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 pointM o = 0 (M at o ) o = longitude of central meridian (for UTM zone) k o = 0.9996 (scale factor at the central meridian)
26
Embed
4.6 Transformation Between Geographic and UTM Coordinates 4.6.1 Conversion from Geographic to UTM Coordinates
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
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)
111
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
113
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
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
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
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
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
Projectionsurface
hHN
CentralMeridian
ko = 0.9996
119
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
123
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.