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Chapter 4 – Angle and Directions

One of the basic purposes of surveying is to determine the relative positions of

points on or near the earth’s surface. Assigning coordinates to a given point is a

useful and common way to indicate its position. Angles are usually measured to

compute the coordinates of a particular point.

Vertical angle

A vertical angle between two lines of sight is measured in a plane that is vertical at the

point of observation. Sometimes the two points sighted do not lie in the same plane.

Angle of elevation – angle measured upward from a horizontal reference line and is

considered a positive (+) angle.

Angle of depression – angle measured downward from the horizon and is considered to

be a negative (-) angle.

Zenith direction – an upward vertical direction usually used as a reference for measuring

vertical angles.

Zenith angle – an angle measured with respect to zenith direction.

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Horizontal angles

An interior angle – is measured on the inside of a closed polygon

An exterior angle – is measured outside of the closed polygon

At any point, the sum of the interior and exterior angles must equal 360°.

The sum of all interior angles in a closed polygon is equal to (180°)(n – 2), where n is the

number of sides.

The sum of the exterior angles must equal (180°)(n + 2).

An angle measured in a clockwise direction, from the rear to the forward point or station,

is called an angle to the right.

An angle turned counterclockwise from the rear to the forward station is called an angle

to the left.

A horizontal angle between the extension of a back or preceding line and the succeeding

or next line forward is called a deflection angle. Deflection angles are always less than

180°; they must be clearly identified as being turned either to the left (counterclockwise)

or to the right (clockwise), using letters L or R, respectively.

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Azimuth and Bearing of a line

The direction of any line may be described either by its azimuth angle or by its bearing.

Azimuth directions are usually preferred by surveyors; they are purely numerical and

help to simplify office work by allowing a simple routine for computations.

Bearings required two letter symbols as well as a numerical value, and each bearing

computation requires an individual analysis with a sketch. Bearings are almost always

used to indicate the direction of boundary lines in legal land descriptions.

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Azimuths

The azimuth of a line is the clockwise horizontal angle between the line and a given

reference direction or meridian.

Usually, north is the reference direction, south is sometimes used as a reference for

geodetic surveys that cover large areas.

Any azimuth angle will have a positive value between 0 and 360°.

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Bearings

A bearing of a line is the angle from the north (N) or the south (S) end of the meridian.

It has the added designation of east (E) or west (W), whichever applies.

The directions due east and due west are perpendicular to the north-south meridian.

A line may fall in one of four quadrants: northeast (NE), south-east (SE), southwest

(SW), or northwest (NW).

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A bearing may be measured either in a clockwise or in a counterclockwise direction,

depending on which quadrant the line is in. A bearing angle is always an acute angle (less

than 90°).

It must always be accompanied by the two letters that indicate the quadrant of the line.

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Directions

Every line actually has two directions, a forward direction and a back direction. The

difference depends on which way the line is being observed.

Generally, the forward direction is taken in the same sense with which the field work was

carried out. For example, the forward direction of line AB can be taken as the direction

the surveyor faces when occupying point A and sighting toward point B.

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The back direction of that line would be that which is observed when standing on B and

looking toward A. Calling the line “AB” implies its forward direction; calling the line

“BA” implies its back direction.

For connected lines, it is necessary to be consistent in designating forward or back

direction. For example, line BC should be considered a forward direction so that it is

consistent with the direction of AB.

The back azimuth of a line is determined simply by adding (or subtracting) 180° to the

forward azimuth; when the forward azimuth is more than 180°, 180° is subtracted so that

the numerical value of the back azimuth does not exceed 360°. To determine the back

bearing of a line, it is only necessary to reverse the letters; the numerical value does not

change

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Azimuth computations

Many types of surveying problems involve the computation of the azimuths or bearings

of adjoining lines, given a starting direction and a series of measured angles. These

computations are particularly important for traverse surveys.

Before azimuths are computed:

1. It is usual to check that the figure is geometrically closed (the sum of the interior

angles = (n – 2) 180°.

2. Compute the azimuth in clockwise direction or counter-clockwise direction.

3. Clockwise direction – When computations are to proceed around the traverse in a

clockwise direction, subtract the interior angle from the back azimuth of the

previous course.

4. Counter clockwise direction – When computations are to proceed around the

traverse in a counter-clockwise direction, add the interior angle to the back

azimuth of the previous course.

Example 1

The azimuth of side 1-2 is given for the three-sided traverse shown. The three interior

angles are also given. Determine the azimuth direction for sides 2-3 and 3-1.

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Solution

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Exercise

Sketch for azimuth calculations. Computations to be staged (i) clockwise, and (ii)

counter-clockwise.

Solution

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Bearing computations

There is no systematic method of directly computing bearings, each bearing computation

will be regarded as a separate problem.

The sketch of each individual bearing computation should show the appropriate interior

angle together with one bearing angle. The required bearing angle should also be shown

clearly.

Example 2

In the traverse shown, the bearing of side CA and angles A and B are given. Determine

the bearings of side AB and side BC. Check by re-computing the bearing of CA.

Solution

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Example 3

The bearings of two adjoining lines, EF and FG, are N46°30’E and S14°45’E,

respectively. Determine the deflection angles formed at the point of intersection, station

F.

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Exercise

Sketch for bearing computations.

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Solution