Chapter -6- Angles, Bearings and Azimuths. Ishik University Sulaimani Civil Engineering Department Surveying II CE Introduction 1/28/2018

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When an angle is measured in a horizontal plane it is horizontal angle. When measured in a vertical plane it is vertical angle.

1 Ishik University Sulaimani Civil Engineering Department Surveying II CE 215 Chapter -6- Angles, Bearings and Azimuths 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 1 1. Introduction Measurement of angles is basic to any survey operation. When an angle is measured in a horizontal plane it is horizontal angle. When measured in a vertical plane it is vertical angle. Angle measurements involve three steps: (i) Reference or starting line; (ii) Direction of turning; (iii) Angular value (Value of the angle). 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 2 1

2 Different Types Of Horizontal Angles horizontal angles can be ; (i) Interior angles, or (ii) Deflection angles Interior angles can be clockwise when the direction of turning is clockwise, or counterclockwise when the direction of turning is counter-clockwise. Similarly deflection angles are measured clockwise or towards right and counter-clockwise or towards left. 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 3 (i) Interior angles, They are measured on the inside of a closed polygon. Normally the angle at the apex with in a polygon is measured. Then a check can be made on their value because the sum of all angles equals (n-2)180. Polygons (closed traverse) commonly used for boundary surveys. Interior angles of a polygon Note: Exterior angles are located outside a closed polygon and they provide a check, i.e. the sum of interior and exterior angles at any station must total /28/2018 Assistant Lecturer / Asmaa Abdulmajeed 4 2

3 ii. Deflection angles They are measured from an extension of the back line, to the forward station. They are measured to the right (c.w or +) or to the left (c.c.w or -) depending on the direction of the route. They are always smaller than 180, and the direction of turning is identified by R or L. Deflection angles 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 5 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 6 3

4 Different types of angles: (a) Closed polygon-instrument station; A, B, C, D and E, all angles measured clockwise. (b) Closed polygon instrument stations A, B, C, D and E all angles measured counter-clockwise. (c) Deflection Angles. 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 7 It is customary to use small letters in the Greek alphabet to symbolize angle measurement. alpha beta gamma theta phi delta 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 8 4

5 We can use a coordinate system with angles by putting the initial side along the positive x- axis with the vertex at the origin. Quadrant I angle positive Quadrant II angle negative Initial Side If the terminal side is along an axis it is called a quadrantal angle. Quadrant IV angle We say the angle lies in whatever quadrant the terminal side lies in. 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 9 We will be using two different units of measure when talking about angles: Degrees and Radians = 360 = 90 = - 90 If we start with the initial side and go all of the way around in a counterclockwise direction we have 360 degrees If we went 1/4 of the way in a clockwise direction the angle would measure -90 You are probably already familiar with a right angle that measures 1/4 of the way around or 90 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 10 5

6 What is the measure of this angle? You could measure in the positive direction and go around another rotation which would be another 360 = = = 45 You could measure in the negative direction You could measure in the positive direction = = 405 There are many ways to express the given angle. Whichever way you express it, it is still a Quadrant I angle since the terminal side is in Quadrant I. 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 11 Direction of a Line The direction of a line is the horizontal angle between it and an arbitrary closed reference line called a Meridian. (It is a line on the mean surface of the earth joining the north and south poles). Different meridians are used for specifying a direction; I. True Meridian: It is the north-south reference line that passes through the earth s geographic poles. II. Magnetic Meridians: defined by a freely suspended magnetic needle that is influenced by earth s magnetic field only. III. An Assumed Meridian: can be established by merely assigning any arbitrary directions. For example, taking certain street line to be true north. 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 12 6

7 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 13 There are two types : - a) POLYGON or LOOP TRAVERSE b) LINK TRAVERSE A F B A C B C E D F E G D 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 14 7

8 A F X B RIGHT HAND ANGLES A C LEFT HAND ANGLES B D E C D E Y F G a) is obviously closed b) must start and finish at points whose co-ordinates are known, and must also start and finish with angle observations to other known points. Working in the direction A to B to C etc is the FORWARD DIRECTION This gives two possible angles at each station. 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed Units of Angle Measurement In the United States and many countries; o The sexagesimal system: degrees, minutes and seconds with the last unit further divided decimally. o The circumference of circles is divided into 360 parts of degrees; each degree is further divided into minutes and seconds. In Europe; o Centismal system: The circumference of circles is divided into 400 parts called gon (prevously called grads) 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 16 8

9 If the angle is not exactly to the next degree it can be expressed as a decimal (most common in math) or in degrees, minutes and seconds (common in surveying and some navigation). 1 degree = 60 minutes degrees 1 minute = 60 seconds = " minutes seconds To convert to decimal form use conversion fractions. These are fractions where the numerator = denominator but two different units. Put unit on top you want to convert to and put unit on bottom you want to get rid of. Let's convert the seconds to minutes 30" 1' 60" = 0.5' 1 degree = 60 minutes 1 minute = 60 seconds = 25 48'30" = ' = Now let's use another conversion fraction to get rid of minutes. 48.5' 1 60' =.808 9

10 1 = 60 = 3600" 1 circle = 360 =2 π rad 1 rad =360 / 2 π = Example 1.: Convert these angles from decimal number to deg, min, sec, and vice versa. 1) =? Solution : 4) =? Solution : = = ) =? 5) =? Solution : Solution : 3) =? Solution : 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed Bearing Represent one system for designating directions of lines. A bearing is defined as the acute horizontal angle between a reference meridian and the line. The angle is measured from either the north or south towards the east or west, to give a reading smaller than 90. It is measured in relation to the north or south ends and are placed in one of the quadrants (NE, NW, SE, SW). True bearings/magnetic bearings/assumed bearings are measured from True /Magnetic /Assumed meridians. 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 20 10

11 3. Bearing Quadrants NE, SE, SW, NW Always < 90 Written N23 15 W Exterior Angles (Interior Angle = 360 Exterior Angle) Back Bearing change directions Bearing AB = N23 15 W Back bearing AB = bearing BA =S23 15 E 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed Bearing Assume that a compass is setup successively at points A, B, C and D and bearings read on lines AB, BA, BC, CB, CD and DC. Bearings AB, BC and CD are called Forward bearings and BA, CB, DC are Back bearings. 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 22 11

12 Designation of Bearings: 1. The Whole Circle Bearing System (W.C.B) 2. The Quadrantal Bearing System (Q.B) 3. Bearing 1. The W. C. B System: The W. C. B system is also sometimes known as Azimuthal System. In this system, bearing of a line is measured from the true north or magnetic north in clockwise direction. The value of a bearing may vary from 0 to 360, utilizing the whole circle of graduations. Prismatic Compass is graduated on whole circle bearing system. 2. The Q. B System: In Q. B System, bearings of survey lines are measured eastward or westward from North and South whichever is nearer. In this system, both north and south directions are used as reference meridians & bearings are reckoned either clockwise or anticlockwise, depending upon the position of the line. The quadrant in which a line lies is mentioned to specify the location of the line. Surveyor s compass is graduated in quadrantal bearing system. Bearings designated by Q.B. System are sometime called Reduced Bearings. 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed Bearing Conversion Of W.C.B into Q.B (R.B.) CASE W.C.B between Rule for Q.B (R.B.) Quadrant I 0 and 90 W.C.B N.E. II 90 and W.C.B S.E. III 180 and 270 W.C.B S.W. IV 270 and W.C.B N.W. 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 24 12

13 3. Bearing Conversion Of Q.B. (R.B.) into W.C.B CASE R.B Rule for W.C.B W.C.B between I N α E R.B 0 and 90 II S β E R.B 90 and 180 III S γ W 180 +R.B 180 and 270 IV N δ W R.B 270 and 360 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 25 Conversion of bearing from one system to the other: Example 2. Convert the following WCBs to RBs (a) WCB of AB = (Ans ) (b) WCB of BC = (Ans = ) Example 3. Convert the following whole circle bearing to R.B.: a b c d Example 4. Convert the following reduced bearing to W.C.B.: a. N E b. S E c. S W d. N W 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 26 13

14 FORE and BACK Bearings: Every line may be defined by two bearings, one observed at either end of the line. Both the bearings expressed in W.C.B System differ each other by 180. The bearing of a line in the direction of the progress of survey, is called Fore or Forward Bearing (F.B) while the bearing in the opposite direction of the progress of survey is known as Reverse or Back Bearing (B.B). Relationship Between Fore and Back Bearings: a). W.C.B System: Back bearing = Fore Bearing Positive sign is used when fore bearing is less than 180 and negative sign is used when the fore bearing is greater than 180. b). Q.B System: To convert the fore bearing of a line into its back bearing in Q.B system, replace N by S, S by N, E by W and W by E, without changing the numerical value of the bearing. 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 27 Example 5. The following are the observed fore bearing of the lines: a. AB, ; BC, ; CD, and DE, b. Find their back bearings. Example 6. The fore bearings of the lines are as follows: a. AB: N E; BC: S E; CD: S W; DE: N W. b. Find their back bearings 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 28 14

15 4. Azimuth Azimuths are angles measured clockwise from any reference meridian. They are measured from the north and vary from 0 to 360' and do not require letters to identify their quadrant. Example 7. 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed Azimuth 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 30 15

16 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 31 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 32 16

17 Example 8. Example 9. 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 33 Example 10. Convert the following azimuths to bearings 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 34 17

18 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 35 Example 11. 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 36 18

19 Example 12. Converting Bearing to Azimuth: North East: azimuth angle equals bearing Angle Ex: N 76 o 30 E = 76 o 30 South East: azimuth angle equals 180 minus bearing angle Ex: S 42 o 28 E = 180 o 42 o 28 = 137 o 32 South West: azimuth angle equals 180 plus bearing angle Ex: S 36 o 47 W = 180 o + 36 o 47 = 216 o 47 North West: azimuth angle equals 360 minus bearing angle Ex: N 62 o 56 W = 360 o - 62 o 56 = 297 o 04 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 37 Example 13. Converting Azimuth to Bearing: North East: the prefix N and the suffix E must be added Ex: 76 o 30 = N 76 o 30 E South East: the azimuth angle is subtracted from 180 o and S and E are added Ex: 168 o 40 = 180 o 168 o 40 = S 11 o 20 E South West: 180 o is subtracted from the azimuth angle and S and W are added Ex: 195 o 22 = 195 o o = S 15 o 22 W North West: the azimuth angle is subtracted from 360 o and N and W are added Ex: 314 o 35 = 360 o 314 o 35 = N 45 o 25 W 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 38 19

20 Example 14. N N 90 o 0 0 E Δ C L The center line is always located in the middle of the highway, with the exception of the PI for horizontal curves. Convert bearing to azimuth: S 75 o E = 104 o N 65 o E = 65 o Compute angle: Δ = 104 o o = 39 o Δ = 40 o L (L for left) 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 39 1/28/2018 Assistant Lecturer / Asmaa Abdulmajeed 40 20

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