Surveying
(805210)
Prepared by:
Dr. Aslam Al-Omari
Chapter 5: Angle Measurement
5.1) Introduction
5.2) Units of Angle Measurements
5.3) Horizontal, Vertical, and Zenith Angles
5.4) True Bearings
5.5) Magnetic Bearings and Declination
5.6) Azimuths
5.7) Back Bearing and Back Azimuth
5.8) & 5.9) Magnetic and Gyro Compass
Chapter 5: Angle Measurement
5.10) Principal Elements of an Angle-Measuring
Instrument
5.11) Surveying Telescope
5.12) Parts of a Vernier Transit
5.19)-5.21) Scale-Reading, Digital, and Electronic
Theodolites
5.22) Setting up a Theodolite
5.1) Introduction
„
Distance and angular measurements are required to
fix the position of a point
„
Angular measurements: Horizontal & Vertical
„
Instruments: Transit or Theodolite
5.2) Units of Angle
Measurements
„
„
Sexagesimal System:
¾ Circumference is divided into 360 degrees (360°)
¾ One degree is divided into 60 minutes (1°=60′)
¾ One minute is divided into 60 seconds (1′=60″)
Centesimal System:
¾ Circumference is divided into 400 grads or grades
(400g)
¾ One grad is divided into 100 centesimal minutes
(1g=100c)
¾ One centesimal minute is divided into 100
centesimal seconds (1c=100cc)
5.2) Units of Angle
Measurements
„
The Radian (Rad):
„ One radian is defined as the angle at the center of
a circle that is subtended by an arc having exactly
the same length as the radius.
Arc Length = r γ
¾ r: Radius
¾ g: Central Angle
(in radians)
Circumference=2πr
¾ S:
„
5.2) Units of Angle
Measurements
„
To convert among the three systems:
360 D = 400 g = 2π (rad )
„
Example 5.1: Find the sum of these three angles
Centesimal System
100.4527g
251.7590g
312.0314g
Sum = 664.2431g
Or
264.2431g
Sexagesimal System
75°51′23″
207°18′41″
340°39′57″
Sum = 623°50′01″
Or
263°50′01″
5.2) Units of Angle
Measurements
„
Example:
D
⎛ 51 ⎞ ⎛ 23 ⎞
75 51′ 23′′ = 75 + ⎜ ⎟ + ⎜
⎟
⎝ 60 ⎠ ⎝ 60 × 60 ⎠
= 75 D + 0.85 D + 0.00638 D
D
D
= 75.85638
D
D
5.2) Units of Angle
Measurements
„
Example 5.2:
What is the sexagesimal equivalent of 264.2431g?
D
360
D
237
.
81879
=
264.2431 g = 264.2431 g ×
400 g
60 ′ ⎞
⎛
D
D
= 237 + ⎜ 0.81879 × D ⎟
1 ⎠
⎝
= 237 D + 49.1274 ′
60 ′′ ⎞
⎛
= 237 + 49 ′ + ⎜ 0.1274 ′ ×
⎟
1′ ⎠
⎝
= 237 D + 49 ′ + 7.644 ′′
D
= 237 D 49 ′7.644 ′′
5.2) Units of Angle
Measurements
„
Example 5.3:
What is the grad equivalent of 263°50′01″?
D
D
′
⎛
⎞
⎛
⎞
1
1
1
D
D
⎟⎟ + ⎜⎜ 01′′ ×
⎟⎟
×
263 50 ′01′′ = 263 + ⎜⎜ 50 ′ ×
60 ′ ⎠ ⎝
60 ′′ 60 ′ ⎠
⎝
= 263 D + 0.83333 D + 0.00028 D
= 263.83361 D
g
400
= 263.83361 D ×
360 D
= 293.1485 g
= 293 g14 c 85 cc
5.2) Units of Angle
Measurements
„
Example:
Find the sexagesimal equivalent of 1 rad?
360 D
1(rad ) = 1(rad ) ×
= 57.2957795 D
2π (rad )
60 ′ ⎞
⎛
D
D
= 57 + ⎜ 0.2957795 × D ⎟
1 ⎠
⎝
= 57 D + 17.746770 7 ′
= 57 + 17 ′ + 0.746770 7 ′ ×
D
= 57 D + 17 ′ + 44.80624 3 ′′
= 57 D17 ′44.80624 3 ′′
60 ′′
1′
5.2) Units of Angle
Measurements
„
Example:
Find the centesimal equivalent of 1 rad?
400 g
= 63.6619772 g
1(rad ) = 1(rad ) ×
2π (rad )
c
⎛
100
= 63 g + ⎜⎜ 0.6619772 g × g
1
⎝
= 63 g + 66.19772 c
⎞
⎟⎟
⎠
cc
100
= 63 g + 66 c + 0.19772 c × c
1
= 63 g + 66 c + 19.772 cc
5.2) Units of Angle
Measurements
„
Example:
What is the length of the arc that corresponds to a
central angle of 45° if the radius is 150m?
Arc Length (S ) = rγ (rad )
2π (rad ) ⎞
⎛
= 150m × ⎜ 45 ×
⎟
360 ⎠
⎝
= 150m × 0.785398163(rad )
D
D
= 117.810m
5.3) Horizontal, Vertical, and
Zenith Angles
„
Horizontal Angles (Figure 5.1):
¾ Angles measured on horizontal plane
¾ In Figure 5.1, points A′, B′, & C′, are the
projections of points A, B, & C, respectively.
¾ Angles A′B′C′, B′C′A′, and C′A′B′ are the
horizontal angles
5.3) Horizontal, Vertical, and
Zenith Angles
Figure 5.1: Horizontal Angles
5.3) Horizontal, Vertical, and
Zenith Angles
„
„
Vertical Angle (Figure 5.2):
¾ Measured in a vertical plane
¾ Uses the horizontal plane as reference plane
¾ It is +ve (-ve) if the point being sited on is above
(below) the horizontal plane
¾ Its value can range from -90° to +90°
Zenith Angle or Zenith Distance (Figure 5.2):
¾ Also, measured in a vertical plane
¾ Uses the overhead extension of the plumb line as
reference line.
¾ Its value ranges from 0° to +180°
5.3) Horizontal, Vertical, and
Zenith Angles
Figure 5.2: Vertical and Zenith Angles
5.4) True Bearings
Bearings:
For OA: N 70° E
For OB: S 44° E
For OC:
S 81°20′ W
For OD:
N 32°45′ W
Figure 5.3: True Bearing
A
5.5) Magnetic Bearings
Figure 5.4: Magnetic Bearing
5.6) Azimuths
Azimuths:
For OA: 70°
For OB: 136°
For OC: 261°20′
For OD: 327°15′
Figure 5.6: Azimuth of a Line
True
North
5.7) Back Bearing &
Back Azimuth
N
Back bearing of line OA=
Bearing of line AO = S 30° W
Back azimuth of line OA=
Azimuth of line AO = 210°
A
30°
N
30°
O
210°
5.10) Principal Elements of An
Angle-Measuring Device
Four Common
Types of anglemeasuring
devices:
1) Vernier
Transit
2) Scale reading
Theodolite
3) Digital
Theodolite
4) Electronic
Theodolite
5.10) Principal Elements of An
Angle-Measuring Device
Basic Elements:
• A line of sight
• A horizontal
axis
• A vertical axis
• A graduated
vertical circle
• A graduated
horizontal
circle
5.12) Parts of A Vernier
Transit
These parts are:
1) Leveling Head
2) Lower Plate
3) Upper Plate
5.16) Setting Up A Transit