Angle Measurement and Line Direction

Angle Measurement

• Angles are defined as the difference in direction between two convergent lines.
• Horizontal angles are formed by the directions to two objects in a horizontal plane.
• Vertical angles are formed by two intersecting lines in a vertical plane, one being horizontal.
• Interior angles are measured inside a closed polygon, and their sum equals (n - 2) × 180°, where n is the number of angles.
• Exterior angles are located outside a closed polygon and are the supplement of interior angles. Their sum, plus the interior angles, equals 360°.
• Deflection angles are measured from an extension of the preceding course to the ahead line, either right (R) or left (L).

Direction of a Line

• Direction of a line is defined by the horizontal angle between the line and an arbitrarily chosen reference line (meridian).
• Geodetic meridian is a north-south reference line passing through Earth's geographic poles.
• Astronomic meridian passes through instantaneous position of Earth's geographic poles.
• Magnetic meridian is defined by a freely suspended magnetic needle, only affected by Earth's magnetic field.
• Grid meridian is used in surveys based on a coordinate system, parallel to geodetic north for a central meridian.
• Record meridian refers to directional references used in recorded documents from a previous survey.
• Assumed meridian is an arbitrarily assigned direction, often using a reference line.

Bearings and Azimuths

• Azimuths are horizontal angles measured clockwise from a reference meridian.
• In surveying, azimuths are usually observed from north.
• Bearings are another system for designating directions of lines, defining the acute horizontal angle between a reference meridian and the line, measured from north or south towards east or west.
• Bearings are less than 90° and use letters (N, S, E, W) to indicate the quadrant.

Comparison of Azimuths and Bearings

• Azimuths vary from 0° to 360°.
• Bearings vary from 0° to 90°.
• Azimuth requires only a numerical value.
• Bearings require two letters and a numerical value.

Back Azimuths and Back Bearings

• The back azimuth/bearing of a line is in the reverse direction of the forward azimuth/bearing.
• Back azimuth is calculated based on the forward azimuth:
  • Adding 180° if forward azimuth is less than 180°, or
  • Subtracting 180° if forward azimuth is greater than 180°.
• Back bearing is derived from the forward bearing by switching the first and second letters respectively (e.g., N to S, E to W).

Theodolite Survey

• Theodolites are surveying instruments used to measure horizontal and vertical angles.
• Theodolites work by combining optical plummets (or plumb bobs), a spirit level, and graduated circles.
• They can be classified as Optical-reading, Optical-reading repeating, Optical-reading directional, and Electronic-Digital.

Traverse

• A traverse is a series of consecutive lines whose lengths and directions are determined from observations.
• Closed traverses return to the starting point, forming a closed figure. Examples are polygon and link.
• Open traverses do not return to the starting point and need greater accuracy.

Angle Misclosure

• Angular misclosure is the difference between the observed angles and the geometrically correct total for a polygon traverse.
• Using (n-2)*180° for interior angles and (n+2)*180° for exterior angles where n is the number of sides.

Balancing Angles

• Angle balancing involves adjusting observed interior angles to match the geometric total of a closed traverse, using corrections based on poor observation conditions (if any).
• Two main methods are used: applying an average correction to all angles or making larger corrections to angles with poor observing conditions.

Departures and Latitudes

• Departures of a course are its orthographic projection along the east-west axis, and latitudes are its projection along the north-south axis.
• Calculated using the line's length and azimuth/bearing.

Coordinates

• Plane rectangular coordinate system.
• X-axis is east-west.
• Y-axis is north-south.
• Used to define the location of traverse stations.

Inversing

• Methods relate departures, latitudes, azimuths and bearings, enabling calculation of length and azimuth if departure and latitude are known.