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:Circumference is divided into 360 degrees (360°)One degree is divided into 60 minutes (1°=60′)One minute is divided into 60 seconds (1′=60″)

Centesimal System: Circumference is divided into 400 grads or grades (400g)One grad is divided into 100 centesimal minutes(1g=100c)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.

S: Arc Length = r γr: Radiusg: Central Angle

(in radians)Circumference=2πr

5.2) Units of Angle Measurements

To convert among the three systems:

Example 5.1: Find the sum of these three angles

( )radg 2400360 π==

Centesimal System Sexagesimal System100.4527g

251.7590g

312.0314g

75°51′23″207°18′41″340°39′57″

Sum = 664.2431g

Or 264.2431gSum = 623°50′01″Or 263°50′01″

5.2) Units of Angle Measurements

Example:

85638.75 00638.085.075

606023

605175321575

=

++=

⎟⎠⎞

⎜⎝⎛

×+⎟

⎠⎞

⎜⎝⎛+=′′′

5.2) Units of Angle Measurements

Example 5.2: What is the sexagesimal equivalent of 264.2431g?

464.794237 464.794237

1064127.094237

4127.49237 10681879.0237

81879.2374003602431.2642431.264 g

gg

′′′=

′′+′+=

⎟⎠⎞

⎜⎝⎛

′′′

×′+′+=

′+=

⎟⎠⎞

⎜⎝⎛ ′

×+=

=×=

5.2) Units of Angle Measurements

Example 5.3: What is the grad equivalent of 263°50′01″?

cccg

g

g

8514293 1485.293

36040083361.263

83361.263 00028.083333.0263

061

06110

061052631005263

=

=

×=

=

++=

⎟⎟⎠

⎞⎜⎜⎝

⎛′

×′′′

×′′+⎟⎟⎠

⎞⎜⎜⎝

⎛′

×′+=′′′

5.2) Units of Angle Measurements

Example: Find the sexagesimal equivalent of 1 rad?

( ) ( ) ( )

380624.447157 380624.447157

1067746770.07157

7746770.1757 1062957795.057

2957795.57rad2

360rad1rad1

′′′=

′′+′+=′′′

×′+′+=

′+=

⎟⎠⎞

⎜⎝⎛ ′

×+=

=×=π

5.2) Units of Angle Measurements

Example: Find the centesimal equivalent of 1 rad?

( ) ( ) ( )

cccg

c

ccccg

cg

g

cgg

gg

772.196663 1

10019772.06663

19772.6663 1

1006619772.063

6619772.63rad2

400rad1rad1

++=

×++=

+=

⎟⎟⎠

⎞⎜⎜⎝

⎛×+=

=×=π

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?

( ) ( )( )

( )m

radm

radm

radrSLengthArc

8101177853981630150

360245150

. .

=×=

⎟⎠⎞

⎜⎝⎛ ××=

=

πγ

5.3) Horizontal, Vertical, and Zenith Angles

Horizontal Angles (Figure 5.1): Angles measured on horizontal planeIn Figure 5.1, points A′, B′, & C′, are the projections of points A, B, & C, respectively.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):Measured in a vertical planeUses the horizontal plane as reference planeIt is +ve (-ve) if the point being sited on is above(below) the horizontal planeIts value can range from -90° to +90°

Zenith Angle or Zenith Distance (Figure 5.2):Also, measured in a vertical planeUses the overhead extension of the plumb line as reference line.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° EFor OB: S 44° EFor OC: S 81°20′ WFor OD: N 32°45′ W

A

Figure 5.3: True Bearing

5.5) Magnetic Bearings

Figure 5.4: Magnetic Bearing

5.6) Azimuths

True North

Figure 5.6: Azimuth of a Line

Azimuths:For OA: 70°For OB: 136°For OC: 261°20′For OD: 327°15′

5.7) Back Bearing & Back Azimuth

N

N

O

A

30°

30°

210°

Back bearing of line OA= Bearing of line AO = S 30° W

Back azimuth of line OA= Azimuth of line AO = 210°

5.10) Principal Elements of An Angle-Measuring Device

Four Common Types of angle-measuring devices:

1) VernierTransit

2) Scale readingTheodolite

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 Head2) Lower Plate3) Upper Plate

5.16) Setting Up A Transit