CHAPTER 2 COORDINATE SYSTEMS CE

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1 CHAPTER 2 COORDINATE SYSTEMS CE

2 2.0 Co-Ordinate Systems 2.1 RECTANGULAR (Cartesian) SYSTEMS Used to define positions and orientation D3 2 1 A B C D E Descartes Cartesian co-ordinates are denoted as (X, Y), or in this case (Numbers, Letters). Rene Descartes 17 th C 12 French scientist and philosopher

3 2.1.1 Mine Coordinates UTM, Arbitrary, etc 13

4 2.2 Geographic Coordinates Used to define position and orientation, change in location or relative distance between points on a sphere Based on latitude and longitude Equator: formed by the intersection of a plain bisection of the earth at right angles to the axis of rotation Meridians: formed by the intersection of vertical planes passing through the center of the earth and the sphere 14

5 2.2 Geographic Coordinates 15

6 2.2.1 Latitude Measured in degrees north or south of the equator 0 to 90 North 0 to 90 South 16

7 2.2.2 Longitude Measured in degrees west or east of the Prime Meridian The Meridian, which passes through Greenwich, England, is known as the Prime Meridian for most of the world 17

8 1899 Map of Scotland

9 2.3 Introduction to Co-ordinate Systems Simplification of the earth s surfaces Used for most forms of early navigation Used for most land surveying applications USA Canada boundary Provincial Boundaries 19

10 2.4 History of the Radius of the Earth Greek scholar Eratosthenese 220 B.C. Keeper of the scrolls for Egypt - Alexandria 3 basic requirements for determining radius of the earth: BC 1) Precisely measured N S line 2) One of the baselines on the Tropic of Cancer (summer solstice) 3) Length of noon shadow a minimum at the other end of the baseline 20

11 2.4 History of the Radius of the Earth (250,000 stadia) (6,403,657 m) Poseidonius B.C. C earth = 24, 000 mi (R=6,147,316 m) Greek philosopher Ptolemy C earth = 18,000 mi (R=4,610,430 m) Columbus most likely used Ptolemy s value R ~ 6,367,272 m 21

12 2.5 Astronomical Coordinates Used for navigating and early mapping Positions determined on a celestial sphere Celestial Equator Plane made by Earth s Equator on the celestial sphere 22

13 2.5 Astronomic Coordinates (Equatorial Coordinate System) Sun s center is directly over the earth s equator Ecliptic Path made by the Earth travelling around the sun Perihelion (Jan 4) and Aphelion (July 5) Points on Ecliptic when the Earth is farthest away from the Sun Celestial Equator: The Vernal equinox and Autumnal equinox are points of intersection between the ecliptic and the celestial equator. This plane also passes through the center of the sun 23

14 Nadir Zenith Dependant Astronomic Coordinates Position of stars based on this method continually changing Azimuth and Altitude Azimuth = angle from the north Altitude h = angular position above the horizon To Polaris Polaris Saskatoon HORIZON h h N.P. h f 90-h S.P. EARTH Equator 24

15 2.5.2 Independent Astronomic Coordinates All stars are basically fixed in position on the celestial sphere Right ascension (R.A.) and declination (d) y Star R.A. Upper Transit Hour Angle Pole Observer Lower Transit R.A. measured in hours, minutes, and seconds westward along the celestial equator from the vernal equinox to the intersection of the meridian passing through the star. 1 hour = 15 o [A measure of time] Declination measured north (+) or south (-) from the celestial equator along the meridian to the star REF: 25

16 2.5.2 Independent Astronomic Coordinates Values of R.A. and declination are provided in solar ephemeris or Star Almanacs Position of stars based on this method essentially fixed 26

17 2.5.2 Independent Astronomic Coordinates Independent position of the stars Note: Sidereal Hour Angle = Distance measure eastward in degrees from vernal equinox. 27

18 2.5.2 Independent Astronomic Coordinates Meade 8 LX200 Telescope 28

19 2.6 Relationship between Astronomic and geographic coordinate systems Stars are used to provide and determine latitude, longitude, and time Latitude Saskatoon HORIZON E ZENITH S h Polaris T N.P. O EARTH S.P. Equator NADIR 29

20 2.6.1 Latitude Polaris d N.C.P. Measured altitude of star h, at L.C. North Celestial Pole U.C. o 90 - d o 90 - d L.C. Polaris = h +( S.C.P. o 90 - d ) 90-d Night horizon looking north 30

21 2.6.2 Azimuth Azimuth of reference line = measured angle from Polaris W.E. Polaris E.E. 31

22 2.6.3 Time/Longitude Solar Day The interval between two successive lower transits of the sun s center over the same meridian 360 o Rotation in 24 hr. 15 o per hour Upper Transit Sun 32

23 2.6.3 Time/Longitude In 1759, an obsessed English clockmaker created a watch so excruciatingly precise that he received a prize worth one million dollars (U.S.). The prize Measuring was awarded Time because with this watch, the clockmaker, John Harrison, had solved the problem of finding Mogul Caliph in Jaipur, India longitude at sea. But what does time have to do with knowing where you are? John Harrison ( ) H1 Sir Cloudesley Shovell s fleet wrecked on the Isles, Artist unknown. Harrison's linked balance mechanism negates the effects of motion of the clock. 33 Animation National Maritime Museum

24 2.6.3 Time/Longitude Standard Time Sir Sandford Fleming, The Father of Standard Time. Proposed Standard Time Prime Meridian Conference s 144 time zones in N.A. 12 mi E and 12 mi. W, 1 minute difference in time. Railways (i.e. Baltimore time) Irish travel guide Slago 5:35 pm, misprint, 5:35 a.m. Fleming was born on July 7, 1827, in Scotland, and emigrated to Quebec at the age of 17. In 1858 as the chief engineer of the Northwest Railway, he first proposed a railway to the pacific. He was in charge of the initial survey for the first Canadian railway to span the continent. Fleming also designed the first Canadian postage stamp. Issued in 1851, it cost 3 cents and depicted the beaver. Fleming took an active part in the intellectual and scientific life of Canada, throughout his long career and received many honours. He died in Halifax in

25 2.6.3 Time/Longitude Time Zones 35

26 2.6.3 Time/Longitude 36

27 2.6.3 Time/Longitude Greenwich Mean Time Time at Greenwich, England Sometimes called Universal time (useful for astronomers, astronauts and people dealing with satellite data) Daylight Savings Time An Hour shift which is seasonally inserted in some Time Zones. Daylight savings time begins when clocks are set from 2:00 a.m. to 3:00 a.m. on the first Sunday in April and ends when clocks are set from 3:00 a.m. to 2:00 a.m. on last Sunday in October U.S./Canada Second Sunday in March first Sunday in November Europe 2 nd last Sunday in March last Sunday in 1:00 a.m. GMT Note: Only Saudi Arabia uses local times because of religious considerations 37

28 2.6.3 Time/Longitude International Date Line Imaginary line that separates two calendar dates Matter of convenience, no force in international law 38

29 2.6.3 Time/Longitude Sidereal Time The interval between two successive upper transits of the vernal equinox over the same meridian Mean Solar Days per year Sidereal Days per year One Sidereal Day = O.9973 Solar Days = 23 hr. 56 m sec solar days Upper Transit y Star R.A. Hour Angle Pole Lower Transit 39

30 2.7 Convergency Between Meridians Correction required when trying to place a rectangular coordinate system on a sphere Assumptions: The earth is a sphere (R = 20,890,000 ft., = 6,367,272 m.) CE = 40,006,

31 2.7.1 Angular Convergence Definitions q = angular convergence of meridians f = latitude d = distance between meridians Dl = angular distance between meridians 41

32 q = sin l = Angular Convergence q = CD / / CD fn : ( D l and / ' / ( radians q = ( CD / DE ) = D l = f = d DE DO DE R cos f q = D l sin f = ( d [ subst f ) ) DO ' ( radians. ) DO ' D l = D l sin f DO ' / sin f into 1 ) / R )(sin f / cos f ) = 7 ( d / R )(tan f ) North D B B O 1. 1 ) ) q ( rads ) = x 10 d tan f, for d in meters. (1/6,367,272) = 1.571E-7 [ Note : 2 p radians in 360 o, or 1 rad = o ] E P O 360 deg. q " = 2 p rads x q " = d tan f ( 3600 " / deg.) d ( miles 20, 890, 000 ft [ Note : for q in " and d in miles ]... ) 5280 ft / mile tan f 1. 3 ) 42

33 2.7.1 Angular Convergence Definitions L = length of meridian between parallel of latitudes q = angular convergence between meridians at latitude s = distance along parallel between meridians c = linear convergence along parallel 43

34 2.7.1 Angular Convergence Derivation for Linear Convergence c q = ( Subst. int o Eq ) L c s tan f = L R ( sl tan f ) c = ) R If d and L are expressed in miles and R is the radius L c q North Meridian q Mean Distance s Parallel North of the earth in feet, then : c = 4 3 s L tan f ) 44

35 2.8 Measurement of Distance on the Earth s Surface Nautical Miles 90 o N.P. C E = 40,006, m Earth Equator Origin of the meter 1 Nautical Mile = 1 arc S.P. 1 Nautical Mile = C E /(360x60) equal to one ten-millionth of the distance from the equator to the pole measured on a meridian: 1 Nautical Mile ft.= 1,852.16m [for C E = 40,006,749.88m] 45

36 2.8 Measurement of Distance on the Earth s Surface

37 2.8 Measurement of Distance on the Earth s Surface Spherical Trigonometry B Sin Law and Cosine Law B a sin a sin b = = sin A sin B sin c sin C cos a = cos b cos c + sin b sin c Cos A ) 1. 7 ) O a b c C A M P P c A b C Example The latitude and longitude of Boston and Cape Town are know to be: Boston: f = 42 o 21 North and l = 71 o 04 West Cape Town f= 33 o 56 South and l =18 o 29 East 47

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