Chapter 4 Angular Measurements PDF

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Chapter 4 Angular Measurements

Objectives: On completion of this chapter, student should be able to: • • • • • • •

obtain the relative direction of lines; identify the different systems of designating directions of lines; distinguish between polar and rectangular coordinates; define angles, bearings, and azimuths; convert bearings to azimuth (v.v.) determine the forward and back bearing/azimuth; and define reference systems for angular measurement such as true, grid, magnetic and assumed meridian.

Introduction: The location of points and the orientation of lines frequently depend upon the measurement of angles and directions. In surveying, directions are given by bearings and azimuths. This module, will introduce the students to the basic principle of measuring reckoning angles and directions. Bearings and azimuths represent two common systems of designating the directions of lines and are encountered in most survey operations. In surveying practice it is customary to refer directions with respect to both the north and south ends of a meridian and also both to the east and west. Observe and computed values of bearings and azimuth are related values such that an azimuth maybe computed from the bearings or vice versa.

4.1 Polar/Rectangular Coordinates Polar coordinates are represented by a distance and an angle (d,  ) from a plane. Rectangular coordinates are represented by two distance components (x,y) that are at right angles to each other (perpendicular).

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The relationship between polar/rectangular coordinates is best shown by a diagram (see figure).

 d

•

x

Rectangular (x,y) y

Polar (d,  )

Converting Polar to Rectangular Coordinates: From the figure, x and y can be solving by: x y cos  = sin  = d d x = d cos  y = d sin 

Converting Rectangular and Polar Coordinates: From the figure above, d and  can be solved by Pythagorean theorem: •

y x

d 2 = x2 + y2

tan  =

d = x2 + y2

 = tan −1

y x

4.2 Meridian Meridian – a real or an imaginary reference line of fixed direction.

Types of Meridian • • •

True meridian – line passes thru the north-south poles or the geographical poles of the earth. grid meridian Magnetic meridian – it lies parallel with the magnetic lines of force of the earth. Its direction is not constant because the magnetic poles is constantly changing. 54

•

Assumed meridian – an arbitrary or chosen meridian.

4.3 Angles and Directions Angles and directions may be defined by means of bearings, azimuths, deflection angles, angles to the right, or interior angles. The values of which can be observed when obtained in the field and calculated when obtained indirectly by computation. Conversion from one to the other, is a matter of sketching to show the existing relations. By Deflection angles – The angle between a line and the prolongation of the preceding line. It is recorded as right or left according as the line to which measurement is taken lies to the right (clockwise) or left (counterclockwise) of the prolongation of the preceding line. The values may be between 0o to 180o.

22o R

B

33o L

A

In some cases, a positive ( + ) sign is placed before the angle instead of “R” after the angle; and a negative ( - ) sign for “L”. That is, for this example, 22o and 33o. For any closed polygon, the algebraic sum of the deflection angle is 360o. By Angles to right (or azimuth from back line) – the angles measured clockwise from the preceding line to the following line.

B A

By Interior angles - it is the angles between the adjacent sides or lines of any closed polygon. If n is the number of sides in a closed polygon, the sum of the interior angles is (n-2) 180.

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4.4 Bearings and Azimuths. Bearings – an acute angle between the reference meridian and the line. It may be true, magnetic, grid and assumed bearings. The bearing of a line is indicated by the quadrant in which the line falls and the acute angle which the line makes with the meridian Nin that quadrant. N

W NE E

W S S

 E

 W S

If the bearing of a line coincides with the North-meridian, it is said to be “due North”; South-meridian, “due South”; East, “due East”; and West, “due West”.

Whole-circle bearings - The direction of survey lines is generally expressed as an angle measured from a reference meridian, generally north, commencing from 0 degrees (0°) and increasing clockwise to 360 degrees (359°59'60"). Bearings are never expressed as "North, X degrees East". (This term is adopted in foreign countries but not in the Philippines).

Forward and Back Bearing When the bearing of a line is observed in the direction in which the survey progresses, it is referred to as “forward bearing”, if the bearing of the same line is observed in an opposite direction it is referred to as “back bearing”.

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B o

44

44o

67o

67o

A C

In the figure above, assume that an instrument is set up successively at stations A, B and C, and the bearings read on lines AB, BA, BC, and CB. The observed bearings of lines AB, and BC are called forward bearings. The bearings of BA and CB are called back bearings.. It can be readily seen that back bearings can be obtained from the forward bearings by simply changing the letter N to S and also changing E to W, and vice versa.

Azimuths – the angle between the reference meridian and the line measured in clockwise direction from either North or South branch of the meridian. The values are ranging from 0 o to 360o, and letters (such as, NE, NW, SE, and SW) are not required to identify quadrants. Some mistakes made in using azimuths and bearings are: 1. 2. 3. 4.

Confusing magnetic and other reference bearings. Mixing clockwise and counterclockwise angles. Interchanging bearings for azimuths. Failing to change bearing letters when using the back bearing of a line. 5. Using an angle at the wrong end of a line in computing bearings – that is, using angle A instead of angle B when starting with line AB as a reference.

Forward and Back Azimuth Rules in getting the value of back azimuth when forward azimuth is given: 1. If forward azimuth is equal or greater than 180o, subtract 180o, to obtain back azimuth.

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Aback = Aforward – 180o

2. If forward azimuth is less than 180o, add 180o, to determine the back azimuth. Aback = Aforward + 180o

Converting Azimuth to Bearing (V.V.) (Assuming azimuth is from south branch) N Bearing = 180o – Azimuth

Bearing = Azimuth – 180 o

W

E Bearing = Azimuth

Bearing = 360o – Azimuth

W

Examples: 1. Convert the following azimuth to bearing. a. 225o b. 200

c. 345o d. 90o

Solutions: a. bearing = 225o – 180o = Ñ 45o E b. bearing = S 20o W c. bearing = 360o – 345o = S 15o E

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d. due west

2. Convert the following to azimuth (from south) a. due south b. Ñ 45o W

c. S 30o E d. S 60o W

Solutions: a. 0o b. 135o

c. 330o d. 60o

4.5 Magnetic Declination Magnetic declination (or variation) – the angle between the true meridian and the magnetic meridian. The rate of change is published yearly by the “Almanac for Geodetic Engineers of the Philippines.” An east declination exists if the magnetic meridian is east of (geodetic) true north; west declination occurs if it is west of the true north.

TN

MN

East declination

MN TN

West declination

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Magnetic declinations vary at any point over the time. These variations can be categorized as secular, daily, annual, and irregular variations. Secular variations – this is the most important variations because of its magnitude. Daily variations – daily variation of the magnetic needle’s declination causes it to swing through an arc averaging approximately 8’ for the U.S. Annual variations – this periodic swing is less than 1 min of arc and can be neglected. Irregular variations – unpredictable magnetic disturbances and storms can cause short-term irregular variations of a degree or more. One way of determining the magnetic declination at a point is to interpolate it from an isogonic chart. An isogonic chart shows magnetic declinations in a certain region for a specific epoch of time. Lines on such maps connecting points that have the same declination are called isogonic lines. The isogonic line along which the declination is zero (where the magnetic needle defines geodetic north as well ass magnetic north) is termed the agonic line.

Examples: 1.

The magnetic bearing of line XY is S 45o15’ E, compute its true bearing if the magnetic declination at that instant is 01o15’ east.

Solutions:

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TN MN

1o 15’

45o 15’

  = 45o15' − 1o15'  = 44o 00' therefore, the true bearing of line XY is S 44o00’ E.

2.

The magnetic bearing of a property line was recorded as S 43 o 30’ E in 1862. At that time the magnetic declination at the survey location was 3o15’ W. What geodetic bearing is needed for a subdivision property plan?

Solutions: TN MN

3o15

43o 30’



 = 43o 30' + 3o15'  = 46 o 45' therefore, true bearing of the property line is S 46 o 45’ E

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In 1771 a line AB had a magnetic bearing of Ñ 22o15’ W. The declination of the needle at that place in 1771 was 02 025’ west. In 1976 the declination was 04o25’ east. What was the magnetic bearing of the line in 1976? Its true bearing?

3.

Solutions: MN 1771

TN

MN 1976





4o25’

22o15’ 2o25’

true bearing:

 = 22o 15 + 2o 25'  = 24 o 40'

magnetic bearing in 1976:  =  + 4 o 25'

 = 24o 40' + 4o 25'  = 29 o 05'

therefore, true bearing of the line is N 24o 40’ W and the magnetic bearing in 1976 was N 29o 05’ W.

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