1. 110 4.6 Transformation Between Geographic and UTM Coordinates 4.6.1 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 = 0.9996 (scale factor at the central meridian)

2. 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

3. 112 A cos'eC tanT 22 2    4.6 Transformation Between Geographic and UTM Coordinates 4.6.1 Conversion from Geographic to UTM Coordinates

4. 113 Northing and Easting          6 2 4 22 2 720 1614861 24 '28134245 2 11 A TT A eCCT A Ck k o           5 22 3 N 120 '5872185 6 1 A eCTT A CTARk x o          22 2 22 o N 2 cos'11k k Rk x e o               6 22 4 2 2 N 720 '3306005861 24 495 2 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 4.6.1 Conversion from Geographic to UTM Coordinates

5. 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 4.6.2 Conversion from UTM to Geographic Coordinates

6. 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    4 1 3 1 8sin 512 e 10976sin 96 e 151                      eee 4 1 2 1 3 11 1 4sin 32 55 16 e 212sin 32 27 2 3                    4.6 Transformation Between Geographic and UTM Coordinates 4.6.2 Conversion from UTM to Geographic Coordinates e e e 2 2 1 11 11    eee a M 642 256 5 64 3 4 1            o o k y MM 

7. 116    1 2 22 3 11 cos 6 21  5 120 D 1111 24'832825 TeCTC D CTD o     1 1 11 tan N   R R   2 2 6 720 D 4 24 D D             2 1 22 111 3'252452989061 CeTCT  22 111 '941035 eCCT              4.6 Transformation Between Geographic and UTM Coordinates 4.6.2 Conversion from UTM to Geographic Coordinates

8. 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 4.6.3 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

9. 118 4.6 Transformation Between Geographic and UTM Coordinates 4.6.3 UTM Map Scale Factor [Review] R Ellipsoid surface Ground surface Mean sea level Projection surface h H N Central Meridian k o = 0.9996 k o = 1.00

10. 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 = 740.66m 4.6 Transformation Between Geographic and UTM Coordinates 4.6.4 EXAMPLE A FIND: a)UTM coordinates for point A, where: a = 6,378,206.4 m 1/f = 294.9786982  = 49°15’N = 49.25° = 0.859575 radians  = 104°20’W = -104.3333° = -1.82096 radians (UTM zone 13) o = 105° W k o = 0.9996  o = 0°

11. 120 Example 4.6.4 EXAMPLE A 4.6 Transformation Between Geographic and UTM Coordinates

12. 121 4.6 Transformation Between Geographic and UTM Coordinates 4.6.4 EXAMPLE A 00681478.0 1 ' 00676865.02 2 2 2 22 606.5457211      0  o M6874.630,390,  m  842.127,372,6m e e e ffe 34689285.1tan 2  T   sin1 22   N e a NR   1 1 2 3 22 2    M e e aR  00290374.0cos' 22  eC  N used in this equation is not to be confused with geiodal height.

13. 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 4.6.4 EXAMPLE A    A eCTT A CTANk x o 120 '5872185 6 1 5 2 2 3         A eCTT A CCT A NMMky oo 720 '3306005861 24 495 2 tan 6 22 4 2 2             m 5439. 48,518  add a false easting of 500,000m E = 548,518.544 m N = 5,455,242.563 m

14. 123 http://www.geod.nrcan.gc.ca/apps/gsrug/geo_e.php 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: 548518.544 meters EAST Northing: 5455242.563 meters NORTH GSRUG UTM coordinates: UTM Zone: 13 Easting: 548518.573 meters EAST Northing: 5455242.533 meters NORTH 4.7 Application of UTM Coordinates

15. 124 FIND: b) Latitude, longitude and height of point A with respect to NAD 83 ellipsoid a’ = 6378137m1/f’ = 298.257 GIVEN dx = 4m dy = 159mdz = 188m for Saskatchewan 4.7 Application of UTM Coordinates 4.7.1 EXAMPLE B Note: dx =  x

16. 125 4.7 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   rad0004874.0105059.8 6     m e e aR M 842.6372127 sin1 1 2 3 22 2       m e a R N 874.6390630 sin1 22     fff1072587.3 979.294 1 257.298 1 ' 5    maaa4.694.63782066378137'  = 49 o 15’ 0.16” N = 104 o 20’ 1.75 W = 714.956m = 0.155” rad 10 306.4105149.7 57    deg. = 1.75” = - 25.704m  hR yx N cos sin          R R N N                       h R cossinf1- f R f a cossine azcossinysincosxsin 2 M M

17. 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 00.000000 seconds NORTH LONGITUDE: 104 degrees 20 minutes 00.000000 seconds WEST Transformation: NAD27 -> NAD83 NAD 83 Output data: LATITUDE: 49 o 15’ 0.11403“ N Shift: 0.11403 seconds Standard deviation: 0.078m LONGITUDE: 104 o 20’ 1.87927” W Shift: 1.87927 seconds Standard Deviation: 0.208m 4.7 Application of UTM Coordinates 4.7.1 EXAMPLE B con’t. http://www.geod.nrcan.gc.ca/apps/ntv2/ntv2_utm_e.php

18. 127 4.7 Application of UTM Coordinates 4.7.1 EXAMPLE B con’t. National Geodetic Survey http://www.ngs.noaa.gov/cgi-bin/nadcon.prl Works up to 50 o N In Western Canada

19. 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 = 294.979 o = 105°W k o = 0.9996 M o = 0°  4.7 Application of UTM Coordinates 4.7.2 EXAMPLE C

20. 129 4.7 Application of UTM Coordinates 4.7.2 EXAMPLE C

21. 130 4.7 Application of UTM Coordinates 4.7.2 EXAMPLE C    720 D C3'e252T45C298T9061 24 D 'e9C4C10T35 2 D R tan N 0093924.0 kN x D 000,60)easting false(Xx 0028879.0cos'eC 35976.1tanT 6 2 1 22 111 4 22 111 2 1 1 11 o1 1 22 1 1 2 1  -104.17337  rad 818168.1  0144274.0rad 832596.1  rad8618735.0                               .38171349     cos 6 D CT21D 1 1111 3 11 o  T24'e8C3T28C25 222     D 120 5 " 168.54' 22 49  N 10'24.134"104  W

22. 131 GSRUG - Geodetic Survey Routine: UTM and Geographic UTM to Geographic computation output Input Geographic Coordinates UTM Zone: 13 Northing: 5470000 meters Easting: 560000 meters ELLIPSOID: CLARKE 1866 ZONE WIDTH: 6 Degree UTM Output geographic coordinates: LATITUDE: 49 o 22’ 54.168061” N LONGITUDE: 104 o 10’ 24.134352” W Calculated geographic coordinates: LATITUDE: 49 o 22’ 54.168” N LONGITUDE: 104 o 10’ 24.134” W 4.7 Application of UTM Coordinates 4.7.2 EXAMPLE C

23. 132  Given: A-B has a calculated grid Azimuth  = 37 o 52’ 59.5” Corrected (Astronomic) Azimuth A to B  = 37 o 52’ 59.5 + 0 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)(13158333.49sin"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

24. 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’ 59.5 + 0 o 33’ 58” = 38 o 26’ 57.5” factor Scale UTM Precise scale Factor (S.F.) -17.95mN 722.71mh 66.740H :software v.2H-GPS Canada Geodetics From  m   99963.0 '28134245 2 11 22 2      k  1614861 2  720 6    A TT 24 4  A eCCT A Ckk o 925.6379250 )(sin Precise Elevation Factor (E.F.) cos 2 2    R R R R  M M    R N R N 999887.0    hR R   Precise Elevation Factor (E.F.) 999517.0999887.099963.0  True grid factor = S.F. X E.F. “A”

25. 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 999513.0999884.099963.0 factor grid eApproximat  999884.0 740272,367,6 272,367,6 Factor (E.F.)Elevation   "1.24'300"1.18241606.115.3013.52     '1549tan 5280 2808.35.48518 13.52tan13.52  d 99963.0 272,367,62 5.548,48 19996.0 2 1 2 2 2 2                     R x kM op Rough Conversion

26. 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’ 59.5 + 0 o 33’ 58” = 38 o 26’ 57.5”