Angles and Directions

Angles and Directions Angles and Directions Definitions Systems of Angle Measurement Angle Arithmetic Horizontal and Vertical Angles Angle and Direction Measuring Equipment 1

Angle Definitions A measure of the divergence of two intersecting lines Direction A heading or course with respect to a reference line Expressed as a Bearing or an Azimuth Definitions Cont. Azimuth = AZ 0º # AZ # 360 º CW from North Example: 100 º45 Bearing Angle = β 0º # β # 90º CW and CCW from North and South Quadrants: NE, SE, etc. Ex: N15 30 W 2

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Relationship between bearing angle (β) and Azimuth (AZ) when (N = 0 ) Reference Line or Meridian Magnetic North Geodetic North Astronomic North Grid North Assumed North Definitions Cont. 4

Systems of Angle Measurement Sexegesimal Centesimal Military System Radian Conversions Systems of Angle Meas. Cont. Sexegesimal system Unit: degree ( ) minute ( ) second ( ) D.MS 1 circle = 360 Circumference = π D = 2 π R = C 1 1 = C 360 5

Systems of Angle Meas. Cont. Sexegesimal system cont. 1 = 60 1 = 60 1 60 60 = 1 = 3600 Systems of Angle Meas. Cont. Sexegesimal system cont. Angles expressed in ( ) D.MS. Do not report angles in decimal degrees Calc. carried to nearest ( ) Note that 5 decimal places required to consistently prevent round off error when converting to seconds 6

Systems of Angle Meas. Cont. Sexegesimal system cont. Conversion from D.MS to decimal degrees (D.DDD) Convert 5 08 02 (5.0802) to D.DDD 8 2 D. DDD = 5 + + = 5. 13389 60 3600 8 + 2 60 D. DDD = 5 + = 5. 13389 60 Systems of Angle Meas. Cont. Sexegesimal system cont. Conversion from D.DDD to D.MS Convert 54.83257 to D.MS D. DDD = 54. 83257 D. MS = 54 ( 0. 83257 60) = 54 49. 9542 D. MS = 54 49 ( 0. 9542 60) DMS. = 544957 7

Systems of Angle Meas. Cont. Sexegesimal system cont. Calculator use many calculators have a conversion key to change from D.MS to D.DDD and vice versa some calculators do not require a conversion to use trig functions etc. know how your calculator works. Systems of Angle Meas. Cont. Made these conversions D.MS 5.0425 98.2742 120.1308 D.DDD 6.84231 42.15674 187.5714 D.DDD D.MS 8

Systems of Angle Meas. Cont. Answers to conversions D.MS 5.0425 98.2742 120.1308 D.DDD 5.07361 98.46167 120.21889 D.DDD 6.84231 42.15674 187.57139 D.MS 6.5032 42.0924 187.3417 Systems of Angle Meas. Cont. Centesimal system 1 circle = 400 grads g 1 1 = 400 C g c' 1 = 100 = 100 centesimal minutes c' c" 1 = 100 = 100 centesimal seconds g c' c" 54 75 85 g = 54. 7585 ( decimal grads) 9

Systems of Angle Meas. Cont. Military system 1 circle = 6400 mils 1mil = 1 6400 C The subunit of the mil is the decimal mil 1 mil is the angle subtending, approximately, 1 unit in 1000 units. Systems of Angle Meas. Cont. Radian measure angle ( radians) = S θ ( rad) = R C 1 circle = = R arc length radius 2R π = 2π rad R 10

Systems of Angle Meas. Cont. Conversions 360 = 2π rad 2π rad π rad 1 = = 360 180 360 180 1rad= = 2π π Systems of Angle Meas. Cont. Other conversions may be developed using: 360 = 400 360 = 6400 mil 2π rad = 6400 mil etc. g 11

Systems of Angle Meas. Cont. Make these conversions: 24 05'07" 2.18762 rad 280.42318 mil 1 mil (rad) ( ' ") ( ' ") (rad) Systems of Angle Meas. Cont. Answers to conversions: 24 05'07" 2.18762 rad 280.42318 mil 1 mil 0.420367 rad 125 20'29" 15 46'26" 0.000981rad 12

Angle Arithmetic 2 methods Adding two angles by first converting to decimal degrees 13

Adding two angles using the unit for unit method Subtracting 1 angle from another by first converting to D.DDD 14

Subtracting 1 angle from another on unit for unit basis 15

Horizontal Angles Angle Requirements Types of Horizontal Angles Interior angles Exterior angles Angles-to-the-right Deflection angles Closing the Horizon Horizontal Angles Cont. Angle Requirements Reference or starting line Direction of turning Angular distance (value of the angle) 16

Horizontal Angles Cont. Interior Angles polygon or closed traverse No direction associated with int. angles, but must be supplied as supplemental info to correctly represent a traverse. - Σ int. = ( n-2 ) 180 n = no. of sides Horizontal Angles Cont. Exterior Angles polygon or closed traverse No direction, but again, direction must be provided as supplemental information. - Σ = (n + 2) 180 n = no. of sides 17

Horizontal Angles Cont. Angle-to-the-Right Name indicates direction of turning Used with open or closed traverse When closed traverse, same as int. or ext. angle Azimuth from the back line - index on point last occupied Horizontal Angles Cont. Deflection Angles Angle to the right or left of the prolongation of the preceding line. Angles have designation of R or L Right Deflections are (+), Left Deflections are (-) For closed traverse: Σ defl. Angles = ± 360 regardless of number of sides 18

Horizontal Angles Cont. Closing the Horizon Term for measuring all the angles about a point. Σ α i = 360 Vertical Angles Direct vs Reverse Vertical Angle Meas. Reference Line is Horizontal Reference Line is Vertical Conversion Formulas Index Error 19

Vertical Angles Cont. First, consider direct and reverse positions of the telescope. Telescope in the illustration is in the direct position, vertical circle is on left (face left) and telescope tube level is under the telescope Vertical Angles Cont. Reference Line Horizontal Angle of elev. (+α) 0 α +90 Angle of depression (-α) 90 α 0 All true for Direct (α d ) and Reverse (α r ) measurements to the same point. 20

Vertical Angles Cont. Reference Line Vertical Called Zenith Angle (ZA) sometimes zenith distance (ZD) Always positive 0 ΖΑ 360 Direct (ZA d ) and Reverse (ZA r ) measurements to same point will be different ZA d + ZA r = 360 Vertical Angles Cont. Conversion Formulas When α is measured Direct or when 0 ZA 180 Use: α d = 90 - ZA d When α is measured Reverse or when 180 ZA 360 Use: α r = ZA r 270 21

Vertical Angles Cont. Find the missing values α d ZA d α r ZA r +10 265 160 +40-15 350 95-20 Vertical Angles Cont. Answers to missing values α d ZA d α r ZA r +10 80-5 265-70 160 +40 310-15 105 +80 350-5 95-20 250 22

Index Error Vertical Angles Cont. Results from improper (lack of adjustment) indexing of the vertical circle with respect to the direction of gravity. Eliminated by double sighting i.e., measuring angles once direct and once reverse and then averaging or calculating a correction. Vertical Angles Cont. Index Error cont. For angles measured as ±α Adjusted α = (α d + α r ) 2 For zenith angles (ZA) Closure = (ZA d + ZA r ) 360 C A = -Closure 2 Adjusted ZA d = ZA d + C A and Adjusted ZA r = ZA r + C A 23

Vertical Angles Cont. Correct for Vertical Index Error ZA d ZA r Sum Clos. C A Adj. ZA d 084 42 15 275 17 35 135 12 48 224 46 30 092 54 37 267 05 53 Vertical Angles Cont. Answers to Vertical Index correction ZA d ZA r Sum Clos. C A Adj. ZA d 084 42 15 275 17 35 359 59 50 135 12 48 224 46 30 359 59 18 092 54 37 267 05 53 360 00 30 24

Vertical Angles Cont. Answers to Vertical Index correction ZA d ZA r Sum Clos. C A Adj. ZA d 084 42 15 275 17 35 359 59 50-10 +5 084 42 20 135 12 48 224 46 30 359 59 18-42 +21 135 13 09 092 54 37 267 05 53 360 00 30 +30-15 092 54 22 Angle Measuring Equipment American Transit Theodolite Total Stations Instrument Errors Natural Errors Personal Errors Gross Blunders 25

Angle Measuring Equipment. Cont. American Transit Resolution 1 to 15 Mechanical instr. Graduated brass plate circles Included a compass Two horizontal motions: Repeating, double angle measurements Angle Measuring Equipment. Cont. Am. Transit Cont. Verniers: increased the angle reading resolution Least count = S n S = smallest grad. on main scale n = no. of div. on vernier Left side for CW angles and Right side for CCW angles. 26

Angle Measuring Equipment Cont. Theodolite Resolution: 0.1 to 6 Optical-mechanical Glass circles Compass was an accessory Included both repeating and directional instrument types Angle Measuring Equip. Cont. Theodolites Cont. Directional = 1 horizontal motion Optical plummets and tribrachs with 3 leveling screws were common Optical micrometers increase angle resolution Note the complex optical system in the repeating instrument to the left 27

Angle Measuring Equip. Cont. Theodolites Cont. Various circle reading methods, includes optical micrometers and direct scale reading Angle Measuring Equipment Cont. Total Stations Resolution: 1 to 6 Electro-mechanical Most are directional with one horizontal motion, but can simulate two horizontal motions for double angle measurement. 28

Angle Measuring Equipment Cont. Total Stations Cont. Digital display Distance measuring Data recording Minimum Skill to use Robotic remote control. Angle Measuring Equipment Cont. Instrumental Errors Line of sight axis perpendicular to horizontal (transit or tilting) axis Horizontal axis perpendicular to vertical axis 29

Angle Measuring Equipment Cont. Inst. Errors. Cont. Plate bubbles adjusted so horizontal circle will be in horz. plane Vertical circle indexed using gravity as a reference Plate centers and verniers concentric Angle Measuring Equipment Cont. Inst. Errors Cont. Horizontal and vertical cross wires must lie in respective planes. Telescope tube level axis must be parallel with the line of sight 30

Angle Measuring Equipment Cont. Natural Errors Wind Temperature changes Refraction Settling of tripod Angle Measuring Equipment Cont. Personal Errors Centering vertical axis over station Leveling (centering bubble) Pointing or centering on target, rod, or plumb line Vernier interpolation or matching marks with optical micrometers 31

Angle Measuring Equipment Cont. Gross Blunders Sighting on or setting up over the wrong sta. Careless plumbing and leveling over stations Careless sighting and centering on targets, rods, or plumb lines Reading the wrong circle or digital display Reading or recording wrong angles Turning the wrong tangent screw Direction Measuring Equipment Compass Hand Compasses Instrument Errors Natural Errors Personal Errors Gross Blunders North Seeking Gyroscope 32

Direction Measuring Equipment Compass Length of the needle was a factor in accuracy This one was developed by F. Henry Sipe in the early 1970s as a way to reduce the cost of rural land surveys. It had a 6- inch needle and a vernier reading to 1 of arc. Direction Meas. Equip. Cont. Compass Cont. Most compass circles are graduated to 1 or perhaps 30 Estimate to nearest 15 Compasses were mounted on tripods or a Jacob s staff. Level with a bull s eye level. 33

Direction Meas. Equip. Cont. Compass Cont. Sight from vane with slit to one with hair Counter weight on South end of needle East and West reversed on compass face Direction Meas. Equip. Cont. Hand Compasses 34

Direction Meas. Equip. Cont. Instrument Errors The pivot, needle, or sight vanes bent Magnetism of needle weak Natural Errors Local attraction Magnetic variations Magnetic declination angle between magnetic north and astronomic north Direction Meas. Equip. Cont. Natural Errors Cont. Magnetic variations Cont. Magnetic declination angle between magnetic north and astronomic north, changes with time. Annual change change in one year Secular change change in 2 or more years Personal Errors Parallax, reading from side of needle not along it from the south end 35

Direction Meas. Equip. Cont. Gross Blunders in compass work Reading the wrong end of the needle Setting the declination off on the wrong side Setting declination off when magnetic bearings required Compass not leveled Failure to check for local attraction Direction Meas. Equip. Cont. North Seeking Gyroscope Accuracy is 20 36