110 4.6 Transformation Between Geographic and UTM Coordinates 4.6.1 Conversion from Geographic to...
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Transcript of 110 4.6 Transformation Between Geographic and UTM Coordinates 4.6.1 Conversion from Geographic to...
1
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
Mo = 0 (M at o)
o = longitude of central meridian (for UTM zone)
ko = 0.9996 (scale factor at the central meridian)
2
22 cos)(sin N
N
RR
RRR
m
m
Radius of curvature at a given azimuth
22N
sin1 e
aRN
Radius of curvature on the plane of the prime vertical
FROM EQUATION SHEET
4.6 Transformation Between Geographic and UTM Coordinates
sin1
1
2
322
2
e
eaRm
Radius of Curvature in the plane of the meridian
Rm
RN
R
3
A
cos'eC
tanT22
2
radians in is here w
6sin3072
e354sin
1024
e45
256
e15
2sin1024
e45
32
e3
8
e3
256
e5
64
e3
4
e1
aM664
642642
4.6 Transformation Between Geographic and UTM Coordinates
4.6.1 Conversion from Geographic to UTM Coordinates
radiansin areand wherecos)( oo
4
Northing and Easting
62
422
2
7201614861
24'28134245
211
ATT
AeCCT
ACkk o
522
3
N120
'58721856
1A
eCTTA
CTARkx o
22
222
oN2
cos'11kkRk
xe
o
622
42
2
N720
'330600586124
4952
tanA
eCTTA
CCTA
RMMky 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
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
6okR
xD
1N
1m1111 NN )( latitudefootprint at the calculated R and ,R T, C, are R and ,R ,T ,C
radiansin is here w
41
31 8sin
512
e10976sin
96
e151
eee 41
21
311
1 4sin32
5516
e212sin
3227
23
4.6 Transformation Between Geographic and UTM Coordinates
4.6.2 Conversion from UTM to Geographic Coordinates
e
ee
2
2
111
11
eeea
M642
2565
643
41
oo k
yMM
7
1
2223
11
cos6
21
5
120
D1111 24'832825 TeCTC
DCTD
o
1
111
tanN
R
R
2
26
720
D
4
24
DD
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
8
earth.) theof radius average the using approximated be alsocan factor scale UTM(The
factor Grid
distance Grid Distance Ground
(Elevation factor)factor) X (scaleFactor G rid
factor S UTMApproximate cale
level) sea factor 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.
hR
R
HR
R
E
E
Elevation factor Approx. Elevation factor
2
2
oR2
x1kk
valuesor true approximate either the using
computed becan then maps for UTMfactor scale grid The
9
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
h
H
N
CentralMeridian
ko = 0.9996
k o =
1.0
0k
o = 1.00
10
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 ko= 0.9996 o = 0°
12
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 m
ee
e
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
e
aNR
sin1
1
23
22
2
Me
eaR
00290374.0cos' 22 eC
N used in this equation is not to be confused with geiodal height.
13Meridian 3o West of Control Meridian
Equator
Meridian 3o East of Control Meridian
UTM Coordinates
O m North
Ce
ntr
al
Me
rid
ian
50
0,0
00
m E
as
t
6o Zonex
Y
4.6 Transformation Between Geographic and UTM Coordinates
4.6.4 EXAMPLE A
AeCTTA
CTANkx o120
'58721856
15
223
AeCTTA
CCTA
NMMky oo720
'330600586124
4952
tan6
224
22
m5439.48,518
add a false easting of 500,000m
E = 548,518.544 m
N = 5,455,242.563 m
14
http://www.geod.nrcan.gc.ca/apps/gsrug/geo_e.phpGSRUG - 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
FIND:
b) Latitude, longitude and height of point A with respect to NAD 83 ellipsoid
a’ = 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
16
4.7 Application of UTM CoordinatesEXAMPLE B
mmhhh )704.25(66.740' )"75.1('20104'
"155.0'1549' h
mRffR
aazyxh N
Nsin1sinsincoscoscos 2
rad 0004874.0105059.8 6
m
e
eaRM 842.6372127
sin1
1
2
322
2
m
e
aRN 874.6390630
sin1 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
hR
yx
N cos
cossin
RR
NN
hR
cossinf1-f1-
Rf
a
cossineazcossinysincosxsin
2
M
M
17
Calculated Output data:
LATITUDE: 49o 15’ 0.155“ N
LONGITUDE: 104o 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: 49o 15’ 0.11403“ N
Shift: 0.11403 seconds
Standard deviation: 0.078m
LONGITUDE: 104o 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
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
19
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
21
4.7 Application of UTM Coordinates
4.7.2 EXAMPLE C
720
DC3'e252T45C298T9061
24
D'e9C4C10T35
2
D
R
tanN
0093924.0kN
xD
000,60)easting false(Xx
0028879.0cos'eC
35976.1tanT
621
22111
422
111
2
1
111
o1
122
1
12
1
-104.17337 rad 818168.10144274.0rad 832596.1
rad8618735.0
.38171349
cos
6
DCT21D
1
1111
3
11
o
T24'e8C3T28C25 222
D120
5
"168.54'22 49 N
10'24.134"104 W
22
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: 49o 22’ 54.168061” N
LONGITUDE: 104o 10’ 24.134352” W
Calculated geographic coordinates:
LATITUDE: 49o 22’ 54.168” N
LONGITUDE: 104o 10’ 24.134” W
4.7 Application of UTM Coordinates
4.7.2 EXAMPLE C
23
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
“Tru
e” N
ort
h
= 49o 22’ 54” N=104o 10’ 24” W
F)(2
secsinθΔα 3
m
F)(13158333.49sin"2688θΔα 3
"1sincossin12
1F 22
mm "1sincossin12
1F 22
mm
Gri
d N
ort
hGri
d N
ort
h
Cen
tral
Mer
idia
n
=10
5o
W = 2038 “ = 0o 33’ 58”
38o 26’ 57.5”
B
A = 49o 15’ N=104o 20” W
24
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”
factor Scale UTMPrecise scale Factor (S.F.)
-17.95mN 722.71mh 66.740H
:software v.2H-GPS Canada Geodetics From
m
99963.0
'281342452
11 222
k
1614861 2720
6
ATT
24
4
AeCCT
ACkk o
925.6379250)(sin
Precise Elevation Factor (E.F.)
cos22
RR
RR
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”
25
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
999884.0740272,367,6
272,367,6
Factor (E.F.)Elevation
"1.24'300"1.18241606.115.3013.52
'1549tan5280
2808.35.4851813.52tan13.52 d
99963.0272,367,62
5.548,4819996.0
21 2
2
2
2
R
xkM op
Rough Conversion