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Coordinates and Traverse computations
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Su
rve
yin
g
Chapter 7
Coordinate Geometry &
Traverse Surveying
Dr. Mazen Abualtayef
7.1 Introduction
7.2 Coordinate Geometry
7.3 Traverse Surveying
Content
7.1 Introduction
The engineering planning and design made the use of
coordinates to define geographic positions of survey points a
necessity.
This book uses the coordinate system utilized by the
Palestinian survey department where x-axis is taken to
coincide with the north direction, while the y-axis coincides
with the east direction.
y
x
i j
i( y i ,x i )
j( y j ,x j )
Horizontal coordinates only
7.2 Coordinate Geometry
7.2.1 The Inverse Problem
If the X and Y coordinate of two points are known, the
horizontal distance and the azimuth of the line joining them
can be computed as following:
dij = (xj xi) + (yj yi)
ij = tan-1 ((yj yi ) /( xj xi )) + C
C = 0 if y is positive and x is positive (1st quadrant).
C = 180 if y is positive and x is negative (2nd quadrant).
C = 180 if y is negative and x is negative (3rd quadrant).
C = 360 if y is negative and x is positive (4th quadrant).
y
x
1st quadrant
2nd quadrant3rd quadrant
4th quadrant
i j
i( y i ,x i )
j( y j ,x j )
Example 7.1
Given the following horizontal coordinates for points i & j
Xi = 181680.76 m. Yi = 174410.56 m.
Xj = 181810.22 m. Yj = 174205.31 m.
Compute the horizontal distance (dij) and azimuth (ij)
Solution
xj xi = 181810.22 181680.76 = 129.46 m
yj yi = 174205.31 174410.56 = -205.25 m
dij = (-205.25) + (129.46) = 242.67 m.
ij = tan-1 (-205.25 / 129.46)
= 21 33' 42" + 360 = 302 14' 29" (4th quadrant, C=360)
i
j
7.2.2 Location by angle and distance
i and j are two points of known coordinates, the
horizontal coordinate of a new point such as k can be
determined by measuring the horizontal angle and the
distance dik
ik = ij + (if it is larger than 360 then subtract 360)
xk = xi + dik sin ikyk = yi + dik cos ik
ij
ik
i
k
j
y
x
dik
Example 7.2
Given The information in example 7.1 and
= 111 27' 45" dik = 318.10 m
Compute the horizontal coordinates of point k.
Solution
ij = 302 14' 29"
ik = ij + = 302 14' 29" + 111 27' 45" = 413 27' 29"
= 413 27' 29" - 360 = 53 42' 14"
Yk = 174410.56 + 318.10 sin(53 42' 14") = 174666.94 m
Xk = 181680.76 + 318.10 cos(53 42' 14") = 181869.06 m
58
54
i
j
k
7.2.3 Locating the North direction at a point
Suppose you are standing with Theodolite or Total
Station at point i (with known coordinates) and you would
like to locate the direction of the north at it toward point j
(with known coordinate), perform the following steps:
1. Calculate the azimuth of line ij (ij).
2. Let the horizontal circle reading of your instrument
read the value of ij while sighting point j.
3. Rotate the instrument in a counterclockwise direction
till you read 0. It will be point at the north direction.
7.2.4 Locating by Distance and Offset
y
x
i
jp
k
o2o1
m
n
ij
If the point lie to the left of line ij, then the coordinates of point p
can be calculated from the following equations:
Yp = Yi + dim sin ij + o1 sin (ij 90) = Yi + dim sin ij - o1 cos ijXp = Xi + dim cos ij + o1 cos (ij 90) = Xi + dim cos ij + o1 sin ij
If the point lie to the right of line ij, then the coordinates of point
k calculated from the following equations:
Yk = Yi + din sin ij + o2 sin (ij + 90) = Yi + din sin ij + o2 cos ijXk = Xi + din cos ij + o2 cos (ij + 90) = Xi + din cos ij - o2 sin ij
Example 7.3
Given the following horizontal coordinates for points i & j
Xi = 1000.00 m Yi = 1000.00 m
Xj = 975.00 m Yj = 1050.00 m
An edge of a building k is located at a distance of 30.00 m
and an offset of 10.00 m to the right of line ij. Compute the
coordinate of point k.
Solution
Yk = 1000.0 + 30.0 sin(116 33 54) + 10.0 cos(116 33 54)
= 1022.36 m
Xk = 1000.0 + 30.0 cos(116 33 54) + 10.0 sin(116 33 54)
= 977.64 m
"54'3311618000.100000.975
00.100000.1050tan 1
ij
7.2.5 Intersection by Angles
The coordinate of a new point (k) can be determine
by measuring horizontal angles ( & ) from two
points of known coordinates ( i & j )
dik / sin = djk / sin = dij / sin (180--)
Yk = Yi + dik sin ikXk = Xi + dik cos ik
Or
Xk = Xj + djk sin jkYk = Yj+ djk sin jk
x
y
i
j
kij ik
dik
jk
djk
Example similar to 7.4
In the figure:
Xi = 5329.41 ft Yi = 4672.66 ft
Xj = 6321.75 ft Yj = 5188.24 ft
= 31 26' 30" = 42 33' 41"
Compute the horizontal coordinates Xk & Yk
Solution
Xj - Xi = 6321.75 5329.41 = 992.34 ft
Yj Yi = 5188.24 4672.66 = 515.58 ft
dij = (922.34) + (515.58) = 1118.29 ft
ij = tan-1 (992.34 / 515.58) = 62 32' 44"
ik = ij + = 62 32' 44" + 31 26' 30" = 93 59' 14"
180 - = 180 31 26' 30" 42 33' 41" = 105 59' 49"
dik = 1118.29 sin (42 33' 41) / sin (105 59' 49) = 786.86 ft
Xk = 5329.41 + 786.86 sin (93 59' 14" ) = 6114.37 ft
Yk = 4672.66 + 786.86 cos (93 59' 14" ) = 4617.95 ft
x
y
i
j
kij ik
dik
jk
djk
7.2.6 Intersection by distances
i
j
kij ik
dik
jk
djk
The coordinate of a new point (k) can be determined by
measuring distances (dik & djk) from two points of known
coordinates i & j
djk = dij + dik - 2 dij dik cos
= cos-1 (dij + dik - djk ) / 2 dij dik
Then the coordinates of k can be computed by section 7.2.2
y
x
P.S. Read example 7.5
7.2.7 Resection
As in the following figure, the horizontal position of a new
point like (P) can be determined by measuring the horizontal
angles to three points of known coordinates like: A & B & C
A
P
CB
NM
c b
R
Let J = + then J = 360 ( M+N+R )
Let H = sin / sin
The following steps to compute point P
coordinates:
1- compute AB & AC & b & c & R from the
known coordinates of points: A,B,C.
2- compute J = 360 ( M+ N+ R )
3- compute H = b sin M / c sin N
4- compute ( tan = sin J / (H + cos J ))
5- compute = 180 - N
6- compute AP = AC +
7- compute AP = b sin / sin N
8- compute Xp & Yp
Xp = XA + AP sin APYp = YA + AP cos AP
P.S. Read example 7.6
7.2.8 Mapping Details using EDM
Example 7.7
Example 7.7
i = 1.50 m, t = 1.60 m
Example 7.7
7.2.8 Mapping Details using EDM
7.2.8 Mapping Details using EDM
7.2.9 Transformation of Coordinates
7.2.9 Transformation of Coordinates
7.2.9 Transformation of Coordinates
7.2.9 Transformation of Coordinates
Example 7.8
Example 7.8
7.3 Traverse Surveying
Def: Traverse is one of the most commonly used
methods for determining the relative positions of a
number of survey points.
7.3.1 Purpose of the Traverse:
1- Property survey to establish boundaries.
2- Location and construction layout surveys for
highways, railways and other works.
3- Ground control surveys for photogrammetric mapping.
7.3 Traverse Surveying
7.3.2 Types of Traverse:
a- Open Traverse:
b- Closed Traverse:
7.3.3 Choice of Traverse Stations:
1- Traverse stations should be as close as possible to
the details to be surveyed.
2- Distances between traverse stations should be
approximately equal.
3- Stations should be chosen on firm ground .
4- When standing on one station, it should be easy to
see the BS and FS stations.
7.3.4 Traverse Computations and correction of errors
If B coordinates are known, then
C coordinates are:
xC = xB + dBC sin 2yC = yB + dBC cos 2
A- Azimuth of a line:
1- when ( 1 + f ) > 180
2 = f - ( 180 1) = f + 1 - 180
2- when ( 1 + f ) < 180
2 = f + 180 + 1 = f + 1 + 180
Checks and correction of errors:-B
X last point X first point = X all lines
Y last point Y first point = y all lines
In order to meet the previous two conditions, the following corrections are performed:
1- Angle correction:
a- Closed loop traverse:
For a closed traverse of n sides,
sum of internal angles = (n 2) 180
error = sum of measured angles ((n 2) 180)
correction = - error / no of internal angles
b- For both loop and connecting closed traverse: If the azimuth of the last line
in the traverse is known, then the error = c (calculated azimuth) - n (known azimuth)
correction / angle = - / n
the corrected azimuth i = i (initially computed azimuth) i ( / n)
2- Position correction:
IF the calculated and known coordinates of last point are:
(Xc,Yc) and (Xn,Yn) respectively, then
Closure error in x-direction (x) = Xc Xn
Closure error in y-direction (y) = Yc Yn
Closure error in the position of the last points = (x + y)How to correct and distribute this error?
Compass (Bowditch ) Rule: used for position correction as follow:
Correction to departure of line ij (x) = -(length of line ij / total length of traverse)( x)
Correction to departure of line ij (y) = -(length of line ij / total length of traverse)( y)
Correction can be done directly to coordinates:
Cxi = - (Li / D) ( x ) & Cyi = - (Li / D) ( y ) Where:
Li = the cumulative traverse distance up to station i
D = total length of the traverse
The corrected coordinates of station i ( x'i , y'i ) are:
X'i = Xi + Cxi & Y'i = Yi + Cyi
7.3.5 Allowable error in Traverse surveying
the following figure:
Example 7.9
y x
y
x
x y
Preliminary coordinates
Corrected coordinates
Final results
y y
x x
y x
Example 7.10
y y
x x