Transformation Between Geographic and UTM Coordinates 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 meridian Where  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  M o = 0 (M at  o ) o = longitude of central meridian (for UTM zone) k o = (scale factor at the central meridian)

111 Radius of curvature at a given azimuth Radius of curvature on the plane of the prime vertical FROM EQUATION SHEET 4.6 Transformation Between Geographic and UTM Coordinates Radius of Curvature in the plane of the meridian RmRm RNRN RR

112 A cos'eC tanT 22 2    4.6 Transformation Between Geographic and UTM Coordinates Conversion from Geographic to UTM Coordinates

113 Northing and Easting          ' A TT A eCCT A Ck k o           N 120 ' A eCTT A CTARk x o          o N 2 cos'11k k Rk x e o               N 720 ' tan A eCTT A CCT A RMMk y oo  UTM Scale Factor Or in terms of Latitude and Longitude 4.6 Transformation Between Geographic and UTM Coordinates Conversion from Geographic to UTM Coordinates

114 Where  1 = footprint latitude which is the latitude at the central meridian which has the same y coordinate of the point  = the rectifying latitude Used for converting UTM coordinates on an ellipsoid of known f and a, to  and. Negative values are used for western longitudes. These equations are not as accurate as the geographic to UTM conversion 4.6 Transformation Between Geographic and UTM Coordinates Conversion from UTM to Geographic Coordinates

115 o kR x D 1 N  1m1111 NN )( latitudefootprint at the calculated R and,R T, C, are R and,R,T,C radiansin is here w    sin 512 e 10976sin 96 e 151                      eee sin e 212sin                    4.6 Transformation Between Geographic and UTM Coordinates Conversion from UTM to Geographic Coordinates e e e    eee a M            o o k y MM 

116    cos 6 21  D ' TeCTC D CTD o     tan N   R R   D 4 24 D D             ' CeTCT  ' eCCT              4.6 Transformation Between Geographic and UTM Coordinates Conversion from UTM to Geographic Coordinates

117 earth.) theof radius average the using approximated be alsocanfactor scale UTM(The factor Grid distance Grid Distance  Ground (Elevation factor) factor) X (scaleFactor G  rid factor S UTMApproximatecale level) sea factor to (ScalefactorElevation 4.6 Transformation Between Geographic and UTM Coordinates UTM Map Scale Factor The 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 N in comparison to H, and because the geoid height is usually used for elevation. Elevation factor Approx. Elevation factor valuesor true approximate either the using computed becan then maps for UTMfactor scale grid The

Transformation Between Geographic and UTM Coordinates UTM Map Scale Factor [Review] R Ellipsoid surface Ground surface Mean sea level Projection surface h H N Central Meridian k o = k o = 1.00

119 GIVEN: Points on map from geodetic bench marks Map: NAD27, 1:250,000 NTS map of 72H (Willow Bunch Lake)  = 49°15’N  = 104°20’W Approx. Elevation h = 2430 ft = m 4.6 Transformation Between Geographic and UTM Coordinates EXAMPLE A FIND: a)UTM coordinates for point A, where: a = 6,378,206.4 m 1/f =  = 49°15’N = 49.25° = radians  = 104°20’W = ° = radians (UTM zone 13) o = 105° W k o =  o = 0°

120 Example EXAMPLE A 4.6 Transformation Between Geographic and UTM Coordinates

Transformation Between Geographic and UTM Coordinates EXAMPLE A '      0  o M ,390,  m  ,372,6m e e e ffe tan 2  T   sin1 22   N e a NR      M e e aR  cos' 22  eC  N used in this equation is not to be confused with geiodal height.

122 Meridian 3 o West of Control Meridian Equator Meridian 3 o East of Control Meridian UTM Coordinates O m North Central Meridian 500,000 m East 6 o Zone x Y 4.6 Transformation Between Geographic and UTM Coordinates EXAMPLE A    A eCTT A CTANk x o 120 '         A eCTT A CCT A NMMky oo 720 ' tan             m ,518  add a false easting of 500,000m E = 548, m N = 5,455, m

123 GSRUG - Geodetic Survey Routine: UTM and Geographic This program will compute the conversion between Geographic coordinates, latitude and longitude and Transverse Mercator Grid coordinates. 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 output Input Geographic Coordinates LATITUDE: 49 degrees 15 minutes 0 seconds NORTH LONGITUDE: 104 degrees 20 minutes 0 seconds WEST ELLIPSOID: CLARKE 1866 ZONE WIDTH: 6 Degree UTM Output- Calculated UTM coordinates: UTM Zone: 13 Easting: meters EAST Northing: meters NORTH GSRUG UTM coordinates: UTM Zone: 13 Easting: meters EAST Northing: meters NORTH 4.7 Application of UTM Coordinates

124 FIND: b) Latitude, longitude and height of point A with respect to NAD 83 ellipsoid a’ = m1/f’ = GIVEN dx = 4m dy = 159mdz = 188m for Saskatchewan 4.7 Application of UTM Coordinates EXAMPLE B Note: dx =  x

Application of UTM Coordinates EXAMPLE B mmhhh)704.25(66.740'  )"75.1('20104'  "155.0'1549'  h   m Rff R a azyxh N N sin1 cos 2   rad     m e e aR M sin       m e a R N sin1 22     fff ' 5    maaa '  = 49 o 15’ 0.16” N = 104 o 20’ 1.75 W = m = 0.155” rad    deg. = 1.75” = m  hR yx N cos sin          R R N N                       h R cossinf1- f R f a cossine azcossinysincosxsin 2 M M

126 Calculated Output data: LATITUDE: 49 o 15’ 0.155“ N LONGITUDE: 104 o 20’ 1.75” W National Transformation: NAD27 - NAD83 (NTv2), NTv2 Computation output Input Coordinates LATITUDE: 49 degrees 15 minutes seconds NORTH LONGITUDE: 104 degrees 20 minutes seconds WEST Transformation: NAD27 -> NAD83 NAD 83 Output data: LATITUDE: 49 o 15’ “ N Shift: seconds Standard deviation: 0.078m LONGITUDE: 104 o 20’ ” W Shift: seconds Standard Deviation: 0.208m 4.7 Application of UTM Coordinates EXAMPLE B con’t.

Application of UTM Coordinates EXAMPLE B con’t. National Geodetic Survey Works up to 50 o N In Western Canada

128 FIND: c) Latitude and longitude of point B with respect to NAD 27 E = 560,000mN = 5,470,000m a = 6,378,206.41/f = o = 105°W k o = M o = 0°  4.7 Application of UTM Coordinates EXAMPLE C

Application of UTM Coordinates EXAMPLE C

Application of UTM Coordinates EXAMPLE C    720 D C3'e252T45C298T D 'e9C4C10T35 2 D R tan N kN x D 000,60)easting false(Xx cos'eC tanT o   rad  rad  rad                                    cos 6 D CT21D o  T24'e8C3T28C     D " '  N 10'24.134"104  W

131 GSRUG - Geodetic Survey Routine: UTM and Geographic UTM to Geographic computation output Input Geographic Coordinates UTM Zone: 13 Northing: meters Easting: meters ELLIPSOID: CLARKE 1866 ZONE WIDTH: 6 Degree UTM Output geographic coordinates: LATITUDE: 49 o 22’ ” N LONGITUDE: 104 o 10’ ” W Calculated geographic coordinates: LATITUDE: 49 o 22’ ” N LONGITUDE: 104 o 10’ ” W 4.7 Application of UTM Coordinates EXAMPLE C

132  Given: A-B has a calculated grid Azimuth  = 37 o 52’ 59.5” Corrected (Astronomic) Azimuth A to B  = 37 o 52’ o 33’ 58” = Find: “True” Azimuth of line from A to B” (seconds) 4.8 Map Azimuth and Scale Factors of Line A m   “True” North   = 49 o 22’ 54” N  =104 o 10’ 24” W F)( 2 secsinθΔα 3       m F)( sin"2688θΔα 3   "1sincossin 12 1 F 22 mm  "1sincossin 12 1 F 22 mm  Grid North Central Meridian  =105 o W  = 2038 “ = 0 o 33’ 58” 38 o 26’ 57.5” B A  = 49 o 15’ N  =104 o 20” W

133 MAP AZIMUTH AND SCALE FACTORS OF LINE A - B 4.8 Map Azimuth and Scale Factors of Line A Corrected (Astronomic) Azimuth A to B  = 37 o 52’ o 33’ 58” = 38 o 26’ 57.5” factor Scale UTM Precise scale Factor (S.F.) mN mh H :software v.2H-GPS Canada Geodetics From  m   '      k      A TT 24 4  A eCCT A Ckk o )(sin Precise Elevation Factor (E.F.) cos 2 2    R R R R  M M    R N R N    hR R   Precise Elevation Factor (E.F.)  True grid factor = S.F. X E.F. “A”

134 MAP AZIMUTH AND SCALE FACTORS OF LINE A - B 4.8 Map Azimuth and Scale Factors of Line A Factor (S.F.) Scale UTM earth spherical aon based factors scale eApproximat factor grid eApproximat  ,367,6 272,367,6 Factor (E.F.)Elevation   "1.24'300"     '1549tan tan13.52  d ,367, ,                     R x kM op Rough Conversion

135 MAP AZIMUTH AND SCALE FACTORS OF LINE A - B 4.8 Map Azimuth and Scale Factors of Line A More Precise Spherical Conversion Corrected (Astronomic) Azimuth A to B  = 37 o 52’ o 33’ 58” = 38 o 26’ 57.5”