CHAPTER 4: ANGLES & DIRECTIONS Definitions Meridians longitude lines True meridians converge to meet at the pole. Grid meridians parallel to the central (true) meridian. Magnetic meridian using magnetized needles that points to the magnetic North. Horizontal angles deviation from the North direction or meridians. Measured by Vernier transit (20") Theodolite (1") Vertical Angles are referenced to: The horizon by plus (up) or minus (down) angles. The zenith: directly above the observer. The nadir: directly below the observer. Q 1. Indicate which angles shown in the above figure are horizontal angles. Vertical Angles are used in slope distance corrections in height determination Traverse: is a continues series of measured lines. Lines are measured by lengths and angles, and defined by coordinates. Closed polygons (traverse) Sum of interior angles = (n - 2) 180 Sum of exterior angles = (n + 2) 180 281 10’ + 217 11’ + 220 59’ + 284 21’ + 256 19’ = 1258 120’ = 1260 Deflection angle: From the prolongation of the back line to the forward line measured either to left (L) or to right (R). Azimuths & Bearings Azimuth: Direction of a line given by the angle measured clockwise from the North meridian. Range: 0 - 360 Q1. Calculate the Azimuths of lines 1-4. Bearing: Acute angle between N-S meridian and the line measured clockwise or counterclockwise. (1) N # # # E (2) S # # # E (3) S # # # W (4) N # # # W Q2. Calculate the Bearings of lines 5 - 8. Relationship between Bearings and Azimuths To convert from Azimuths to Bearings NE quadrant: Bearing = Azimuth SE quadrant: Bearing = 180 - Azimuth SW quadrant: Bearing = Azimuth - 180 NW quadrant: Bearing = 360 - Azimuth To convert from Bearings to Azimuths NE quadrant: Azimuth = Bearing SE quadrant: Azimuth = 180 - Bearing SW quadrant: Azimuth = 180 + Bearing NW quadrant: Azimuth = 360 - Bearing Q3. Convert calculated Azimuths of lines 1 – 4 to Bearings, and convert calculated Bearings of lines 5 – 8 to Azimuths. Reverse directions of lines To reverse Bearing: Reverse direction letters AB BA N S S N E W W E and angles stay as is. To reverse Azimuth if Azimuth < 180 add 180 if Azimuth 180 Subtract 180 Azimuth Computations 1: Check interior angles sum = (n - 2) 180 2: Counterclockwise (recommended) a- reverse Azimuth b- add next interior angle c- go to start and check 3: Clockwise a- find the Azimuth of the starting line (going clockwise) b- reverse Azimuth c- subtract interior angle d- go to start to check Note: you may need to add 360 to computations to facilitate subtraction. Example: Find the azimuths of all the lines of the traverse. 1- Check sum of interior angles = (n - 2) 180 of internal angles = 7850’ + 14249’ + 1391’ + 7539’ + 10341’ = 54000’ (n - 2) * 180 = (5 – 2) *180 = 54000’ OK 2: Counterclockwise solution Line AB Az AB = 360 00’ - 7850’ = 28110’ Az BA = 28110’ - 18000’ = 10110’ + B = 14249’ Az BC = 24359’ Az CB = 24359’ - 18000’ = 6359’ + C = 13901’ Az CD = 20300’ Az DC = 20300’ - 18000’ = 2300’ + D = 7539’ Az DE = 9839’ Az ED = 9839’ + 18000’ = 27839’ E = 10341’ Az EA = 38220’ = 2220’ Az AE = 2220’ + 18000’ = 20220’ A = 7850’ Az AB 28110’ OK. Q1. Solve the same problem in a clockwise solution. Bearing Computations Computations solely depend on the trigonometric relationships. The solution can proceed in either clockwise or anticlockwise procedures. There is no systematic method for bearings computations. Each bearing computation is regarded asd a separate problem. A neat and a well labeled diagram should accompany each computation. Example: Find the azimuths of all the lines of the traverse. Soln.: 1- Line AB N 58 W 2- Line BC <X = 180 – 140 = 40 Bearing Angle = 180 – 58 - 40 = 82 BC = S 82 W 3- Line CD <X = 180 – 128 = 52 Bearing Angle = 180 – 98 - 52 = 30 BC = S 30 W 4- Line DE <X = 30 Why? Bearing Angle = 180 – 84 - 30 = 66 BC = S 66 E 5- Line EA <X = 66 Why? <Y = 180 - 120 = 60 Bearing Angle = 180 – 60 - 66 = 54 BC = N 54 E Q1. Solve the same problem in a clockwise solution, and compare your answers with the obtained ones. Magnetic Direction Compass will always point in the direction of magnetic North Magnetic North is usually not the geographic North. The Magnetic North Pole is located about 1,000 miles south of the Geographic Pole. Magnetic declination: Horizontal angle between direction taken by compass and the geographic North. Magnetic direction is used only in the lowest order of survey. Maps are available to convert from Magnetic North to Geographic North. Movement of the magnetic North with time For extra information, here are some suggested web sites: http://members.tripod.com/norpolar/magno.html http://antwrp.gsfc.nasa.gov/apod/ap991019.html http://www.geolab.nrcan.gc.ca/geomag/northpole_e.shtml Chapter 5: Theodolites/Transits Older versions were called Transits. Nowadays, both words (Transits and Theodolites) are used interchangeably. Usage Measure horizontal angles (deviation from the North). Measure vertical angles (deviation from horizon, Nadir, or Zenith). Establish straight lines. Establish horizontal and vertical distances by using stadia. Establish difference in elevation when used as leveling machine. Major Parts Alidade Assembly: includes telescope, vertical circle and vernier, horizontal verniers to read horizontal angle, and clamps. Circle Assembly: consists of a horizontal circle that has a hole to fit the spindle of the alidade into it. Leveling Head: were the circle assembly fits on. Note: the circle assembly has two clamps Upper clamp: to tighten the alidade to the circle Lower clamp: to tighten the circle to the leveling head. Targets: are plates having their center marked. Types of Theodolites In terms of measuring operation: Repeating instruments: Can be zeroed, measure 1, 2, 3, … The circle assembly has two clamps (upper & lower) Direction instruments: Can not be zeroed. The circle assembly has just one clamp (upper) In terms of model: Engineer transit: Old USA Horizontal setting 0zenith Optical theodolite (Repeating): New USA & other countries Horizontal setting 90 or 270 zenith 0 zenith could be at the zenith or Nadir Electronic Theodolites: Similar to optical theodolites Precision is high Digital readouts (no interpolation) Zero-set buttons Horizontal angles can be turned left or right Automatic repeat - angle averaging Add EDM Total Station How to check if the theodolite is measuring in Nadir, Zenith, or from Horizon? Put telescope in a horizontal position and tilt it slightly up and check reading: If reading close to zero Reading from horizon If < 90 Zenith If > 90 Nadir Measuring Horizontal Angles Turning the angle at least twice (plunging\transiting the telescope) will eliminate mistakes, most instrument errors, and increase precision. A. Directional Theodolites: Directional Theodolites can’t be zeroed. 1. Theodolite at A 2. While instrument at Face-Left (FL), vertical circle on the left side of surveyor, target telescope at "L" point and record reading in the column FL (a) corresponding to point L. Position I Position II ST PT Mean Angle (FL) (FR) 276° 14’23” 96° 14’ 34” L 276° 14’ 28” (a) (d) A 31° 37’ 09” 307° 51’ 33” 127° 51’ 41” R 307° 51’ 37” (b) (c) 3. Go clockwise and target at "R" point and record reading in the column FL (b) corresponding to point R. The difference in the readings in the “FL” Column will be nearly equal to the value of the angle. 4. Plunge (transit) the telescope, now the instrument is Face Right (FR), vertical circle on the right side of surveyor. 5. While still targeting on "R", record reading in the column FR corresponding to point R (c). The difference between the FL and FR readings for the same point should be around 180. 6. Go anticlockwise and target on point "L" and record reading in the column FR corresponding to point L. 8. In the “Mean” column, take the mean of the minutes and seconds for each point and take the degrees for that point either from the FL or FR column. You have to stick to one of the positions, FL or FR, in the whole table for taking the degrees values. 9. The angle value is calculated by getting the difference between the two values in the “Mean” column. 10. The angle LÂR can be obtained by calculating difference in angels from position I (FL) and position II (FR). ST. PT. POSITION I (F.L.) POSITION II (F.R.) L 276° 14’ 22” 96° 14’ 34” A R 307° 51’ 33” 127° 51’ 41” Difference 31° 37’ 11” 31° 37’ 07” Mean 31° 37’ 09” B. Repeating Theodolites Repeating theodolites can be zeroed. 1. Theodolite at A 2. Zero instrument and target L 3. Go clockwise and target R 4. Record in (Direct) 5. Plunge telescope disengage lower motion gear 6. Target at L, record in (Direct) 7. Go clockwise and target R and record double 8. Take mean of double Mean = ST Direct Double Angle ’ A 13° 20’12” 26° 40 28” 13° 20’ 14” Example 1: Mean = ST Direct Double Angle 39’ A 78° 49’23” 157° 08” 78° 49’ 34” ’ B 142° 49’53” 285° 38 28” 142° 49’ 14” ’ C 139° 00’17” 278° 01 56” 139° 00’ 48” ’ D 75° 39’12” 151° 17 56” 75° 38’ 58” ’ E 103° 41’10” 207° 22 28” 103° 41’ 14” Summation 539° 59’ 58” Correct Summation of angles = (n-2) * 180 = 3 * 180 = 540° 00’ Angular error of closure = 540° 00’ 00" - 539° 59’ 58” = 02" Measuring Vertical Angles Vertical angles are angles measured in the vertical plane with zero or reference being a horizontal or a vertical line. That is, a vertical angle is not measured from a low point to a high point, but from the horizontal to the high point, a (+ve) vertical angle or an angle of elevation, and from the horizontal to the low point, a (-ve) vertical angle or an angle of depression.