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Angle Measurement Determining the locations of points and orientations of lines frequently depends on the observation of angles and directions. In surveying, directions are given by azimuths and bearings. - An angle is defined as the difference in direction between two convergent lines. - A horizont...

Angle Measurement Determining the locations of points and orientations of lines frequently depends on the observation of angles and directions. In surveying, directions are given by azimuths and bearings. - An angle is defined as the difference in direction between two convergent lines. - A horizontal angle is formed by the directions to two objects in a horizontal plane. Horizontal angles are the basic observations needed for determining bearings and azimuths. - A vertical angle is formed by two intersecting lines in a vertical plane, one of these lines horizontal. Vertical angles are used in trigonometric leveling, stadia, and for reducing slope distances to horizontal. Types of Measured Angles - Interior angles are observed on the inside of a closed polygon. Normally the angle at each apex within the polygon is measured. Then, a check can be made on their values because the sum of all interior angles in any polygon must equal ( n - 2 ) 180° where n is the number of angles. Polygons are commonly used for boundary surveys and many other types of work. Surveyors (geomatics engineers) normally refer to them as closed traverses. - Exterior angles, located outside a closed polygon, are explements of interior angles. The advantage to be gained by observing them is their use as another check, since the sum of the interior and exterior angles at any station must total 360°. - Deflection angles, right or left, are measured from an extension of the preceding course and the ahead line. It must be noted when the deflection is right (R) or left (L).  Deflection angles. Direction of a Line The direction of a line is defined by the horizontal angle between the line and an arbitrarily chosen reference line called a meridian. Different meridians are used for specifying directions including (a) geodetic (also often called true), (b) astronomic, (c) magnetic, (d) grid, (e) record, and (f) assumed. - The geodetic meridian is the north-south reference line that passes through a mean position of the Earth’s geographic poles. - The astronomic meridian is the north-south reference line that passes through the instantaneous position of the Earth’s geographic poles. Astronomic meridians consists in making observations on the celestial objects. Geodetic and astronomic meridians are very nearly the same, and the former can be computed from the latter by making small corrections. - A magnetic meridian is defined by a freely suspended magnetic needle that is only influenced by the Earth’s magnetic field. - Surveys based on a state or other plane coordinate system employ a grid meridian for reference. Grid north is the direction of geodetic north for a selected central meridian and held parallel to it over the entire area covered by a plane coordinate system. - In boundary surveys, the term record meridian refers to directional references quoted in the recorded documents from a previous survey of a particular parcel of land. - An assumed meridian can be established by merely assigning any arbitrary direction— for example, taking a certain street line to be north. The directions of all other lines are then found in relation to it. Bearings and Azimuths Azimuths are horizontal angles observed clockwise from any reference meridian. In plane surveying, azimuths are generally observed from north, but astronomers and the military have used south as the reference direction. Examples of azimuths observed from north are shown in below figure. As illustrated, they can range from 0° to 360° in value. Thus the azimuth of OA is 70°; of OB, 145°; of OC, 235°; and of OD, 330°. Azimuths may be geodetic, astronomic, magnetic, grid, record, or assumed, depending on the reference meridian used. Bearings are another system for designating directions of lines. The bearing of a line is defined as the acute horizontal angle between a reference meridian and the line. The angle is observed from either the north or south toward the east or west, to give a reading smaller than 90°.The letter N or S preceding the angle, and E or W following it shows the proper quadrant. Thus, a properly expressed bearing includes quadrant letters and an angular value. An example is N80°E. In below figure: Comparison of Azimuths and Bearings Example: The first course of a boundary survey is written as N37°13.W. What is its equivalent azimuth? Solution Since the bearing is in the northwest quadrant, the azimuth is 360° - 37°13. = 322°47. Back Azimuths and Back Bearings The back azimuth or back bearing of a line is the azimuth or bearing of a line running in the reverse direction. The azimuth or bearing of a line in the direction in which a survey is progressing is called the forward azimuth or forward bearing. The azimuth or bearing of the line in the direction opposite to that of progress is called the back azimuth or back bearing. The back azimuth can be obtained by adding 180° if the azimuth is less than 180° or by subtracting 180° if the azimuth is greater than 180°. The back bearing can be obtained from the forward bearing by changing the first letter from N to S or from S to N and the second letter from E to W or from W to E. The bearing of the line A-B is N 68° E. The bearing of the line B-A is S 68° W. The azimuth of the line A-B is 68°. The azimuth of the line B-A is 248°. Example: Solution Example 2: determine azimuth for each line? Example 3: The angles at the stations of a closed traverse ABCDEFA were observed as given below: Traverse station Interior angle A 120°35′00′′ B 89°24′20′′ C 131°01′00′′ D 128°02′20′′ E 94°54′40′′ F 155°59′20′′ calculate the azimuth of the traverse lines in the following systems if azimuth of the line AB is 42°: Magnetic declination is the horizontal angle observed from the geodetic meridian to the magnetic meridian. Navigators call this angle variation of the compass; the armed forces use the term deviation. An east declination exists if the magnetic meridian is east of geodetic north; a west declination occurs if it is west of geodetic north. East declinations are considered positive and west declinations negative. The relationship between geodetic north, magnetic north, and magnetic declination is given by the expression Geodetic azimuth = magnetic azimuth + magnetic declination Methods of Measuring Angles Angles normally are measured with a theodolite or total station system, but also can be determined by means of a tape or a compass. Tapes: Here the angle is not directly measured rather calculated from the measurement of distance. The accuracy depends on the accuracy of measurement of the distance. For acute angles on level ground, the error needs not exceed 05’to 10’. For obtuse angle, the corresponding acute angle should be determined. This method is generally slow and is used in absence of direct measuring instruments and as check. Theodolite: are used for measuring both vertical and horizontal angles with an accuracy of 10’’for horizontal and 1’ for vertical angles. Total station: incorporate both angle and distance measurement. The accuracy of measuring angles is 20’’to 1’’. Compass: In compass survey, the direction of the survey line is measured by the use of a magnetic compass while the lengths are by chaining or taping. Where the area to be surveyed is comparatively large, the compass survey is preferred. However, where the compass survey is used, care must be taken to make sure that magnetic disturbances are not present. The two major primary types of survey compass are: the prismatic compass and surveyors compass Compass surveys are mainly used for the rapid filling of the detail in larger surveys and for explanatory works. It does not provide a very accurate determination of the bearing of a line as the compass needle aligns itself to the earth’s magnetic field which does not provide a constant reference point. 1. Prismatic compass: This is an instrument used for the measurement of magnetic bearings. It is small and portable usually carried on the hand. This Prismatic Compass is one of the two main kinds of magnetic compasses included in the collection for the purpose of measuring magnetic bearings, with the other being the Surveyor's Compass. The main difference between the two instruments is that the surveyor's compass is usually larger and more accurate instrument, and is generally used on a stand or tripod. 2. Surveyor’s Compass: Similar to the prismatic compass but with few modifications, the surveyors compass is an old form of compass used by surveyors. It is used to determine the magnetic bearing of a given line and is usually used in connection with the chain or compass survey.  Introduction Kinds of Horizontal Angles Reference Meridian Bearing and Azimuth Angle Angle Measuring Instruments. 1- Introduction An angle is determined by three elements: 1. reference line. 2. direction of turning; clockwise (cw) or counterclockwise (ccw). 3. angle magnitude. Units of angle measurement are three: 1. Sexagesimal units of degrees, minutes, and seconds. 2. centesimal units of grads(gons). 3. radians. 2- Kinds of Horizontal Angles A- Angles to the right: Measured clockwise from the rear station to the forward station with magnitudes from 0 to 360. B- Angles to the left: measured counter clockwise with similar magnitudes. C- Interior angles: on the inside of a closed polygon, they can be angles to the right or to the left. (Interior angles) = (n-2) x180 D- Exterior angles are on the outside of a closed polygon, they can be angles to the right or to the left. (Exterior angles) =(n+2) x180 E- Deflection angles: Measured clockwise (R), or counterclockwise (L) from an extension of the back line to the forward station. Magnitudes are from 0 to180 preceded by R or L letters Direction A- Reference Meridian Direction It is the north-south reference line to which directions of lines are referred. Four kinds of meridians are used: 1. True(astronomic): it defines the direction of earth's geographic poles. 2. Magnetic meridian: it is defined by a freely suspended magnetic needle. 3. Grid meridian: it is the direction of true north for a selected central meridian of any coordinate system. 4. Assumed meridian: it is defined by the user in an arbitrary manner B- Bearing & Backward Bearing Direction It is the acute horizontal angle measured from either the north or south. Bearing is written as a number ≤ 90º with letter N or S preceding it, and with letter E or W following it. C- Azimuth & Back word Azimuth Direction It is the clockwise angle measured from the north direction of the reference meridian. Azimuth angle can take values from 0 to 360.  3- Angle Measuring Instruments Compass Sextant Theodolite Total station THEODOLITE SURVEY Theodolite Survey: Use of theodolite, temporary adjustment, measuring horizontal and vertical angles, theodolite traversing. Theodolites are perhaps the most universal surveying instruments; their primary use is for accurate measurement or layout of horizontal and vertical angles. A similar instrument common in the U.S.A. is called "Transit". How Does a Theodolite Work? A theodolite works by combining optical plummets (or plumb bobs), a spirit (bubble level), and graduated circles to find vertical and horizontal angles in surveying. An optical plummet ensures the theodolite is placed as close to exactly vertical above the survey point. The internal spirit level makes sure the device is level to the horizon. The graduated circles, one vertical and one horizontal, allow the user to actually survey for angles. Theodolites may be classified as: Optical-reading theodolites Optical-reading repeating theodolites Optical-reading directional theodolites Electronic Digital Theodolite BASIC OPERATION PRINCIPLE OF A THEODOLITE The main principle of every theodolite operation is a selected basic axial configuration according to certain requirements. Basic Axes of theodolite A. Theodolite Vertical Axis B. Theodolite Horizontal Axis C. Theodolite Collimation Axis APPLICATIONS Measuring horizontal and vertical angles. Locating points on a line. Prolonging survey lines. Finding difference of level. Setting out grades Ranging curves Tacheometric Survey Measurements of Bearings Measuring angles Repetition method Directional method Closing the horizon method (Reiteration method) EX 1 Angle to the Theo st. Obs st Telescope D or R H.C.R Right ABC D 0° 00´ 00´´ A R 180° 00´ 05´´ D 75° 09´ 30´´ 75° 09´ 30´´ C R 255° 09´ 38´´ 75° 09´ 33´´ B D 90° 00´ 00´´ A R 269° 59´ 51´´ C D 165° 09´ 32´´ 75° 09´ 32´´ R 345° 09´ 28´´ 75° 09´ 37´´ ‫ وكذلك الخطا القياسي المحتمل لها بأتباع التالي‬ABC ‫باالمكان حساب افضل قيمة للزاوية االفقية‬ ‫نحسب المعدل الحسابي للزاوية‬ X1 + X2 + X3 + X4 75° 09´ 30´´ + 75° 09´ 33´´ + 75° 09´ 32´´ + 75° 09´ 37´´ ̅= X = 𝑛 4 ABC = 75° 09´ 32´´ H1 = X1 - X̅ = 75° 09´ 30´´ - 75° 09´ 33´´=-3 H2 = X2 - X̅ = 75° 09´ 33´´ - 75° 09´ 33´´= 0 H3 = X3 - X̅ = 75° 09´ 32´´ - 75° 09´ 33´´=-1 H4 = X4 - X̅ = 75° 09´ 37´´ - 75° 09´ 33´´=+4 √(𝑣1 2 + 𝑣2 2 + 𝑣3 2 + 𝑣4 2 √(−32 + 02 + −12 + +42 𝛿𝑥𝑖 = = = ±2.944 𝑛−1 4−1 𝛿𝑥𝑖 2.944 δx = =± = ±1.47´´ √𝑛 √4 ∴ 𝐻 𝐴𝐵𝐶 = 75° 09´ 33´´ ± 1.5´´ Ex2: Vertical angle Theo st. Obs st Telescope D or R V.C.R (v) D 70 00 10 19 59 50 R 289 59 44 19 59 44 B A D 70 00 12 19 59 48 R 289 59 42 19 59 42 X̅1+ X̅2 + X̅3 + X̅4 ̅ X= = 19 59 46 𝑛 V1 = X1 - X̅ = 4 V2 = X2 - X̅ = -2 V3 = X3 - X̅ = -2 V4 = X4 - X̅ = -4 √(𝑣1 2 + 𝑣2 2 + 𝑣3 2 + 𝑣4 2 𝛿𝑥𝑖 = = ±3.65 𝑛−1 𝛿𝑥𝑖 3.65 δx = =± = ±1.8´´ √𝑛 √4 ∴ 𝑉𝐴 = 19° 59´ 46´´ ± 1.8´´  Traversing Definition of a traverse A traverse is a series of consecutive lines whose ends have been marked in the field and whose lengths and directions have been determined from observations. In traditional surveying by ground methods, traversing, the act of marking the lines, that is, establishing traverse stations and making the necessary observations, is one of the most basic and widely practiced means of determining the relative locations of points. - There are two kinds of traverses: closed and open. Two categories of closed traverses exist: polygon and link. - In the polygon traverse, the lines return to the starting point, thus forming a closed figure that is both geometrically and mathematically closed. Link traverses finish upon another station that should have a positional accuracy equal to or greater than that of the starting point. The link type (geometrically open, mathematically closed) must have a closing reference direction, - An open traverse (geometrically and mathematically open) consists of a series of lines that are connected but do not return to the starting point or close upon a point of equal or greater order accuracy. Observation of Traverse Angles or Directions The methods used in observing angles or directions of traverse lines vary and include (1) interior angles, (2) angles to the right, (3) deflection angles, and (4) azimuths. These are described in the following subsections. 1. Traversing by Interior Angles Although interior angles could be observed either clockwise or counterclockwise, to reduce mistakes in reading, recording, and computing, they should always be turned clockwise from the backsight station to the foresight station. For example, angle EAB of below figure was observed at station A, with the backsight on station E and the foresight at station B. - Interior angles may be improved by averaging equal numbers of direct and reversed readings. 2. Traversing by Angles to the Right Depending on the direction of the traversing, angles to the right may be interior or exterior angles in a polygon traverse. If the direction of traversing is counter clockwise around the figure, then clockwise interior angles will be observed. However, if the direction of traversing is clockwise, then exterior angles will be observed. Data collectors generally follow this convention when traversing. Thus, in below figure, for example, the direction from A to B, B to C, C to D, etc., is forward. By averaging equal numbers of direct and reversed readings, observed angles to the right can also be checked and their accuracy improved. 3. Traversing by Deflection Angles A deflection angle is not complete without a designation R or L, and, of course, it cannot exceed 180°. Each angle should be doubled or quadrupled, and an average value determined. The angles should be observed an equal number of times in face left and face right to reduce instrumental errors. Deflection angles can be obtained by subtracting 180° from angles to the right. Positive values so obtained denote right deflection angles; negative ones are left. 4. Traversing by Azimuths As shown in the below figure, azimuths are observed clockwise from the north end of the meridian through the angle points. The instrument is oriented at each setup by sighting on the previous station with either the back azimuth on the circle (if angles to the right are turned) or the azimuth (if deflection angles are turned).Then the forward station is sighted. The resulting reading on the horizontal circle will be the forward line’s azimuth. Observation of Traverse Lengths The length of each traverse line (also called a course) must be observed, and this is usually done by the simplest and most economical method such as tape, total station etc. In closed traverses, each course is observed and recorded as a separate distance. On long link traverses for highways and railroads, distances are carried along continuously from the starting point using stationing. for example, beginning with station 0+00 at point A, 100-m stations (1+00, 2+00, 3+00) are marked until hub B at station 4+00 is reached., and the end 8 + 19.60. The length of a line in a stationed link traverse is the difference between stationing at its end points; thus, the length of line BC is 819.60 - 400.00 = 419.60 m. Referencing Traverse Stations Traverse stations often must be found and reoccupied months or even years after they are established. Also they may be destroyed through construction or other activity. Therefore, it is important that they be referenced by creating observational ties to them so that they can be relocated if obscured or re-established if destroyed Angle Misclosure The angular misclosure for an interior-angle traverse is the difference between the sum of the observed angles and the geometrically correct total for the polygon. The sum, Σ, of the interior angles of a closed polygon should be Σ = (n - 2) 180° Where n is the number of sides, or angles, in the polygon. The sum of the angles in a triangle is 180°; in a rectangle, 360°; and in a pentagon, 540°.Thus, each side added to the three required for a triangle increases the sum of the angles by 180°. If the direction about a traverse is clockwise when observing angles to the right, exterior angles will be observed. In this case, the sum of the exterior angles will be Σ = (n + 2) 180° Figure shows a five-sided figure in which, if the sum of the observed interior angles equals 540°00'05" the angular misclosure is 5". Misclosures result from the accumulation of random errors in the angle observations. Permissible misclosure can be computed by the formula 𝑐 = 𝐾√𝑛 Where n is the number of angles, and K a constant that depends on the level of accuracy specified for the survey. Sources of Error in Traversing Some sources of error in running a traverse are: 1. Poor selection of stations, resulting in bad sighting conditions caused by (a) alternate sun and shadow, (b) visibility of only the rod’s top, (c) line of sight passing too close to the ground, (d) lines that are too short, and (e) sighting into the sun. 2. Errors in observations of angles and distances. 3. Failure to observe angles an equal number of times direct and reversed. Mistakes in Traversing Some mistakes in traversing are: 1. Occupying or sighting on the wrong station. 2. Incorrect orientation. 3. Confusing angles to the right and left. 4. Mistakes in note taking. 5. Misidentification of the sighted station. Balancing Angles In elementary methods of traverse adjustment, the first step is to balance (adjust) the angles to the proper geometric total. For closed traverses, angle balancing is done readily since the total error is known, although its exact distribution is not. Angles of a closed traverse can be adjusted to the correct geometric total by applying one of two methods: 1. Applying an average correction to each angle where observing conditions were approximately the same at all stations. The correction for each angle is found by dividing the total angular misclosure by the number of angles. 2. Making larger corrections to angles where poor observing conditions were present. Of these two methods, the first is almost always applied. Example For the traverse of the figure, the observed interior angles are given in the table. Compute the adjusted angles using methods 1 and 2. Departures and Latitudes After balancing the angles and calculating preliminary azimuths (or bearings), traverse closure is checked by computing the departure and latitude of each line. As illustrated in the figure, the departure of a course is its orthographic projection on the east-west axis of the survey and is equal to the length of the course multiplied by the sine of its azimuth (or bearing) angle. Departures are sometimes called eastings or westings. The latitude of a course is its orthographic projection on the north-south axis of the survey, and is equal to the course length multiplied by the cosine of its azimuth (or bearing) angle. Latitude is also called northing or southing. In equation form, the departure and latitude of a line are: 𝐷𝑒𝑝𝑎𝑟𝑡𝑢𝑟𝑒 = L × sin 𝛼 L𝑎𝑡𝑖𝑡𝑢𝑑𝑒 = L × cos 𝛼  - In traverse calculations, east departures and north latitudes are considered plus; west departures and south latitudes, minus. Traverse Linear Misclosure and Relative Precision - All angles and distances were measured perfectly, the algebraic sum of the departures of all courses in the traverse should equal zero. Likewise, the algebraic sum of all latitudes should equal zero. - Because the observations are not perfect and errors exist in the angles and distances, the conditions just stated rarely occur. The amounts by which they fail to be met are termed departure misclosure and latitude misclosure. Their values are computed by algebraically summing the departures and latitudes, and comparing the totals to the required conditions. - The linear misclosure of the traverse is calculated from the following formula: - The relative precision of a traverse is expressed by a fraction that has the linear misclosure as its numerator and the traverse perimeter or total length as its denominator, or Example Based on the preliminary azimuths from table and lengths shown in the below, calculate the departures and latitudes, linear misclosure, and relative precision of the traverse. The relative precision for this traverse is TRAVERSE ADJUSTMENT For any closed traverse, the linear misclosure must be adjusted (or distributed) throughout the traverse to “close” or “balance” the figure. This is true even though the misclosure is negligible in plotting the traverse at map scale. There are several methods available for traverse adjustment, but the one most commonly used is the compass rule (Bowditch method). As noted earlier, adjustment by least squares is a more advanced technique that can also be used. The compass, or Bowditch, rule adjusts the departures and latitudes of traverse courses in proportion to their lengths. Corrections by this method are made according to the following rules: Example Using the preliminary azimuths and lengths from previous table, compute departures and latitudes, linear misclosure, and relative precision. Balance the departures and latitudes using the compass rule. Solution: The correction in departure for AB is And by Equation (10.6) the correction for the latitude of AB COORDINATES Normally, plane rectangular coordinate system (Cartesian plane) having x-axis in east-west direction and y-axis in north-south direction, is used to define the location of the traverse stations. The y-axis is taken as the reference axis and it can be (a) true north, (b) magnetic north, (a) National Grid north, or (d) a chosen arbitrary direction. Usually, the origin of the coordinate system is so placed that the entire traverse falls in the first quadrant of the coordinate system and all the traverse stations have positive coordinates as shown in figure. Given the X and Y coordinates of any starting point A, the X coordinate of the next point B is obtained by adding the adjusted departure of course AB to XA. Likewise, the Y coordinate of B is the adjusted latitude of AB added to YA. In equation form this is Alternative Methods for Making Traverse Computations A. Balancing Angles by Adjusting Azimuths or Bearings Example: Table lists observed angles to the right for the traverse of figure. The azimuths of lines A-AzMk1 and E-AzMk2 have known values of 139°05' 45" and 86°20'47" respectively. Compute unadjusted azimuths and balance them to obtain geometric closure.  B. Balancing Departures and Latitudes by Adjusting Coordinates Example The table lists the preliminary azimuths and observed lengths for the traverse of previous example. The known coordinates of stations A and E are XA = 12,765.48, YA = 43,280.21, XE = 14,797.12, and YE = 44,384.51. Adjust this traverse for departure and latitude misclosures by making corrections to preliminary coordinates. Solution: Inversing If the departure and latitude of a line AB are known, its length and azimuth or bearing are readily obtained from the following relationships: 𝑑𝑒𝑝𝑎𝑟𝑡𝑢𝑟𝑒 𝐴𝐵 tan 𝑎𝑧𝑖𝑚𝑢𝑡ℎ (𝑜𝑟 𝑏𝑒𝑎𝑟𝑖𝑛𝑔) 𝐴𝐵 = 𝑙𝑎𝑡𝑖𝑡𝑢𝑑𝑒 𝐴𝐵 𝑑𝑒𝑝𝑎𝑟𝑡𝑢𝑟𝑒 𝐴𝐵 𝑙𝑎𝑡𝑖𝑡𝑢𝑑𝑒 𝐴𝐵 𝑙𝑒𝑛𝑔𝑡ℎ 𝐴𝐵 = = sin 𝑎𝑧𝑖𝑚𝑢𝑡ℎ (𝑜𝑟 𝑏𝑒𝑎𝑟𝑖𝑛𝑔)𝐴𝐵 cos 𝑎𝑧𝑖𝑚𝑢𝑡ℎ (𝑜𝑟 𝑏𝑒𝑎𝑟𝑖𝑛𝑔)𝐴𝐵 𝑙𝑒𝑛𝑔𝑡ℎ 𝐴𝐵 = √(𝑑𝑒𝑝𝑎𝑟𝑡𝑢𝑟𝑒 𝐴𝐵)2 + (𝑙𝑎𝑡𝑖𝑡𝑢𝑑𝑒 𝐴𝐵)2 It can be written to express departures and latitudes in terms of coordinate differences as follows: 𝑑𝑒𝑝𝑎𝑟𝑡𝑢𝑟𝑒 𝐴𝐵 = 𝑋𝐵 − 𝑋𝐴 = ∆𝑋 𝑙𝑎𝑡𝑖𝑡𝑢𝑑𝑒 𝐴𝐵 = 𝑌𝐵 − 𝑌𝐴 = ∆𝑌 ∆𝑋 ∆𝑌 𝑙𝑒𝑛𝑔𝑡ℎ 𝐴𝐵 = = sin 𝑎𝑧𝑖𝑚𝑢𝑡ℎ (𝑜𝑟 𝑏𝑒𝑎𝑟𝑖𝑛𝑔)𝐴𝐵 cos 𝑎𝑧𝑖𝑚𝑢𝑡ℎ (𝑜𝑟 𝑏𝑒𝑎𝑟𝑖𝑛𝑔)𝐴𝐵 𝑙𝑒𝑛𝑔𝑡ℎ 𝐴𝐵 = √(∆𝑋)2 + (∆𝑌)2 Computing Final Adjusted Traverse Lengths and Directions In traverse adjustments, corrections are applied to the computed departures and latitudes to obtain adjusted values. These in turn are used to calculate X and Y coordinates of the traverse stations. By changing departures and latitudes of lines in the adjustment process, their lengths and azimuths (or bearings) also change. Example Calculate the final adjusted lengths and azimuths of the traverse as shown in figure from the adjusted departures and latitudes listed in the table. Example: Using coordinates, calculate adjusted lengths and azimuths for the traverse of previous example. Solution: Example: Compute the length and azimuth of closing line AE and deflection angle α of the figure, given the following observed data: