Measurement of Angles and Directions

Learning Objectives

  • Understand the definition of direction and reference meridians.
  • Differentiate between various angle measurement instruments.
  • Define and convert between azimuths and bearings.
  • Explain magnetic declination and its variations.
  • Understand magnetic dip.
  • Calculate interior, exterior, and deflection angles of closed polygons.
  • Describe the double centering method for prolonging a line.

Direction

The line of sight from one point to another. It is defined by the horizontal angle between the line and a reference meridian.

Instruments for Angle Measurement

Surveying Instruments

  • Transit: The traditional American surveying instrument. It measures horizontal and vertical angles using metallic vernier scales. It is rugged but less precise than modern instruments.
  • Theodolite: A more precise instrument. Originally optical (using glass circles read through microscopes), modern theodolites are electronic, displaying angles digitally.
  • Total Station: The modern standard. It combines an electronic theodolite, an Electronic Distance Measurement (EDM) device, and a microprocessor. It measures angles and distances simultaneously and computes coordinates in real-time.

Interactive Simulation

Rotate the azimuth using the slider to see its corresponding bearing conversion visually represented in the four quadrants.

Azimuth and Bearing Converter

Convert between azimuths and bearings interactively.

Azimuth (Β°)45.0

Results

Azimuth=45.0∘\text{Azimuth} = 45.0^\circ
\text{Bearing} = \text{N 45.0^\circ E}
NSEW

Reference Meridians

Types of Meridians

  • True Meridian: Passes through the true north and south poles (geographic poles). It is fixed and does not change with time. Determined by astronomical observations.
  • Magnetic Meridian: Direction indicated by a freely suspended magnetic compass needle. It varies with location and time due to the earth's magnetic field.
  • Grid Meridian: Parallel lines on a map grid (e.g., UTM). Central meridian is true north; others are parallel to it.
  • Assumed Meridian: Arbitrarily chosen direction for a specific survey (e.g., a street centerline).

Systems of Designating Direction

1. Azimuth

Azimuth Characteristics

The direction of a line as given by an angle measured clockwise from the north (or south) end of a meridian.

  • Range: 0∘0^\circ to 360∘360^\circ.
  • Reference: Usually North (Geodetic) or South (Astronomic).

2. Bearing

Bearing Characteristics

The smallest angle which the line makes with the meridian (north or south).

  • Range: 0∘0^\circ to 90∘90^\circ.
  • Format: N/SN/S (Angle) E/WE/W.
  • Example: N45∘EN 45^\circ E, S30∘WS 30^\circ W.

Interactive Visualization: Azimuth vs. Bearing

Interactive Simulation

Explore the relationship between Azimuth (from North) and Bearing with this interactive tool.

Azimuth vs. Bearing

NSEW

Azimuth

45Β°

Bearing

N 45Β° E

0Β°Adjust Angle360Β°
Rule: Quadrant I (NE): Azimuth = Bearing

Conversion Rules

Azimuth to Bearing Conversion

  • Quadrant I (0-90): Azimuth = Bearing
  • Quadrant II (90-180): Bearing = 180∘180^\circ - Azimuth
  • Quadrant III (180-270): Bearing = Azimuth - 180∘180^\circ
  • Quadrant IV (270-360): Bearing = 360∘360^\circ - Azimuth

Magnetic Declination (DD)

Magnetic Declination Concepts

The horizontal angle between the true meridian and the magnetic meridian.

Magnetic Declination Formula

Calculates magnetic declination from magnetic and true bearings.

D=MagneticΒ Bearingβˆ’TrueΒ Bearing D = \text{Magnetic Bearing} - \text{True Bearing}

Variables

SymbolDescriptionUnit
DDMagnetic declination-
Magnetic Bearing\text{Magnetic Bearing}Bearing relative to magnetic north-
True Bearing\text{True Bearing}Bearing relative to true north-

East Declination Formula

Calculates True North when Magnetic North is East of True North.

TN=MN+D TN = MN + D

Variables

SymbolDescriptionUnit
TNTNTrue North direction-
MNMNMagnetic North direction-
DDEast magnetic declination-

West Declination Formula

Calculates True North when Magnetic North is West of True North.

TN=MNβˆ’D TN = MN - D

Variables

SymbolDescriptionUnit
TNTNTrue North direction-
MNMNMagnetic North direction-
DDWest magnetic declination-

Variations in Magnetic Declination

  • Secular Variation: A slow, continuous change over a long period (centuries). It is the most important variation for surveyors when retracing old boundary lines.
  • Annual Variation: A small periodic change that completes its cycle in one year (typically less than 1 minute of arc). Often negligible in typical surveying.
  • Daily (Diurnal) Variation: A periodic daily swing of the magnetic needle (approx. 3 to 12 minutes of arc). Usually greatest during the day and in summer.
  • Irregular Variation: Unpredictable changes caused by magnetic storms, solar flares, or local magnetic disturbances.

Magnetic Dip

Magnetic Dip

The angle that a freely suspended magnetic needle makes with the horizontal plane. At the magnetic equator, the dip is zero. At the magnetic poles, the dip is 90 degrees.

Angles and Line Operations

Prolongation of a Line

Line Prolongation Overview

Extending a straight line accurately in the field.

Double Centering Method

When using a transit or theodolite, prolonging a line by simply plunging the telescope can introduce instrumental error (collimation error). The correct procedure is Double Centering:

  1. Sight the back point.
  2. Plunge the telescope (reverse) and set a point forward.
  3. Unclamp, rotate the instrument 180 degrees horizontally, and sight the back point again.
  4. Plunge the telescope and set a second point forward.
  5. The true point is exactly midway between the two marked points.

Interior and Exterior Angles

Interior Angle

Angle inside a closed polygon between adjacent sides.

Sum of Interior Angles

Calculates the sum of interior angles in a closed polygon.

Sum=(nβˆ’2)Γ—180∘ \text{Sum} = (n-2) \times 180^\circ

Variables

SymbolDescriptionUnit
Sum\text{Sum}Sum of interior angles-
nnNumber of sides in the polygon-

Exterior Angle

Angle outside a closed polygon.

Sum of Exterior Angles

Calculates the sum of exterior angles in a closed polygon.

Sum=(n+2)Γ—180∘ \text{Sum} = (n+2) \times 180^\circ

Variables

SymbolDescriptionUnit
Sum\text{Sum}Sum of exterior angles-
nnNumber of sides in the polygon-

Deflection Angle

Angle between the prolongation of the preceding line and the succeeding line. Denoted as Right (R) or Left (L).

Sum of Deflection Angles

Calculates the sum of deflection angles in a closed polygon.

Sum=360∘ (Right)βˆ’Sum(Left)=Β±360∘ \text{Sum} = 360^\circ \text{ (Right)} - \text{Sum(Left)} = \pm 360^\circ
Key Takeaways
  • Azimuth: 0βˆ’360∘0-360^\circ clockwise from North/South.
  • Bearing: 0βˆ’90∘0-90^\circ acute angle from North/South.
  • Magnetic Declination: Difference between True North and Magnetic North.
  • Interior Angles Sum: (nβˆ’2)Γ—180∘(n-2) \times 180^\circ.
  • Double Centering: The method to accurately prolong a line and eliminate instrumental collimation error.
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