coordinates and traverse

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

7.2.9 Transformation of Coordinates

Example 7.8


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.


7.3.2 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

coordinates and traverse

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