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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 pointMo = 0 (M at o)o = longitude of central meridian (for UTM zone)ko = 0.9996 (scale factor at the central meridian)
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22 cos)(sin N
N
RRRRR
m
m
Radius of curvature at a given azimuth
22N
sin1 eaRN
Radius of curvature on the plane of the prime vertical
FROM EQUATION SHEET
4.6 Transformation Between Geographic and UTM Coordinates
sin1
1
23
22
2
e
eaRm
Radius of Curvature in the plane of the meridian
Rm
RN
R
112
Acos'eC
tanT22
2
radiansinisherew
6sin3072
e354sin1024
e45256e15
2sin1024
e4532e3
8e3
256e5
64e3
4e1
aM664
642642
4.6 Transformation Between Geographic and UTM Coordinates
4.6.1 Conversion from Geographic to UTM Coordinates
radiansin areand wherecos)( oo
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Northing and Easting
62
422
2
7201614861
24'28134245
211 ATTAeCCTACkk o
522
3
N120
'58721856
1 AeCTTACTARkx o
22
222
oN2
cos'11kkRk
xeo
622
42
2
N720
'330600586124
4952
tan AeCTTACCTARMMky 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
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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
115okR
xD1N
1m1111 NN )(latitudefootprint at thecalculatedRand,RT,C,areRand,R,T,Cradiansin isherew
41
31 8sin
512e10976sin
96e151
eee 41
21
311
1 4sin32
5516e212sin
3227
23
4.6 Transformation Between Geographic and UTM Coordinates
4.6.2 Conversion from UTM to Geographic Coordinates
eee
2
2
11111
eeea
M642
2565
643
41
oo k
yMM
116
1
2223
11
cos6
21
5
120D
1111 24'832825 TeCTCDCTDo
1
111
tanN
R
R
2
26
720D
4
24DD
21
22111 3'252452989061 CeTCT
22111 '941035 eCCT
4.6 Transformation Between Geographic and UTM Coordinates
4.6.2 Conversion from UTM to Geographic Coordinates
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earth.)theofradiusaveragetheusingapproximatedbealsocan factor scaleUTM(The
factorGriddistanceGridDistance Ground
(Elevation factor)factor) X(scaleFactorG rid
factorSUTMApproximate cale
level)seafactor to(Scalefactor Elevation
4.6 Transformation Between Geographic and UTM Coordinates
4.6.3 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 N in comparison to H, and because the geoid height is usually used for elevation.
hRR
HRR
E
E
Elevation factor Approx. Elevation factor
2
2o
R2x1kk
valuesor trueapproximateeither theusingcomputedbecan then mapsfor UTMfactor scalegridThe
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4.6 Transformation Between Geographic and UTM Coordinates
4.6.3 UTM Map Scale Factor[Review]
R
Ellipsoid surface
Ground surface
Mean sea level
Projectionsurface
hHN
CentralMeridian
ko = 0.9996
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GIVEN: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 = 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 ko= 0.9996 o = 0°
120
Example
4.6.4 EXAMPLE A
4.6 Transformation Between Geographic and UTM Coordinates
121
4.6 Transformation Between Geographic and UTM Coordinates
4.6.4 EXAMPLE A
00681478.01
'
00676865.02
2
22
22
606.5457211
0oM
6 874.630,390, m
842.127,372,6 me
ee
ffe
6sin3072
354sin1024
45256
15
2sin1024
4532
38
3256
564
34
1
664
642642
eee
eeeeee
aM
0075952.0cos oA
34689285.1tan2 T
sin1 22N
eaNR
sin1
123
22
2
M
e
eaR
00290374.0cos' 22 eC
N used in this equation is not to be confused with geiodal height.
122Meridian 3o West of
Control Meridian
Equator
Meridian 3o East ofControl Meridian
UTM Coordinates
O m North
Cen
tral
Mer
idia
n50
0,00
0 m
Eas
t
6o Zonex
Y
4.6 Transformation Between Geographic and UTM Coordinates
4.6.4 EXAMPLE A
AeCTTACTANkx o120
'58721856
15
223
AeCTTACCTANMMky oo 720'3306005861
24495
2tan
622
42
2
m5439.48,518
add a false easting of 500,000m
E = 548,518.544 m
N = 5,455,242.563 m
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http://www.geod.nrcan.gc.ca/apps/gsrug/geo_e.phpGSRUG - Geodetic Survey Routine: UTM and GeographicThis 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 outputInput Geographic CoordinatesLATITUDE: 49 degrees 15 minutes 0 seconds NORTHLONGITUDE: 104 degrees 20 minutes 0 seconds WESTELLIPSOID: CLARKE 1866 ZONE WIDTH: 6 Degree UTM
Output- Calculated UTM coordinates:UTM Zone: 13Easting: 548518.544 meters EASTNorthing: 5455242.563 meters NORTH
GSRUG UTM coordinates:UTM Zone: 13Easting: 548518.573 meters EASTNorthing: 5455242.533 meters NORTH
4.7 Application of UTM Coordinates
124
FIND:
b) Latitude, longitude and height of point A with respect to NAD 83ellipsoida’ = 6378137m 1/f’ = 298.257
GIVEN
dx = 4m dy = 159m dz = 188m for Saskatchewan
4.7 Application of UTM Coordinates
4.7.1 EXAMPLE B
Note: dx = x
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4.7 Application of UTM CoordinatesEXAMPLE B
mmhhh )704.25(66.740' )"75.1('20104'
"155.0'1549' h
mRffRaazyxh N
Nsin1sinsincoscoscos 2
rad 0004874.0105059.8 6
m
e
eaRM 842.6372127sin1
1
23
22
2
m
e
aRN 874.6390630sin1 22
fff 1072587.3979.2941
257.2981' 5
maaa 4.694.63782066378137'
= 49o 15’ 0.16” N= 104o 20’ 1.75 W
= 714.956m
= 0.155”rad 10306.4105149.7 57 deg.
= 1.75”
= - 25.704m
hRyx
N coscossin
RRN
N
hR
cossinf1-f1-
Rfa
cossineazcossinysincosxsin2
M
M
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Calculated Output data:LATITUDE: 49o 15’ 0.155“ N
LONGITUDE: 104o 20’ 1.75” W
National Transformation: NAD27 - NAD83 (NTv2), NTv2Computation outputInput CoordinatesLATITUDE: 49 degrees 15 minutes 00.000000 seconds NORTHLONGITUDE: 104 degrees 20 minutes 00.000000 seconds WESTTransformation: NAD27 -> NAD83
NAD 83 Output data:LATITUDE: 49o 15’ 0.11403“ NShift: 0.11403 secondsStandard deviation: 0.078mLONGITUDE: 104o 20’ 1.87927” WShift: 1.87927 secondsStandard 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
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4.7 Application of UTM Coordinates
4.7.1 EXAMPLE B con’t.
National Geodetic Surveyhttp://www.ngs.noaa.gov/cgi-bin/nadcon.prl
Works up to 50o NIn Western Canada
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FIND:c) Latitude and longitude of point B with respect to NAD 27E = 560,000m N = 5,470,000ma = 6,378,206.4 1/f = 294.979o= 105°W ko= 0.9996 Mo= 0°
4.7 Application of UTM Coordinates
4.7.2 EXAMPLE C
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rad 85940716.0
2565
643
41
00681478.01
'
00676865.02
876.5472188
642
2
22
22
eeea
Me
ee
ffe
kyMMo
o
4.7 Application of UTM Coordinates4.7.2 EXAMPLE C
00169791.01111
2
2
1
eee
rad 8619251.08sin 512e10976sin
96e151
4sin32
5516e212sin
3227
23
41
31
41
21
311
1
eee
33.6390681
sin1
77.6372278sin1
1
1221
23
122
2
1
eaR
e
eaR
N
130
4.7 Application of UTM Coordinates
4.7.2 EXAMPLE C
720DC3'e252T45C298T9061
24D'e9C4C10T35
2D
RtanN
0093924.0kNxD
000,60)eastingfalse(Xx0028879.0cos'eC
35976.1tanT
621
22111
422
111
2
1
111
o1
122
1
12
1
-104.17337rad818168.10144274.0rad832596.1
rad8618735.0
.38171349
cos
6DCT21D
1
1111
3
11
o
T24'e8C3T28C25 222
D120
5
"168.54'2249 N
10'24.134"104 W
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GSRUG - Geodetic Survey Routine: UTM and GeographicUTM to Geographic computation outputInput Geographic CoordinatesUTM Zone: 13Northing: 5470000 metersEasting: 560000 metersELLIPSOID: CLARKE 1866 ZONE WIDTH: 6 Degree UTM
Output geographic coordinates:LATITUDE: 49o 22’ 54.168061” NLONGITUDE: 104o 10’ 24.134352” W
Calculated geographic coordinates:LATITUDE: 49o 22’ 54.168” NLONGITUDE: 104o 10’ 24.134” W
4.7 Application of UTM Coordinates
4.7.2 EXAMPLE C
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Given:A-B has a calculated grid Azimuth = 37o 52’ 59.5”
Corrected (Astronomic) Azimuth A to B = 37o 52’ 59.5 + 0o 33’ 58” =
Find:“True” Azimuth of line from A to B” (seconds)
4.8 Map Azimuth and Scale Factors of Line A
m
= 49o 22’ 54” N=104o 10’ 24” W
F)(2
secsinθΔα 3
m
F)(13158333.49sin"2688θΔα 3
"1sincossin121F 22
mm "1sincossin121F 22
mm
Grid
Nor
thGrid
Nor
th
Cen
tral
Mer
idia
n
=105
oW = 2038 “ = 0o 33’ 58”
38o 26’ 57.5”
B
A = 49o 15’ N=104o 20” W
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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 = 37o 52’ 59.5 + 0o 33’ 58” = 38o 26’ 57.5”
factorScaleUTMPrecise scale Factor (S.F.)
-17.95mN722.71mh66.740H:softwarev.2H-GPSCanadaGeodeticsFrom
m
99963.0
'281342452
11 222
k
1614861 2720
6
ATT24
4A
eCCTACkk o
925.6379250)(sin
Precise Elevation Factor (E.F.)
cos22
R
RRR
M
M
RN
RN
999887.0
hR
R
Precise Elevation Factor (E.F.)
999517.0999887.099963.0
True grid factor = S.F. X E.F.
“A”
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MAP AZIMUTH AND SCALE FACTORS OF LINE A - B
4.8 Map Azimuth and Scale Factors of Line A
Factor (S.F.)ScaleUTMearthsphericalaon basedfactorsscaleeApproximat
999513.0999884.099963.0factorgrideApproximat
999884.0740272,367,6
272,367,6Factor (E.F.)Elevation
"1.24'300"1.18241606.115.3013.52
'1549tan5280
2808.35.4851813.52tan13.52 d
99963.0272,367,625.548,4819996.0
21 2
2
2
2
RxkM op
Rough Conversion
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MAP AZIMUTH AND SCALE FACTORS OF LINE A - B
4.8 Map Azimuth and Scale Factors of Line A
2φφsin""θ BA
More Precise Spherical Conversion
2038
238166667.4949.25
sin"2688"θ
Corrected (Astronomic) Azimuth A to B = 37o 52’ 59.5 + 0o 33’ 58” = 38o 26’ 57.5”