S2 4 angle and directions
- 1. 1 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.
- 2. 2 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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- 4. 4 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.
- 5. 5 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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- 7. 7 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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- 9. 9 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.
- 10. 10 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.
- 11. 11 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
- 12. 12 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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- 14. 14 Exercise Sketch for azimuth calculations. Computations to be staged (i) clockwise, and (ii) counter-clockwise. Solution
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- 17. 17 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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- 19. 19 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.
- 20. 20 Exercise Sketch for bearing computations.
- 21. 21 Solution