LOGO

DETERMINING HEIGHT

TOPIC 1.xxx

LOGOIntroduction

For many jobs it is important to be able to determine the height of features. For example: Trees Buildings Etc.

The best equipment and method to use is determined by the desired accuracy and precision of the data.

The equipment and methods used can be divided into two categories: Estimates Measurements.

LOGOEstimate Methods

ShadowLine of sightFixed angle

LOGOEstimating Height- Shadow Method

The shadow length of all objects is proportional to their height. The height of an object can be determined by measuring the

shadow length of an object with a known height and comparing it to the length of the shadow for the unknown height.

A

B=

C

D

LOGOShadow Method-Example

Determine the height of the tree.

m 24.3

H

m 4.5

m 6.0

H(4.5) = (6.0)(24.3)

m 32.4=4.5

)(6.0)(24.3=H

24.3 m

4.5 m

6.0 m

LOGOShadow Method

Advantages1. No surveying

equipment

2. Easy math

Disadvantages1.Requires sunny

day

2.Must have clear space to see shadows.

3.Low precision

LOGOHeight- Line of Sight MethodThe line of sight method is base on the principles of right

triangles.The ratio of the lengths of the sides of a right triangle are the

same as long as the angle is the same.

1.5

0.8= 1.875 = 1.9

2.5

1.3= 1.923 = 1.9

If two lengths of a small triangle and one side of the large triangle are known the length of the other side of the large triangle can be calculated using a ratio.

LOGOHeight- Line of Sight--cont.1. Select a stick of known

height.2. Move away from tree

some distance and place stick in ground. Insure it is plumb

3. Lay on ground and sight across top of stick to the top of the object.

4. Move towards or away from the stick until the sight line is aligned with the top of the stick and the top of the object.

5. Measure the distance from the stick to your eye position.6. Measure the distance from your eye position to the base of the

tree.

LOGOLine of Sight Method--Example

Determine the height of the tower.

The stake and sight position form one triangle, the tower and sight position form a second triangle.

Both triangles have the same angle.

Therefore:

10

9.2=

H

215.3

H =(10)(215.3)

9.2= 234.0 ft

LOGOLine of Sight Method--cont.

Advantages:1. Low tech

2. Doesn’t require sunny day

3. Adaptable to many different objects

4. Easy math

Disadvantages:1.Difficulty

establishing line of sight accurately.

2.Low precision

3.Precision is reduced if stake is not at same elevation as base of the object.

LOGOHeight- Fixed Angle Method The fixed angle method uses a

principle of triangles--the legs of a 45 degree triangle are the same length.

Easy way to get a 45 angle is to fold a piece of paper.

The height is determined by sighting along the hypotenuse of the triangle until the line of sight aligns with the top of the the object.

The height of the object is the distance from the object plus the eye height.

The paper must held horizontal for acceptable results.

LOGOFixed Angle Method-Example

Determine the height of the tree.

m 22.2=

m 5.8+m 16.4=Height

16.4 m

5.8 m

16.4 m

LOGOMeasuring Methods

Transit or theodoliteOthersWith a transit@ theodolite the vertical angle

to the top of the object can be measured using the tangent trig. function.

Knowing the angle, the height of the instrument and the distance from the transit to the object, the height can be calculated.

LOGOHeight Measuring-Theodolite

LOGOHgt. Measuring- Theodolite Example

m 93.73=

m 165.0 x 0.568=

Adjacent x angleTan =Opposite

Adjacent

Opposite=ngleTan A

165m

5.9m

LOGO

LOGO

TOPIC 4: ANGLE AND DIRECTION

MEASUREMENT

MS SITI KAMARIAH MD SA’ATLECTURER

SCHOOL OF BIOPROCESS ENGINEERINGsitikamariah@unimap.edu.my

ERT247/4GEOMATICS ENGINEERING

LOGOIntroduction

An angle is defined as the difference in direction between two convergent lines.

LOGOTypes of Angles

Vertical anglesZenith anglesNadir angles

LOGODefinition

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

A zenith angle is the complementary angle to the vertical angle and is directly above the obeserver

A Nadir angle is below the observer

LOGO

Three Reference Directions - Angles

LOGO Meridians

A line on the mean surface of the earth joining north and south poles is called meridian.

Note: Geographic meridians

are fixed, magnetic meridians vary with time and location.

Relationship between “true” meridian and grid meridians

Figure 4.2

LOGOGeographic and Grid Meridians

LOGOHorizontal Angles

A horizontal angle is formed by the directions to two objects in a horizontal plane. Interior angles Exterior angles Deflection angles

LOGODefinitions:

Interior angles are measured clockwise or counter-clockwise between two adjacent lines on the inside of a closed polygon figure.

Exterior angles are measured clockwise or counter-clockwise between two adjacent lines on the outside of a closed polygon figure.

Deflection angles, right or left, are measured from an extension of the preceding course and the ahead line. It must be noted when the deflection is right (R) or left (L).

LOGOClosed Traverse

Interior Angles

Closed traverse showing the interior angles.

LOGOOpen Traverse

(a)Open traverse showing the interior angles.

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

LOGOTypes of Measured Angles

LOGOAngle Units

Several different units can be used to measure angles.

This class uses two. Decimal Degrees (DD) Degrees Minutes Seconds (DMS)

LOGOAngle Units-DD

DD expresses any part of an angle less than a whole degree as a decimal. 108.24o

Electronic instruments such as total stations and GPS can output angles in DD.

Angles in DD is the system of choice today because it is the easiest form to use with calculators and computer software.

LOGOAngle Units-DMS

DMS is the angle measuring method used on most mechanical instruments. 108o 23’ 40”

In the DMS system there are 60 minutes in each degree and 60 seconds in each minute.

Because both systems are still used, it is useful to know how to convert from one to the other.

LOGODMS to DD

Many calculators have a DMS to DD and DD to DMS conversion key.

It will save a lot of time and reduce mistakes if you learn how to do these conversions on a calculator.

If you cannot do it on a calculator, then you must learn how to do it manually.

To convert from a DMS angle to a DD angle the minutes

and seconds must be converted to a fraction. The

fractions are reduced to decimal equivalents and then the

parts are added.

LOGODMS to DD-cont.

Example: Convert 120o 34’ 45” to DD

120.58or .120.5791..=

0.0125+0.566...120

3,600

45+

60

34+120=45" 34’ 120

o

o

LOGODD to DMS

The manual method from DD to DMS follows the same math principles. The decimal part of the angle must be converted to minutes and seconds.

45.349 = 45o

0.349 x 60 =20.94 =20'

0.94 x 60 =56.4"

Answer 45o 20' 56"

Example: convert the angle 45.349o to DMS

LOGOAdding & Subtracting Angles in DMS

Occasionally when using mechanical instruments it is

necessary to add and subtract angles using DMS.

The addition and subtraction principles are the same,

except units of 60 are carried or subtracted instead of units

of 10.

LOGOAdding Angles in DMS

Example: Add the angles 20o 45’ 27” and 30o 24’ 35”

62" 69’ 50

35" 24’ 30 +

27" 45’ 20

o

o

o

02" 70’ 50 = 62" 69’ 50

needed. if seconds, thereduce tois stepFirst oo

In is not proper to leave an angle measurement with more that 60 minutes or seconds. The answer must be reduced.

02" 10’ 51 = 02" 70’ 50

needed. if minutes, thereduce tois stepnext Theoo

02" 10’ 51 :isanswer The o

LOGOSubtracting Angles In DMS

Subtraction follows the same principles.

Example: Subtract 40o 18’ 50” from 120o 15’ 45”

55" 56’ 79

50" 18’ 40 -

105" 74’ 119

o

o

o

• 45 - 50 and 15 - 18 would result in

negative numbers.

• 120o 15’ 45” must be converted to:

119o 74' 105"

50" 18’ 40 -

45" 15’ 120 o

o

The answer is:

79o 56' 55"

LOGODirections

Azimuth An Azimuth is the direction of a line as given by an

angle measured clockwise (usually) from the north. Azimuth range in magnitude from 0° to 360°.

Bearing Bearing is the direction of a line as given by the acute

angle between the line and a meridian. The bearing angle is always accompanied by letters

that locate the quadrant in which line falls (NE, NW, SE or SW).

Range 0° to 90°.

LOGOBearings and Azimuths

LOGOAzimuths

LOGOAzimuths

LOGOBearing

LOGOBearing

LOGOBearings

LOGORelationships Between Bearings and Azimuths

To convert from azimuths to bearing, a = azimuths b = bearing

Quadrant Angles Conversion

NE 0o 90o a = b

SE 90o 180o a = 180o – b

SW 180o 270o a = b +180o

NW 270o 360o a = 360o – b

LOGOReverse Direction

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

To reverse bearing: reverse the direction

Figure 4.7Reverse Directions

Figure 4.8

Reverse Bearings

Line Bearing

AB N 62o 30’ E

BA S 62o 30’ W

LOGO

LOGOReverse Direction

CD has an azimuths of 128o 20’ DC has an azimuths of 308o 20’

To reverse azimuths: add 180o

Figure 4.8

Reverse Bearings

Line Azimuths

CD 128o 20’

DC 308o 20’

LOGOCounterclockwise Direction (1)

StartGiven

LOGOCounterclockwise Direction (2)

LOGOCounterclockwise Direction (3)

LOGOCounterclockwise Direction (4)

LOGOCounterclockwise Direction (5)

FinishCheck

LOGOSketch for Azimuth Computation

LOGOClockwise Direction (1)

StartGiven

LOGOClockwise Direction (2)

LOGOClockwise Direction (3)

LOGOClockwise Direction (4)

LOGOClockwise Direction (5)

FinishCheck

LOGO

StartGiven

FinishCheck

LOGOAzimuth Computation

When computations are to proceed around the traverse in a clockwise direction,subtract the interior angle from the back azimuth of the previous course.

When computations are to proceed around the traverse in a counter-clockwise direction, add the interior angle to the back azimuth of the previous course.

LOGOAzimuths Computation

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

Course Azimuths Bearing

BC 270o 28’ N 89o 32’ W

CD 209o 05’ S 29o 05’ W

DE 134o 27’ S 45o 33’ E

EA 62o 55’ N 62o 55’ E

AB 330o 00’ N 30o 00’ W

LOGOAzimuths Computation

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

Course Azimuths Bearing

AE 242o 55’ S 62o 55’ W

ED 314o 27’ N 45o 33’ W

DC 29o 25’ N 29o 05’ E

CB 90o 28’ S 89o 32’ E

BA 150o 00’ S 30o 00’ E

LOGOBearing Computation

Prepare a sketch showing the two traverse lines involved, with the meridian drawn through the angle station.

On the sketch, show the interior angle, the bearing angle and the required angle.

LOGOBearing Computation

Computation can proceed in a Clockwise or counterclockwise

Figure 4.11

Sketch for Bearings Computations

LOGOSketch for bearing Computation

LOGOComments on Bearing and Azimuths

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

LOGOMeasuring Angles

There are two methods for measuring existing or laying out new angles. Indirect Direct

Indirect methods measure and lay out angles by utilizing equipment that can not measure angles directly.

Direct measurement and lay out of angles is accomplished by instruments with angle scales.

LOGOAngle Measuring - Indirect

Tapes (or other distance measurement) Using triangle principles Using trigonometry based on slope angles

LOGODetermining Angles – Taping

Need to: measure 90° angle at point X

d d

Lay off distance d either side of X

X

l l

Swing equal lengths (l)

Connect point of intersection and X

LOGODetermining Angles – Taping

A

B

C

Need to: measure angle at point A

Measure distance ABMeasure distance ACMeasure distance BC

Compute angle

)AB)(AC2(

BCABACcos

222

1α

LOGODetermining Angles – Taping

A

B

C

Need to: measure angle at point A

AP

PQtan 1α

Q

Lay off distance APEstablish QP AP

Measure distance QP

Compute angle

P

LOGODetermining Angles – Taping

A

B

C

Need to: measure angle at point A

)AD2(

DE)sin(0.5 α

D

Lay off distance ADLay off distance AE = AD

Measure distance DE

Compute angle E

LOGOAngle Measuring Equipment - Direct

Direct methods of measuring angles involves the use of surveying equipment with angle scales.

The operator must understand how to use each type of instrument.

Examples of Instruments: Sextants Compass Digital theodolites and; Total stations

LOGOAngle Measuring Equipment

LOGOTheodolites

General Background: Theodolites are surveying instruments

designed to precisely measure horizontal and vertical angles.

They are used to establish straight and curved lines.

To establish or measure distance (Stadia) To establish Elevation when used as a level.

(When we set the vertical angle to 90°).

LOGOTheodolites

They have:3 screw level baseGlass horizontal and vertical circles,

read directly or through micrometer.Right angle prism (optical plummet)High precision

LOGOTheodolites

LOGOTheodolites

Electronic read out 1” eliminate mistakes and reading the angles.

Precision varies from 0.5” – 20”Zero is set by a button.Repeated angle averaging.Replacing optical theodolites (It is less

expensive to purchase and maintain).

LOGOTotal Stations

Combined measurements

Digital display

LOGO

Measures and Records:Horizontal Angles

Vertical Anglesand

Slope Distances

CalculatesCalculates::Horizontal Distance

Vertical DistanceAzimuths of LinesX,Y,Z Coordinates

LayoutEtc.

LOGOMeasuring Angles

Instrument handling and setup Discussed in lab

Procedure with repeating instrument

LOGOAngles

All angles have three parts Backsight: The baseline or point used as zero angle. Vertex: Point where the two lines meet. Foresight: The second line or point

LOGOErrors in Angle Measurement

Gross – reading, pointing, setting up over the wrong point, booking

Random – settling of tripod, wind, temperature, refraction

Systematic/instrumental Horizontal axis not perpendicular to the vertical

axis Axis of sight not perpendicular to the horizontal

axis Axis of the plate bubble not perpendicular to the

vertical axis. Vertical index error

LOGO