Angles, Azimuths, and Bearings

Introduction

• Orientation of lines are often dependent on the observations of angles, azimuths, or bearings.
• Today, angles are the most common type of observation
• Three basic requirements to determine an angle
1. You must have a reference or starting line
2. There is a direction of turning. To avoid confusion in surveying, angles are always turned clockwise with the exception of deflection angles
3. An angular distance is recorded (sexagesimal system in U.S.)
   
• Types of Horizontal Angles
• Interior angles - Clockwise angles turned on the interior of a closed polygon traverse. The sum of the angles on a closed polygon traverse is 
                          S angles = (n - 2) 180°
  
  This can be verified by drawing triangles (sum of angle = 180°) in a polygon. There are always 2 less triangles than there are sides to the polygon.
  
• Exterior angles - Clockwise angles turned on the exterior of a closed polygon traverse. Turning both interior and exterior angles is known as closing the horizon which means an angle has been observed about all points of the observer's horizon.
  
• Angles to the right - An angle turned clockwise where the reference line is the previously occupied station in the traverse, and the foresight station is the next station to be occupied in the traverse. This angle is required by data collectors. 
• Deflection angles - An angle measured either clockwise (+) or counter-clockwise (-) from the extension of the previous course in a traverse. These angles were typically used in alignment surveys, but are rarely measured today. The angles were always less than 180° and follow by a R or L for right or left, respectively. E.g. 10°13'R was a clockwise angle of 10°13' right of the extension of the line. 10°13'L was a counter-clockwise angle 10°13' left of the extension of the line.

To avoid confusion, angles should always be measured clockwise, and angles-to-the-right are preferred.

Direction of a Line

• The direction of a line is defined by a horizontal angle between the line and an arbitrarily chosen reference line called a meridan.
• The question of what is the direction's reference meridian can be defined with the following possible options. The types of meridians (and thus directions) are:
1. Geodetic - angle generally measured from geodetic north. Historically south has occasionally been used.
2. Astronomic - angle measured from astronomic north (as determined by the stars - see Chapter 18). This meridian is very close to geodetic, and the two have sometimes been used interchangeably.
3. Magnetic - angle measured from magnetic north. Since magnetic fields fluctuate over time, this meridian is time-dependent.
4. Grid - angle measured from grid (map) north (see Chapter 20). This angle is dependent on the map-projection.
5. Record - angle measured from a meridian defined in a recorded instrument such as a deed.
6. Assumed - angle measured from a meridian assumed by the user. This meridian is often convenient in computations. For instance assume one side of a traverse is due north. Unfortunately if either of the monuments of the line are lost, the meridian is also lost. This type of azimuth should be avoided.
• Types of directions are
1. Azimuths - horizontal angles measured clockwise from a reference meridian. Azimuths can be any of the type above, geodetic azimuth, astronomic azimuth, etc. Examples are
1. 34°, 157°, 235°, 317°
2. Examples are preferred over azimuths since they simplify computations.
2. Bearings - horizontal angles measured from the meridian either east or west. Again they can be geodetic, astronomic, etc. Thus they are designated with nomenclature such as 
1. Require two letters and an acute angle (<90°).
2. Measured both clockwise and counter-clockwise.
3. Can be measured from North or South axis of meridian.
4. N34°E which is a horizontal angle measured from the north end of the meridian going to the east, 
5. S23°E is a horizontal angle measured from the south meridian to the east. (Note that angle is counter-clockwise.)
6. S55°W - a horizontal angle measured from the south meridian to the west
7. N43°W - a horizontal angle measured from the north meridian to the west.

Class practice

Identify the azimuth and bearing of each line in the figure.

Practice Example
Fill in the first four lines and then check your responses by selecting the "Check the above" button. Then complete the next four line and select the "Check last four" button.

Computation of Azimuths

• Back azimuths/bearings are the direction of the line viewed from the opposite end of a line. 
• For azimuth simply add or subtract 180°
• If forward azimuth is 238°55'13", then back azimuth is 238°55'13" - 180° = 58°55'13"
• For bearings simply reverse the direction letters. That is, N for S and E for W 
• If forward bearing is S67°41'E, then the back bearing is N67°41'W
• Computation of azimuths can be performed by analyzing a figure with angles shown
• The tabular method is the most computational efficient procedure. (see Section 7-8 for more on this procedure)
• If azimuths are computed in a counter-clockwise direction around traverse, angles are added.
• If azimuths are computed in a clockwise direction around traverse, angles are subtracted.
• The procedure is to compute a back azimuth, then add/subtract the angle, and so on.

Class Example 

Compute the azimuth of each course for the traverse shown above and the following angles to the right. The azimuth of course AB is 98°39'38"

TABULAR METHOD

Compass and the Earth's Magnetic Field

• Compass was principal instrument used in early U.S. boundary surveys for direction determination.
• Note that E and W are reversed on the compass face so that bearings could be read directly. In the figure, the bearing is N40°E. This represents the direction of line determined by the sighting vanes.
• Many legal descriptions are based on magnetic meridians.
• Thus we must have some understanding of the Earth's magnetic field
• The North magnetic pole is located somewhere in Canada near the Hudson Bay.  Its position changes with time and solar storms. 
• The position of the pole can only be predicted into the near future. 
• Records of the poles position have been maintained since the late 1600's.
• These positions do not take into account local anomalies or solar storms.
• The model is only about 90 percent accurate, and can vary by large amounts in small areas.
• Magnetic declination is the horizontal angle from the geodetic meridian to the magnetic meridian.
• geodetic bearing = magnetic bearing + magnetic declination
• Changes in magnetic declination are broken into annual change and secular change.
• Annual change is the amount of secular variation in one year. There are predictive formulas for these changes, but they are (1) not entirely precise, nor (2) long-term.
• Secular change is the daily permanent change in declination.

Typical Magnetic Declination Problems

• Many older boundary surveys were performed using a compass, and thus are dependent on magnetic declination at time and location of survey. Since these are difficult to model, this was a poor choice of meridian. Yet, today's surveyor must account for this when these deeds are used.
• Thus a typical problem involves a record magnetic bearing that must be converted to a geodetic bearing.

Assume a magnetic bearing of a property line in an 1888 deed on the Hayfield farms is listed as 125°48'. The magnetic declination for this location and time was 4°07' W. What is the geodetic bearing of the line?

Brg_AB = 125°48' - 4°07' = 121°41'

Practice Example. Make a copy of the following table and compute the values in the fields. After you have computed the values, select the "Check" button and check your values against those given.

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Last updated November 04, 2004
Penn State Surveying Program © 2001