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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).
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.
Example: Find the azimuths of all the lines of the traverse.
1- Check sum of interior angles = (n - 2) 180
of internal angles = 78 50’ + 142 49’ + 139 1’ + 75 39’ + 103 41’ = 540 00’
(n - 2) * 180 = (5 – 2) *180 = 540 00’ OK
2: Counterclockwise solution
Line AB Az AB = 360 00’ - 78 50’ = 281 10’
Az BA = 281 10’ - 180 00’ = 101 10’
Az BC = 243 59’
Az CB = 243 59’ - 180 00’ = 63 59’
Az CD = 203 00’
Az DC = 203 00’ - 180 00’ = 23 00’
Az DE = 98 39’
Az ED = 98 39’ + 180 00’ = 278 39’ E = 103 41’
Az EA = 382 20’ = 22 20’
Az AE = 22 20’ + 180 00’ = 202 20’ A = 78 50’
Az AB 281 10’ 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
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.
B. Repeating Theodolites Repeating theodolites can be zeroed.
- Theodolite at A
- Zero instrument and target L
- Go clockwise and target R
- Record in (Direct)
- Plunge telescope disengage lower motion gear
- Target at L, record in (Direct)
- Go clockwise and target R and record double
- Take mean of double
ST Direct Double Mean = Angle
A 13° 20’12”^ 26° 40’^ 28”^ 13° 20’ 14”
Example 1:
ST Direct Double Mean = Angle
A 78° 49’23”^ 157°^ 39’^ 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.
Vertical angles are referred to the vertical line in modern instruments and called zenithal angles (or zenithal distances). If the angle lies between 0° and 90°, it is an angle of elevation (+ve), otherwise it is an angle of depression (-ve) (between 90° and 180°).
Vertical angles are subject to index error which results from:
a. Displacement of the vertical circle b. Lack of adjustment of the vertical circle reading device. The index error is eliminated by sighting in two positions.
3- Plunge (transit) telescope and turn required angle (3120'10"). 4- Locate point C’. 5- Hold the angle reading and sight again on A. 6- Release the angle reading, plunge the telescope and turn it the required angle again. The reading will be double the angle value (6240'20"). 7- Locate point C”. 8- Locate point C which is midway between C’ and C”.
ii) Laying out internal angles:
Laying out internal angles is a special case of external angles.
Q1. Write the procedure for laying out internal angles; include a sketch in your procedure.
II. Prolonging straight lines
Given Line AB required to prolong the line
Procedure: 1- Set Theodolite at B and sight on A. 2- Plunge telescope and locate point C’. 3- Rotate telescope and sight again on A. 4- Plunge telescope and locate point C”. 5- Locate point C which is midway between C’ and C”.
III. Interlining (Balancing in)
Interlining is establishing a straight line between two not inter-visible points.
Procedure: 1- Select a point “C” between the two points that you can see both points from it. 2- Roughly align C’ between A & B. 3- Sight at A. 4- Plunge telescope and locate point B’. 5- Measure BB’ & calculate CC’. 6- Shift Theodolite to C”. 7- Repeat steps 3 – 6 until A, B & C are aligned.
IV. Intersection of two straight lines
This is a common situation when the surveyor is required to locate the intersection point when laying out intersecting streets with the main street.
Procedure: 1- Set Theodolite @ main street center line (T1) and locate tow points (B & C) on it (about 1 m apart) to be on both sides of the intersection point. 2- Stretch a string between points B & C. 3- Move the Theodolite to the center line of the crossing street (T2). 4- Align the telescope with the center line. 5- The instrument man is now in a position to find the intersection point between his view sight and the stretched string.
