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Finding the locations of points and orientationsof lines depends on measurements of angles andof lines depends on measurements of angles anddirections

In surveying, directions are given by azimuthsand bearings.

Angels measured in surveying are classified as Horizontal angels

V i l l Vertical angles Angles in surveying are measured with

Transit / theodolite , or Total station

Three basic requirements determining an angle: Reference or starting line Reference or starting line, Direction of turning, and Angular distance (value of the angle)

Vertical angles are used in slope distance corrections or in height determination are referenced to theg

Horizon by plus or minus angles

Zenith (directly above the observer)

Nadir

Ground Surface

Nadir (directly below the observer)

Zenith and nadir are terms describing points on a celestial sphere (radius with its center at the center of the earth)

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A line on the mean surface of the earth joining the north and south polejoining the north and south pole

Lies in a vertical plane passing through a fixed point of reference and through the observer’s position

All lines of longitude

- Also known as the “Astronomic or Geographic Meridian”

- The line formed by the intersection The line formed by the intersection with the earth’s surface of a plane that includes the earth’s axis of rotation

- Passes through the geographic north and south poles of the earth and the observer’s position

P ll l t th P i (C t l) - Parallel to the Prime (Central) Meridian

- Generally adapted fix reference line in surveying practice

- Used in marking boundaries of land

- A fixed line of reference which lies parallel with the magnetic lines of forces of the earth

- The direction is defined by a freely suspended magnetic needle of a compass held at the observer’s position

- Employed only on rough surveys where a magnetic compass is used where a magnetic compass is used in determining directions

- The direction is not fixed, and varies with time and location

- A fixed line of reference parallel to the central meridian of a system of a plane rectangular coordinates pla e ecta gula coo d ates

- The use of grid meridians is applicable only to plane surveys of limited extent

- All measurements are all projected to a horizontal plane and that all meridians are parallel straight linesmeridians are parallel straight lines

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- Arbitrarily chosen fixed line of reference which is taken for conveniencefor convenience

- Only used on plane surveys of limited extent since they are difficult or may be impossible to re-establish if the original points are lost or obliterated

Symbols used for north points

The magnitude of an angle can be expressed The magnitude of an angle can be expressed in different units, all of which are basically derived from the division of the circumference of a circle

Degree The sexagesimal system is used in which the The sexagesimal system is used in which the circumference of a circle is divided into 360 parts or degrees.

1° = 60’ = 360”

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Grad In the centesimal system the circumference In the centesimal system, the circumference

of a circle is divided into 400 parts

1g = 100c = 1000cc

Mil The circumference is divided into 6400 parts The circumference is divided into 6400 parts Commonly used in military operations as in fire direction of artillery units

1600 mils is equal to 90°

Radian Defined as the angle subtended at the centerDefined as the angle subtended at the center

of a circle by an arc length exactly equal to the radius of the circle

Π = 180°

ConversionConversion

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1. Convert the angle 230°30’50” into its equivalent g qin decimal degrees, in grads, in mils and in radians.

The direction of a line Horizontal angles are usually measured with a Horizontal angles are usually measured with a

theodolite or total stations whose precision can range from 1 second to 20 seconds

The most commonly measured horizontal angles in surveying: Interior Angles and Exterior Angles;

Angles to the Right or to the Left; Angles to the Right or to the Left; Deflection Angles; Bearings; and Azimuths.

Interior Angles: Measured on the inside of a Measured on the inside of a

closed polygon. The sum of all angles in any

polygon must equal (n-2)180°where n is the number of angles

Polygon --> Closed Traverse (used for boundary survey)

(a) Clockwiseinterior angles(angles to the right).

Re-entrant Angle - interior angle which is greater than 180°

(b) Counterclockwiseinterior angles(angles to the left).

Exterior Angles:L t d t id l d l d Located outside a closed polygon and are referred to as explements of interior angles

The advantage to be gained by measuring this angle is their use as another check.

Interior Angel + Exterior Angle = 360°

The sum of the exterior angles equal to (n+2)180°(n+2)180

Explement Angle – the difference between 360° and any one angle.

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Angels to the Right:

M d l k i f h Measured clockwise from the rear to the forward station

As a survey progresses, stations are identified by consecutive alphabetic letters.

Most automatic data collectors require that angles to the right be measured in the field. (a) Clockwise

interior angles(angles to the right).

Angels to the Left:

T d t l k i f th Turned counterclockwise from the rear station.

As seen from the figure, a serious mistake occurs if counterclockwise angles are measured and recorded or assumed to be clockwise.

To avoid this confusion, always , ymeasure angel to the right and note the direction of turning in the filed book with a sketch.

(b) Counterclockwiseinterior angles(angles to the left).

Deflection Angles: Measured from an extension of the

back line, to the forward station. Used principally on the long linear

alignments of route surveys. Deflection angles may be measured

to the right (clockwise) or to the left (counterclockwise) depending upon the direction of the route. Clockwise (+) Clockwise (+) Counterclockwise (-) Deflection angles are always < 180°

The direction of turning is identified by appending an R or L to the numerical value.

AzimuthsA i h h i l l Azimuths are horizontal angles measured clockwise from any reference meridian.

In plane surveying, azimuths are generally measured from north.

Azimuths are used advantageously in boundary, topographic, control, and

h ki d f ll i other kinds of surveys, as well as in computations.

Azimuths range is magnitude from 0 to 360 degrees

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Bearings Bearings are another systems for Bearings are another systems for

designating directions of lines. The bearing of a line is defined as

the acute horizontal angle between a reference meridian and the line.

Measured from either the North or South toward the East or West, to give a reading smaller than 90°.give a reading smaller than 90 .

N70°E, N30°W, S35°E, and S55°W

Bearings Quadrantal System for defining Bearings Quadrantal System for defining Bearings

Q1Q4

Q2Q3

Forward and Back Azimuths

These directions differ by 180° from each other since These directions differ by 180 from each other sincethe back azimuth is the exact azimuth is the exactreverse of the forward azimuth

To determine the back azimuth when the forwardazimuth is known, the following rules are used:1. If the forward azimuth of the line is greater1. If the forward azimuth of the line is greater

than 180 °, subtract 180 ° to obtain the backazimuth.

2. When the forward azimuth of the line is lessthen 180 °, add 180 ° to determine the backazimuth.

Forward and Back Azimuths Forward and Back Azimuths

LineObserved Azim from North

Forward Back

AB 128° 308°

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Forward and Back Bearingg When the bearing of a line observed in the direction

in which the survey progresses – Forward Bearing When the bearing of the same line is opposite

direction – Back Bearing

Line Forward Back Line

AB N 60° E S 60° W BA

Determining Angles from AzimuthsCompute the angle APB θ from the following set of Compute the angle APB,θ, from the following set of

lines whose azimuths are given.

AzimN of line PA, α1 = 40° 48’AzimN of line PB, α2 = 125 ° 30’

N

P

Aα1

α2 θ

S

EWP

B

2 θ

Determining Angles from AzimuthsCompute the angle CPD θ from the following set Compute the angle CPD,θ, from the following set

of lines whose azimuths are given.

AzimS of line PC, α1 = 300° 48’AzimS of line PD, α2 = 80 ° 30’

N

α1

S

EWP

D

C

α2

θ

Determining Angles from BearingsCompute the angle COB θ from the following set Compute the angle COB,θ, from the following set

of lines whose azimuths are given.

Bearing of line CO, α1 = N 20° 48’ EBearing of line BO, α2 = S 80 ° 30’ E

NC

α1

θ

S

EWO

Bα2

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Azimuths BearingsAzimuths Bearings

Vary from 0 to 360° Vary from 0 to 90°

Require only a numerical value Require two letters and a numerical value

Are measured clockwise onlyAre measured clockwise or counterclockwise

Are measured either from north only or from south only on a

Are measured from north and only, or from south only on a particular survey south

May be geodetic, magnetic, grid, assumed, forward or back Same as azimuth

Quadrant Quadrant Letter Numerical Value

To convert from azimuths to bearings (Reckoned from North)

Quadrant Quadrant Letter Numerical Value

0° - 90° NE Bearing = Azimuths

90° - 180° SE Bearing = 180° - Azimuths

180° - 270° SW Bearing = Azimuths - 180°

270° - 360° NW Bearing = 360° - Azimuths

To convert from bearings to azimuths (Reckoned from North)

NE quadrant Azimuth = Bearing

SE quadrant Azimuth = 180° – Bearing

SW quadrant Azimuth = 180° + Bearing

NW quadrant Azimuth = 360° – Bearing

Converting Azimuths to BearingConvert the azimuth to equivalent bearing and Convert the azimuth to equivalent bearing, and

the bearing to equivalent azimuth.

AzimS of line AP, α1 = 40° 48’Bearing of line BP, α2 = S 80 ° 30’ E

N

S

EWP

B

A

α1

α2

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Computation of Azimuths (reckoned from North)North)

Using the deflection angles shown, calculate North azimuths of the lines.

Course Azimuth

AB

BC

CD

DE

EF

Magnetic DeclinationThe magnetic declination in a locality is 10 ° E The magnetic declination in a locality is 10 E.

Determine the true bearing and true azimuth reckoned from north and south of the line whose magnetic bearing is

Bearing of line DO, α1 = S 50 ° 30’ W

TNMN

d

TS

EWO

D

α1

MS

ρ

Computation of Azimuths from North and Bearings for LinesBearings for Lines

Course Azimuth Bearing

BC

CD

DEDE

EA

AB

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Measuring distances alone in surveying does not establish the location of an object. We need to locate the object in 3 dimensions.

1. Horizontal length (distance)

2. Difference in height (elevation)

3. Angular direction

Angle

the difference in direction between two convergent lines

Horizontal angle

is formed by the directions to two objects in a horizontal plane

Vertical angle

is formed by two intersecting lines in a vertical plane, one of these lines horizontal

Zenith angle

is the complementary angle to the vertical angle and is formed by two intersecting lines in a vertical plane, one of these lines directed toward the zenith

Types of Measured Angles

Types of Measured Angles