s2 4 angle and directions
TRANSCRIPT
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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