)2\theta_{\text{true}} = \frac{\theta_{\text{Face Left}} + (\theta_{\text{Face Right}} \pm 180^\circ)}{2}

Test Your Knowledge

A survey boundary line has an Azimuth of 312°15' measured from North. What is its equivalent bearing in the quadrantal system?

A
B
C
D
Test Your Knowledge

The magnetic bearing of a line was observed as S 35°30' E in 1995 when the magnetic declination was 2°15' East. What is the true azimuth (from North) of the line?

A
B
C
D
Test Your Knowledge

In a 5-sided closed interior-angle polygon traverse, four of the measured interior angles are 110°15', 95°40', 132°10', and 108°20'. What must be the value of the fifth interior angle for exact geometric closure?

A
B
C
D
Test Your Knowledge

Which systematic transit/total station error is completely eliminated by taking the numerical average of Direct (Face Left) and Reverse (Face Right) telescope readings?

A
B
C
D
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