4.6 Transformation Between Geographic and UTM …

1 Transformation Between Geographic and UTM Conversion from Geographic to UTM coordinates Used for converting and on an ellipsoid of known f and a, to UTM coordinates . Negative values are used for western longitudes. These equations are accurate to about a centimeter at 7 of longitude from the central meridianWhere o= 0 (latitude of the central meridian at the origin of the x, y coordinates )M = True distance along central meridian from the equator to across from the point Mo= 0 (M at o) o= longitude of central meridian (for UTM zone)ko= (scale factor at the central meridian)111 22cos)

2 (sinNNRRRRRmm Radius of curvature at a given azimuth 22 Nsin1eaRN Radius of curvature on the plane of the prime verticalFROM EQUATION Transformation Between Geographic and UTM coordinates sin1123222 eeaRm Radius of Curvature in the plane of the meridianRmRNR 112 Acos'eCtanT222 radiansinisherew6sin3072e354sin1024e4525 6e152sin1024e4532e38e3256e564e34e1aM6646 42642 Transformation Between Geographic and UTM Conversion from Geographic to UTM Coordinatesradiansin areand wherecos)(oo 113 Northing and Easting 624222720161486124'28134245211 ATTAeCCTACkko 5223N120'587218561 AeCTTACTARkxo 22222oN2cos'11kkRkxeo 622422N720'3306005861244952tanAeCTTACCTA RMM kyoo UTM Scale FactorOr in terms of Latitude and Transformation Between Geographic and UTM Conversion from Geographic to UTM Coordinates114 Where 1= footprint latitude which is the latitude at the central meridian which has the same y coordinate of the point = the rectifying latitudeUsed for converting UTM coordinates on an ellipsoid of known f and a, to and.)

3 Negative values are used for western equations are not as accurate as the Geographic to UTM Transformation Between Geographic and UTM Conversion from UTM to Geographic Coordinates115okRxD1N 1m1111NN)(latitudefootprint at thecalculatedRand,RT,C,areRand,R,T,Cradi ansin isherew 41318sin512e10976sin96e151 eee412131114sin325516e212sin322723 Transformation Between Geographic and UTM Conversion from UTM to Geographic Coordinateseee2211111 eeeaM642256564341 ookyMM 116 1222311cos621 5120D111124'832825 TeCTCDCTDo 1111tanN RR 226720D424DD 21221113'252452989061 CeTCT 22111'941035eCCT Transformation Between Geographic and UTM Conversion from UTM to Geographic Coordinates117earth.

4 Theofradiusaveragetheusingapproximatedbe alsocan factor scaleUTM(ThefactorGriddistanceGridDistan ce Ground(Elevation factor)factor) X(scaleFactorG ridfactorSUTMA pproximatecalelevel)seafactor to(Scalefactor Elevation Transformation Between Geographic and UTM UTM Map Scale FactorThe elevation factor can be approximated using the average radius of the earth (R=6,367,272m) and elevation above the geoid rather than the elevation above the ellipsoid. This is done because of the relatively small value of Nin comparison to H, and because the geoid height is usually used for HRREE Elevation factorApprox.)

5 Elevation factor 22oR2x1kkvaluesor trueapproximateeither theusingcomputedbecan then mapsfor UTMfactor Transformation Between Geographic and UTM UTM Map Scale Factor[Review]REllipsoid surfaceGround surfaceMean sea levelProjectionsurfacehHNCentralMeridian ko= :Points on map from geodetic bench marksMap: NAD27, 1:250,000 NTS map of 72H (Willow Bunch Lake) = 49 15 N = 104 20 WApprox. Elevation h = 2430 ft = Transformation Between Geographic and UTM EXAMPLE AFIND:a) UTM coordinates for point A, where:a = 6,378, m1/f = = 49 15 N = = radians = 104 20 W = = radians (UTM zone 13)

6 O = 105 Wko= o = 0 EXAMPLE Transformation Between Geographic and UTM Transformation Between Geographic and UTM EXAMPLE ' 0 ,390, m ,372,6meeeffe6sin3072354sin102445256152s in10244532383256564341664642642 eeeeeeeeeaM oA T sin122 NeaNR sin1123222 MeeaR '22 eC N used in this equation is not to be confused with geiodal 3oWest ofControl MeridianEquatorMeridian 3oEast ofControl MeridianUTM CoordinatesO m NorthCentral Meridian500,000 m Transformation Between Geographic and UTM EXAMPLE A AeCTTACTANkxo120'5872185615223 AeCTTACCTANMM kyoo720'3306005861244952tan622422 ,518 add a false easting of 500,000mE = 548, mN = 5,455, m123 - Geodetic Survey Routine: UTM and GeographicThis program will compute the conversion Between Geographic coordinates , latitude and longitude and Transverse Mercator Grid coordinates .

7 The user may choose this standard projection or may choose a 3 degree as defined for Canada. The parameters of scale, central meridian, false easting and false northing may define any TM projection and are already defined within the program for two standard projections, UTM and 3 degree. Geographic to UTM computation outputInput Geographic CoordinatesLATITUDE: 49 degrees 15 minutes 0 seconds NORTHLONGITUDE: 104 degrees 20 minutes 0 seconds WESTELLIPSOID: CLARKE 1866 ZONE WIDTH: 6 Degree UTMO utput- Calculated UTM coordinates :UTM Zone: 13 Easting: meters EASTN orthing: meters NORTHGSRUG UTM coordinates :UTM Zone: 13 Easting: meters EASTN orthing: meters Application of UTM Coordinates124 FIND.

8 B) Latitude, longitude and height of point A with respect to NAD 83ellipsoida = 6378137m1/f = = 4mdy = 159mdz = 188m for Application of UTM EXAMPLE BNote: dx = Application of UTM CoordinatesEXAMPLE Bmmhhh) ( ' )" ('20104' " '1549' h mRffRaazyxhNNsin1sinsincoscoscos2 '5 ' = 49o15 N= 104o20 W= = - hRyxNcoscossin RRNN hRcossinf1-f1-Rfacossineazcossinysincosx sin2MM126 Calculated Output data:LATITUDE: 49o 15 NLONGITUDE: 104o20 WNational Transformation : NAD27 - NAD83 (NTv2), NTv2 Computation outputInput CoordinatesLATITUDE: 49 degrees 15 minutes seconds NORTHLONGITUDE: 104 degrees 20 minutes seconds WESTT ransformation: NAD27 -> NAD83 NAD 83 Output data:LATITUDE: 49o15 NShift: secondsStandard deviation: : 104o20 WShift: secondsStandard Deviation: Application of UTM EXAMPLE B con Application of UTM EXAMPLE B con Geodetic up to 50oNIn Western Canada128 FIND.

9 C) Latitude and longitude of point B with respect to NAD 27E = 560,000mN = 5,470,000ma = 6,378, 1/f = o= 105 Wko= 0 Application of UTM EXAMPLE C129rad ' eeeaMeeeffekyMMoo Application of UTM EXAMPLE eee rad 512e10976sin96e151 4sin325516e212sin322723413141213111 eee Application of UTM EXAMPLE C 720DC3'e252T45C298T906124D' ,60)eastingfalse( ' .38171349 cos6 DCT21D11111311o T24'e8C3T28C25222 D1205" '2249 N10' "104 W131 GSRUG - Geodetic Survey Routine: UTM and GeographicUTM to Geographic computation outputInput Geographic CoordinatesUTM Zone: 13 Northing: 5470000 metersEasting: 560000 metersELLIPSOID: CLARKE 1866 ZONE WIDTH: 6 Degree UTMO utput Geographic coordinates :LATITUDE: 49o22 NLONGITUDE: 104o10 WCalculated Geographic coordinates :LATITUDE: 49o22 NLONGITUDE: 104o10 Application of UTM EXAMPLE C132 Given.

10 A-B has a calculated grid Azimuth = 37o52 Corrected (Astronomic) Azimuth A to B = 37o52 + 0o33 58 =Find: True Azimuth of line from A to B (seconds) Map Azimuth and Scale Factors of Line Am = 49o22 54 N =104o10 24 WF)(2secsin 3 mF)( "2688 3 "1sincossin121F22mm "1sincossin121F22mm Grid NorthGrid NorthCentral Meridian =105oW = 2038 = 0o33 58 38o26 BA = 49o15 N =104o20 W133 MAP AZIMUTH AND SCALE FACTORS OF LINE A - Map Azimuth and Scale Factors of Line ACorrected (Astronomic) Azimuth A to B = 37o52 + 0o33 58 =38o26 factorScaleUTMP recise scale Factor ( ) m '28134245211222 k 16148612 7206 ATT244 )(sinPrecise Elevation Factor ( )