ENGINEERING SURVEYING (221 BE)
Distance & Angle
Sr Tan Liat ChoonEmail: [email protected]
Mobile: 016-4975551
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Introduction
Types of Measurements in Surveying:
• Surveying is the art of making suitable measurements in horizontal or vertical planes. This is one of the important subjects of civil engineering. Without taking a survey of the plot where the construction is to be carried out, the work cannot begin
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Introduction
From the above definition, we conclude on two types of measurements in surveying. They are as follows:
• Linear measurements• Angular measurements
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Introduction
Linear measurements are further classified as follows:
• Horizontal Distance• Vertical Distance
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Introduction
Horizontal Distance
• A horizontal distance is measured in horizontal plane if a distance is measured along a slope, it is reduced to its horizontal equivalent
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Introduction
Vertical Distance
• A vertical distance is measured along the direction of gravity at that point. The vertical distance is measured to determine difference in elevations in various points
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Introduction
Angular Measurements
• Two sides meeting at an angle are measured. The angle between them is measured and represented in degrees or radians
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Distance Measurement
Linear measurement is the basis of all surveying and even through angles may be read precisely, the length of at least one line in tract must be measured to supplement the angles in locating points
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Distance Measurement
Is generally regarded as the most fundamental of all surveying observations
Many angles may be read, the length of at least one line must be observed to supplement the angles in locating points
In plane survey, the distance between two points means the horizontal distance
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Distance Measurement
Methods of measuring a horizontal distance:
• Tacheometry (Stadia), Taping, EDM and GPS• Distance from stadia: (High wire – Low wire) *
100 = Distance• EDM & GPS are most common in today’s survey• Direct measurement• Indirect measurement
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Distance Measurement
As simple as ABC:
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New building site – how big is it?
60.159 m
60.159 m
32.579 m
32.579 m
Distance Measurement
Measurement must be straight:
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Distance Measurement
Measurement around obstacle:
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starting point closing pointObstacle to sight line
Distance Measurement
Measurement around obstacle:
Horizontal angles α and β are used to transform the resulting horizontal lengths to an equivalent horizontal length along the measurement line
H 2-4 = h 2-3 cos α + h 3-4 cos β
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1
βα
54
3
2
Distance Measurement
Measurement must be straight:
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Distance Measurement
Measurement must be straight:
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Distance Measurement
Measurement must be straight:
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Distance Measurement
Generally, measurements are made horizontally, but on even, often man-made slopes the distance can be measured directly on the slope, but the vertical or zenith angle must be obtained
V = Vertical DistanceS = Slope DistanceH = Horizontal Distance
18A
SV
H
B
C
Distance Measurement
c2 = a2 + b2
C = 90 = A + Bsin A = a/ccos A = b/ctan A = a/b
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A
ca
b
B
C
Distance Measurement
A + B + C = 180
Sine Rule: a/sin A = b/sin b = c/sin C
Cosine Rule: a2 =b2 + c2 – 2bc cos Acos A = (b2 = c2 - a2)/2bc
Area = 1/2 bc sin A = √ s(s - a) (s - b) (s – c)where s = (a + b + c)/2 20
A
c
b
aB
C
Introduction On Angle Measurement
Measuring distances alone in surveying does not establish the location of an object. We need to locate the object in 3 dimensions. To accomplish that we need:
• Horizontal length (distance)• Difference in height (elevation)• Angular direction
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Introduction On Angle Measurement
Angles measured in surveying are classified as either horizontal or vertical, depending on the plane in which they are observed
Horizontal angles are the basic observations needed for determining bearing and azimuths
Vertical angles are used in trigonometric levelling stadia and for reducing slope distances to horizontal
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Introduction On Angle Measurement
Determining the locations of points and orientations of lines frequently depends on measurements of angles and directions
In surveying, directions are given by azimuths and bearings
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Introduction On Angle Measurement
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
A vertical angle is formed by two intersecting lines in a vertical plane, one of these lines horizontal
A zenith angle is the complementary angle to the vertical angle and is formed by two intersecting lines in a vertical plane, one of these lines directed toward the zenith
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Introduction On Angle Measurement
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Type Of Angle Measurement
Interior angles are measured clockwise or counter-clockwise between two adjacent lines on the inside of a closed polygon figure
Exterior angles are measured clockwise or counter-clockwise between two adjacent lines on the outside of a closed polygon figure
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)
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Type Of Angle Measurement
Angles to the right are turned from the back line in a clockwise or right hand direction to the ahead line
Angles to the left are turned from the back line in a counter-clock wise or left hand direction to the ahead line
Angles are normally measured with a transit or a theodolite, but a compass may be used for reconnaissance work
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Type Of Angle Measurement
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Angle Measurement
Angle is a difference in direction of 2 lines
To turn an angle we need a reference line, direction of turning and angular distance
Angular units:• Degree, minutes, second• Circle divided into 360 degrees• Each degree divided by 60 minutes• Each minute divided into 60 seconds
A check can be made because the sum of all angles in any polygon must equal. (n-2) * 180, where n is the number of angles
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Measuring Horizontal Angle
Set a bearing on the horizontal plate, and lock the upper motion
Release the lower motion, sight the backsight, lock the lower motion, and perfect the sighting with the lower tangent screw
Release the upper motion, turn to the foresight, lock the upper motion, and perfect the sighting
Record the horizontal bearing
Release the lower motion, invert the scope and point to the backsight in the reverse position, lock the lower motion, and perfect the sighting
Release the upper motion, turn to the foresight, lock the upper motion, and perfect the sighting
Record the second bearing30
Measuring Zenith Angle
Point the instrument to the target object in a direct position
Lock the vertical motion, perfect the sighting and record the zenith angle
Loosen both the horizontal and vertical motions, plunge the scope, rotate the alidade 180° and re-point to the target in the reverse position
Lock the vertical motion, perfect the pointing and record the zenith angle
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Measuring Zenith Angle
Direct 83 28’ 16”Reserve 276 31’ 38”
Sum 359 59’ 54”360 Minus Sum 00 00’ 06”Half Value (error) 00 00’ 03”
Plus Original Angle 83 28’ 16”Final Angle 83 28’ 19”
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Useful concepts
Degrees: full circle = 360°
0°
45°
90°
135°
180°
225°
270°
315°
Bearing and Azimuth
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Comparison Of Azimuth and Bearing
Because bearings and azimuths are encountered in so many surveying operations, there is important to know the conversion of these two.
Example 1The azimuth of a boundary line is 128 13’ 46”. Convert this into bearing.
The azimuth places the line in the southeast quadrant. Thus the bearing angle is:180 13’ 46” - 128 13’ 46” = 51 46’ 14”,and the equivalent bearing is S 51 46’ 14” E
Example 2The first course of a boundary survey is written as N 37 13’ W. What is its equivalent azimuth?
Since the bearing is in the northwest quadrant, the azimuth is:360 - 37 13’ = 322 47’
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Comparison Of Azimuth and Bearing
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Azimuths Bearings
Vary from 0° to 360° Vary from 0° to 90°
Require only a numerical value Require two letters and a numerical value
May be geodetic, astronomic, magnetic, grid, assumed, forward or back
May be geodetic, astronomic, magnetic, grid, assumed, forward or back
Are measured clockwise only Are measured clockwise and counterclockwise
Are measured either from north only, or from south only on a particular survey
Are measured from north and south
Comparison Of Azimuth and Bearing
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Quadrant Formulas for computing bearing angles from azimuths
I (NE) Bearing = Azimuth
II (SE) Bearing = 180° - Azimuth
III (SW) Bearing = Azimuth - 180°
IV (NW) Bearing = 360° - Azimuth
Comparison Of Azimuth and Bearing
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Example directions for lines in the four quadrants (Azimuths from north)
Azimuths Bearings
54° N 54° E
112° S 68° E
231° S 51° W
345° N 15° W
Azimuth
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Line Azimuth
O – A 54
O – B 133
O – C 211
O – D 334
Computing Azimuth
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N
B
A
CN
Computing Azimuth41 35’ = AB
+ 180 00’ 221 35’ = BA
+ 129 11’ 350 46’ = BC
- 180 00’ 170 46’ = CB
+ 88 35’ 259 21’ = CD
- 180 00’ 79 21’ = DC
+ 132 30’ 211 51’ = DE
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211 51’ = DE- 180 00’
31 51’ = ED+ 135 42’
167 33’ = EF+ 180 00’
347 33’ = FE+ 118 52’
466 25’ - 360 = 106 25’ = FA+ 180 00’
286 25’ = AF+ 115 10’
401 35’ - 360 = 41 35’ = AB
When a computed azimuth exceeds 360 , the correct azimuth is obtained by merely subtracting 360
Bearing
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Line Bearing
O – A N 54 E
O – B S 47 E
O – C S 31 W
O – D N 26 W
Computing Bearing
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Line Bearing
AB N 25 W
BC N 68 E
CD S 17 W
DA S 62 W
Bearing Bearing of a line is the direction of the line Azimuth of a line is the horizontal angle between 2 lines
Designation of Bearings• Whole circle bearing• Reduced Bearing (RB) or quadrantal bearing (QB)• Fore Bearing (FB) or forward bearing (FB)• Back bearing or Backward bearing (BB)• Calculated bearing
Whole Circle bearing• Bearings measured from north in a clockwise direction is termed as whole circle
bearing• The value varies from 0 degrees to 360 degrees
Reduced bearing/Quadrantal bearing• The bearings measured either from the north or from the south towards east or
west whichever is nearer is known as reduced bearing• The values vary from 0 degrees to 90 degrees for a particular quadrant• It is also known as quadrantal bearing (QB)
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Bearing
Fore Bearing (FB)• The bearings measured in the progress of surveying i.e. in the forward
direction of survey lines is known as fore bearing or forward bearing
Back Bearing (BB)• The bearings measured in opposite to the progress of surveying i.e. in
backward direction of survey line is known as Backward Bearing
Observed Bearing• The bearings taken in a field with an instrument is known as Observed Bearing
Calculated Bearing• The bearings calculated from the field observation is known as calculated
bearing
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Rectangular Coordinate
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Polar Coordinate
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+P (r, θ)
y
x
r
υ
P (x,y)
y
x
Cartesian coordinates
Perpendicular axesOrigin at (0,0)
Coordinates increase right & up of origin
Coordinates decrease down & left of origin
(0,0)x +
+
-
-
Coordinate In A Plane
Cartesian coordinates
Coordinates of point given by bracketed pairs of numbers: (right,up)
(0,0)x
(3,4)x
(-3,-2)x
(x,y)(Easting,Northing)-depending on coordinate system used
Coordinate In A Plane
Often easier to avoid negative values by increasing origin coordinates
+
+
(1000,1000)x
(1004,1006)x
(1001,1002)x
(998,999)x
NOTE: Some countries (incl. Sweden) use on maps:y=East x=North
Others use opposite (e.g. (England, USA)
We’ll use (Easting,Northing)
Coordinate In A Plane
p0x
p1x
Find coordinates of p1 in relation to p0
Easting
Nor
thin
gFinding Coordinate
p0x
p1x
Referencebearing (N)
Instrument
p0 (instrument) has known coordinates
(0,0) for the moment
reference bearing is known (N)
p1 has a unknown coordinates
Finding Coordinate
өNor
thin
g =
d co
s(ө)
Easting = d sin(ө)
Use instrument to measure:d = (horizontal) distance p0-p1ө = angle between North & bearing of p1 from p0
p0
p1x
Referencebearing (N)
ө = bearing from reference
d = distance from p0 to p1
d
with trigonometry...
Polar Coordinates
Finding Coordinate
өNor
thin
g =
d co
s(ө)
Easting = d sin(ө)p0
x
p1x
Referencebearing (N)
d 10m
36.87°
Easting = d sin(ө)= 10 sin(36.87)= 10*0.6= 6m
Northing = d cos(ө)= 10 cos(36.87)= 10*0.8= 8m
ө also called the azimuth
Finding Coordinate
өNor
thin
g =
d co
s(ө)
Easting = d sin(ө)p0
x
p1x
Referencebearing (N)
d 10m
36.87°
(6,8)
p1(Easting) = p0(Easting) + (d sin(ө))
p1(Northing) = p0(Northing) + (d cos(ө))
So if p0=(1000,1000) then
p1(Easting,Northing) = (1006,1008)
Finding Coordinate
Surveying Coordinate
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For surveying we use a slightly different form of notation, instead of x, y,
+P ( E, N)
N
E
D
υ
We use E, N (Easting, Northing)
Surveying Coordinate
Note:
Easting is always quoted first and then Northing
Θ is always measured in a clockwise direction from North
Θ is known as the whole circle bearing
We must be able to convert from Rectangular to Polar and from Polar to Rectangular very quickly
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Surveying Coordinate
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Any line has two bearings:N
N
υPQ QP
Q
P
υ
We consider that the line PQ is a different line to line QP
Surveying Coordinate
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Given the coordinates of two points, calculate the distance and bearingbetween of the line joining them
Q N
υPQ P(E,N)
Q(E, N)
Q(1127.37) – P(937.77) =189.69
Q(850.04) – P(1341.50) = 491.46
b=491.46
a=189.69
sin A = a/cc2 = a2 + b2
c2 = 189.602 + 491.462
c=526.77
c (PQ) = 526.77sin A = 189.60/526.77
A = 21 05’ 45”
Azimuth PQ = 270 + 21 05’ 45” = 291 05’ 45”Bearing PQ = N 68 54’ 15” W
A
Point Easting (E) Northing (N)
P 1341.50 937.77
Q 850.04 1127.37
Local Attraction
The deflection of a magnetic needle from its true position due to the presence of magnetic influencing material such as iron rod, magnetic rock, underground pipeline, electric cables, iron pipes and electric pole in its vicinity is called local attraction
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Method of Correcting the Bearing
There are two methods of correcting the bearing affected by local attraction:
• Included angle method• Error computation method
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Method of Correcting the Bearing
Included angle method:
• The included angles of the traverse are calculated first, then starting from the line which is unaffected by local attraction and using the included angles, the corrected bearings of the traverse are computed
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Method of Correcting the Bearing
Error computation method:
• The direction and the amount of local attraction at each survey station is determined
• Start from the line which is unaffected by local attraction, the corrected bearing of the traverse are computed
• More accurate than the included angle method
• It is adopted by most of the surveyors
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Question
Give the comparison between Azimuth and BearingFrom the figure below, please give the bearings of lines AB and BC.
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N
B
A
CN
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