CE 260CE 260SurveyingSurveying

CHAPTER 4CHAPTER 4

Angles and DirectionsAngles and Directions

4.1 General Background4.1 General Background

Angles in surveying are measured Angles in surveying are measured withwith• A transit / theodolite , orA transit / theodolite , or• Total sectionTotal section

4.2 Reference Directions for 4.2 Reference Directions for Vertical AnglesVertical Angles

Vertical angles are referenced to:Vertical angles are referenced to:1.1. The horizon by up (The horizon by up (++), or down (), or down (--))

2.2. ZenithZenith

3.3. NadirNadir

Note:Note:

ZenithZenith: is directly : is directly aboveabove the observer the observer

NadirNadir : is directly : is directly belowbelow the observer the observer

the three reference directions for vertical angles: horizontal, zenith, and nadir.

Figure 4.1Figure 4.1

4.3 Meridians4.3 Meridians A line on the mean A line on the mean

surface of the earth surface of the earth joining joining northnorth and and southsouth poles is called poles is called meridianmeridian. .

Note: Note: Geographic meridians Geographic meridians

are fixed, magnetic are fixed, magnetic meridians vary with meridians vary with time and location.time and location.

Relationship between “true” meridian and grid meridians

Figure 4.2Figure 4.2

4.4 Horizons Angles4.4 Horizons Angles

Horizontal angles are usually Horizontal angles are usually measured with a theodolite or total measured with a theodolite or total stations whose precision can range stations whose precision can range from 1 second to 20 secondsfrom 1 second to 20 seconds

For all closed polygons of For all closed polygons of nn sides, the sides, the sum of the interior angles will be sum of the interior angles will be

(n-2)x 180(n-2)x 180oo

4.4 Horizons Angles (Cont’d)4.4 Horizons Angles (Cont’d)

Closed traverse showing the interior angles.

Figure 4.3Figure 4.3

A _ 87A _ 87oo 05’ 05’

B _ 120B _ 120oo 28’ 28’

C _ 118C _ 118oo 37’ 37’

D _ 105D _ 105oo 22’ 22’

E _ E _ 108108oo 28’ 28’

538538oo 120’ 120’

= 540= 540oo 00’ 00’

4.4 Horizons Angles (Cont’d)4.4 Horizons Angles (Cont’d)

(a) Open traverse showing the interior angles.

(b) Same traverse showing angle right (202oo 18’) and angle left (157oo 42’)

Figure 4.4Figure 4.4

4.5 Azimuths4.5 Azimuths

An azimuths is direction of line as An azimuths is direction of line as given by an angle measured given by an angle measured clockwise (usually) from the north clockwise (usually) from the north end of a meridian.end of a meridian.

Azimuths range is magnitude from 0Azimuths range is magnitude from 0oo to 360to 360oo

4.6 Bearings4.6 Bearings Bearings is the direction of a line as Bearings is the direction of a line as

given by the acute angle between given by the acute angle between the line and a meridian.the line and a meridian.

The bearing angle, is always The bearing angle, is always accompanied by letters that locate accompanied by letters that locate the quadrant in which the line falls the quadrant in which the line falls (NE, NW, SE, or SW).(NE, NW, SE, or SW).

4.6 Bearings (Cont’d)4.6 Bearings (Cont’d)LineLine BearingsBearings

0 - 50 - 5 N 58N 58oo 20’ E 20’ E

0 - 6 0 - 6 S 24S 24oo 50’ E 50’ E

0 – 70 – 7 S 27S 27oo 30’ W 30’ W

0 – 80 – 8 N 10N 10oo 45’ W 45’ W

Figure 4.6Figure 4.6

Bearings calculated from given data

4.7 Relationships Between Bearings 4.7 Relationships Between Bearings and Azimuthsand Azimuths

To convert from azimuths to bearing,To convert from azimuths to bearing,• a = azimuthsa = azimuths• b = bearingb = bearing

QuadrantQuadrant AnglesAngles ConversionConversion

NENE 00oo 90 90oo b = ab = a

SESE 9090oo 180 180o o b = 180b = 180oo – a – a

SWSW 180180oo 270 270oo b = a -180b = a -180oo

NWNW 270270oo 360 360oo b = 360b = 360oo – a – a

4.7 Relationships Between Bearings 4.7 Relationships Between Bearings and Azimuthsand Azimuths

To convert from azimuths to bearing,To convert from azimuths to bearing,• a = azimuthsa = azimuths• b = bearingb = bearing

QuadrantQuadrant AnglesAngles ConversionConversion

NENE 00oo 90 90oo a = ba = b

SESE 9090oo 180 180o o a = 180a = 180oo – b – b

SWSW 180180oo 270 270oo a = b -180a = b -180oo

NWNW 270270oo 360 360oo a = 360a = 360oo – b – b

4.8 Reverse Direction4.8 Reverse Direction

In figure 4.8 , the line In figure 4.8 , the line • AB has a bearing of N 62AB has a bearing of N 62oo 30’ E 30’ E• BA has a bearing of S 62BA has a bearing of S 62oo 30’ W 30’ W

To reverse bearing: reverse the directionTo reverse bearing: reverse the direction

Figure 4.7Figure 4.7Reverse Directions Figure 4.8Figure 4.8

Reverse Bearings

LineLine BearingBearing

ABAB N 62N 62oo 30’ E 30’ E

BABA S 62S 62oo 30’ W 30’ W

LineLine BearingBearing

ABAB N 62N 62oo 30’ E 30’ E

BABA S 62S 62oo 30’ W 30’ W

4.8 Reverse Direction4.8 Reverse Direction In figure 4.9 , the line In figure 4.9 , the line

• CD has an azimuths of 128CD has an azimuths of 128oo 20’ 20’• DC has an azimuths of 308DC has an azimuths of 308oo 20’ 20’

To reverse azimuths: add 180To reverse azimuths: add 180oo

Figure 4.8Figure 4.8Reverse Bearings

LineLine AzimuthsAzimuths

CDCD 128128oo 20’ 20’

DCDC 308308oo 20’ 20’

4.8 Counterclockwise Direction4.8 Counterclockwise Direction (1) (1)

StartStart

GivenGiven

4.8 Counterclockwise Direction4.8 Counterclockwise Direction (2) (2)

4.8 Counterclockwise Direction4.8 Counterclockwise Direction (3) (3)

4.8 Counterclockwise Direction4.8 Counterclockwise Direction (4) (4)

4.8 Counterclockwise Direction4.8 Counterclockwise Direction (5) (5)FinishFinish

CheckCheck

StartStart

GivenGiven

FinishFinish

CheckCheck

4.8 Clockwise Direction 4.8 Clockwise Direction (1) (1)

StartStart

GivenGiven

4.8 Clockwise Direction 4.8 Clockwise Direction (2) (2)

4.8 Clockwise Direction 4.8 Clockwise Direction (3) (3)

4.8 Clockwise Direction 4.8 Clockwise Direction (4) (4)

4.8 Clockwise Direction 4.8 Clockwise Direction (5) (5)

FinishFinish

CheckCheck

StartStart

GivenGiven

FinishFinish

CheckCheck

4.9 Azimuths Computation4.9 Azimuths Computation

Counterclockwise direction: add the Counterclockwise direction: add the interior angle to the back azimuth of interior angle to the back azimuth of the previous course the previous course

CourseCourse Azimuths Azimuths BearingBearing

BCBC 270270oo 28’ 28’ N 89N 89oo 32’ W 32’ W

CDCD 209209oo 05’ 05’ S 29S 29oo 05’ W 05’ W

DEDE 134134oo 27’ 27’ S 45S 45oo 33’ E 33’ E

EAEA 6262oo 55’ 55’ N 62N 62oo 55’ E 55’ E

ABAB 330330oo 00’ 00’ N 30N 30oo 00’ W 00’ W

4.9 Azimuths Computation4.9 Azimuths Computation

Clockwise direction: subtract the interior Clockwise direction: subtract the interior angle from the back azimuth of the angle from the back azimuth of the previous courseprevious course

CourseCourse Azimuths Azimuths BearingBearing

AEAE 242242oo 55’ 55’ S 62S 62oo 55’ W 55’ W

EDED 314314oo 27’ 27’ N 45N 45oo 33’ W 33’ W

DCDC 2929oo 25’ 25’ N 29N 29oo 05’ E 05’ E

CBCB 9090oo 28’ 28’ S 89S 89oo 32’ E 32’ E

BABA 150150oo 00’ 00’ S 30S 30oo 00’ E 00’ E

4.10 Bearing Computation4.10 Bearing Computation Computation can proceed in a Computation can proceed in a

Clockwise or counterclockwiseClockwise or counterclockwise

Figure 4.11Figure 4.11Sketch for Bearings Computations

4.11 Comments on Bearing and 4.11 Comments on Bearing and AzimuthsAzimuths

Advantage of computing bearings Advantage of computing bearings directly from the given data in a directly from the given data in a closed traverse, is that the final closed traverse, is that the final computation provides a check on all computation provides a check on all the problem, ensuring the the problem, ensuring the correctness of all the computed correctness of all the computed bearingsbearings

Figure 4.12Figure 4.12

Sketch for each Bearings Calculation

4.11 Comments on Bearing and 4.11 Comments on Bearing and AzimuthsAzimuths

Disadvantages associated with Disadvantages associated with computing bearings directly from the computing bearings directly from the data in a closed traverse is that there data in a closed traverse is that there is no systematic approach to the is no systematic approach to the overall solution. Each bearing overall solution. Each bearing computation is unique, requiring computation is unique, requiring individual analysis.individual analysis.

4.11 Comments on Bearing and 4.11 Comments on Bearing and AzimuthsAzimuths

The computation of azimuths The computation of azimuths involves a highly systematic routine: involves a highly systematic routine: add (subtract) the interior angleadd (subtract) the interior angle from the back azimuths of the from the back azimuths of the previous course.previous course.

Figure 4.13Figure 4.13

Summery of Results from clockwise and counterclockwise approaches

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