Download CHAPTER 5 Coordinate Geometry and Traverse Surveying

Dr. Mazen Abualtayef
Surveying
Chapter 5
Angles directions &
angle measuring
equipment
Content
5.1 Introduction
5.2 Horizontal, vertical and zenith angles
5.3 Reference direction
5.4 Reduced bearing of a line
5.5 Azimuth or whole circle bearing
5.6 Back reduced bearing and back azimuth
5.10 Main applications of the Theodolite
5.1 Introduction
Surveying aims to determine the relative location of points on
the surface of the Earth. To locate a point, both distance
and angular measurements are usually required. Angular
measurements are either horizontal or vertical and they are
accomplished with a Theodolites.
5.2 Horizontal, vertical & Zenith
angles
Horizontal Angles:
In the horizontal plane, direction of all lines of a survey are
referenced to the meridian (true or magnetic).
5.2 Horizontal, vertical & Zenith
angles
Vertical Angles:
A vertical angles is measured in a vertical plane in two ways depending
on the reference from which the angle is measured:
1- Elevation or depression angle: using the horizontal plane as a
reference (the value of the angle is between -90º to 90º)
a- when the point being sighted on is above the horizontal plane, the angle is
called an angle of elevation with positive value.
b- when the point being sighted on is below the horizontal plane, the angle is
called an angle of depression with negative value.
2- Zenith angle: using the overhead extension of the plumb line as a
reference line, its value ranges from 0º to 180º.
5.3 Reference Direction
It is convenient to choose a reference line to which
directions of all the lines of surveying are referenced.
Three types of direction lines are used in surveying.
These are:
•
true or geographic north,
•
the magnetic north and
•
the assumed north.
5.3 Reference Direction
5.3.1 True meridian ‫( خط الطول‬at any point): the great
circle that passes through that point and the geographic
north and south poles of the earth.
There are two methods to determine the north:
The watch method
The shadow method
B 10AM
D
C 2PM
5.3 Reference Direction
5.3.2 Magnetic meridian: a direction that the magnetic
needle takes when allowed to come to rest in the earth’s
magnetic needle.
Declination: The angle between the
magnetic and true meridian is called the
magnetic declination and expressed as the
angular distance east or west of the true
meridian, and its value is about 3º.
5.3.3 Assumed meridian: if neither the
geographic north, nor the magnetic north is
known in a certain local area for which the
surveyor is performing measurements, a line can
be arbitrarily chosen and assumed to be in the
direction the north .
5.4 Reduced Bearing of a Line
Bearing is the acute angle that the line makes with the
meridian, it expressed as north or south and how many
degrees to the east or west and its value ranges from 0º to
90º and called true bearing if the true meridian is used and
magnetic bearing if the magnetic meridian is used.
5.5 Azimuth or Whole Circle Bearing
Azimuth is the clockwise horizontal angle that the line
makes with the north end of the reference meridian, and
its value ranges from 0º to 360º. And called true azimuth if
the true meridian is used as the reference and called
magnetic azimuth if the magnetic meridian is used.
5.6 Back Reduced Bearing & Back
Azimuth
Back Reduced Bearing and Back Azimuth: Back
bearing or Back azimuth of a line going from A to B is the
azimuth or Bearing of the same line going from B to A.
N
Reference meridian
N
A
60º
60º
O
5.10 Main applications of the
theodolite
The main use of the theodolite is to measure horizontal
and vertical angles. These angles are used for the
calculation of object heights, distances and point
coordinates.
5.10.1 Measurement of Object Heights
Two cases can be distinguished:
•
The determination of the elevation of a point whose
horizontal distance to Theodolite can be measured
•
The determination of the elevation of a point whose
horizontal distance to Theodolite can’t be measured
5.10.1 Measurement of Object Heights
Case (1): Horizontal distance to Theodolite can be measured
D and  are measured and the elevation of building can be
calculated by
HC  H A  i  D  tan 
H is vertical distance b/w C & C”
 1
1 


H  Dtan α - tan β   D
 tan z1 tan z 2 
5.10.1 Measurement of Object Heights
Case (2): Horizontal distance to Theodolite can’t be measured
A’B’ distance, a, b,
 and  angles are
measured then
elevation of HC can
be calculated by
A' C' B' C' A' B'


sin b sin a sin c
A' B'
A' C' 
sin b
sin c
HC  H A  iA  A' C ' tan 
HC  H B  iB  B' C ' tan 
A' B'
B' C' 
sin a
sin c
5.10.2 Tacheometry
Tacheometry is a system of rapid surveying, by which
the positions, both horizontal and vertical, of points on
the earth surface relatively to one another are
determined without using a chain or tape or a separate
leveling instrument. (http://en.wikipedia.org/wiki/Tacheometry)
Two methods will be discussed here:
• Tangential method and
• Stadia method
5.10.2 Tacheometry
Tangential method
Measure vertical angles θ and Φ on a staff held vertically
at B at two points M and N.
OM  D tan θ
ON  D tan 
OM - ON b  D tan tan  - tan  
b
b
D

tan  - tan    1
1 


 tan z1 tan z 2 
ΔH  i  V  BN  i  D tan  t
5.10.2 Tacheometry
Stadia method
For this method, a Theodolite is equipped with a rectile that
has one vertical and three horizontal cross-wires should be
available.
Vertical Wire
Upper Wire
Middle Wire
Lower Wire
5.10.2 Tacheometry
Stadia method
5.10.2 Tacheometry
Stadia Geometry for Horizontal Sight
d F

r i
 F
D   r  F  C
i
D  kr
F is the focal length of lens.
F and i are constant.
(F+C) = 0.25 to 0.3m & can be neglected.
k is called a stadia coefficient and = 100 for most instruments
5.10.2 Tacheometry
Stadia Geometry for Inclined Sight
m' n'  mn cos , or r'  r cos
S  kr'F  C
5.10.2 Tacheometry
Stadia Geometry for Inclined Sight
V  S sin θ  kr cos sin   F  C sin
D  S cos θ  kr cos   F  C cos
2
1
V  kr sin 2
2
1
2
D  kr cos 
2
1
V  kr sin 2 z
2
1
D  kr sin 2 z
2
Example 5.2
Example 5.2
Example 5.2
Example 5.2