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)
110
 4.6 Transformation Between Geographic
and UTM Coordinates
FROM EQUATION SHEET

Radius of Curvature in the plane of the meridian

1 e2
Rm  a
1  e 
3
2
sin 
2 2

Radius of curvature on the plane of the prime vertical

a
N  RN 
1  e 2 sin 2
 
Rm

R
Radius of curvature at a given azimuth

R m RN
R 
Rm sin 2 ( )  RN cos 2  
RN
111
 4.6 Transformation Between Geographic
and UTM Coordinates
4.6.1 Conversion from Geographic to UTM Coordinates

T  tan2 
C  e'2 cos 2 
A  (  o ) cos   where  and o are in radians

 e 2 e 4
e 6   e2 e 4
e 6  

1   3  5   
  3    3  45   
 sin 2 
 4 64 256   8 32 1024  
M  a 
 e 4
e 6   e 6 
15  45   sin 4   35   sin 6   

 256 
1024   3072 
w here  is in radians

112
 4.6 Transformation Between Geographic
and UTM Coordinates
4.6.1 Conversion from Geographic to UTM Coordinates
Northing and Easting

 
A3
 
5
x  k o RN  A  1  T  C   5  18T  T  72C  58e'
2 2 A

 6 120 
  A2 6 

   
4
2 A 2 A
   
y k o  M M o RN tan     
5 T 9C 4C   
61 58T T 2
 
600C 330 e' 
  2 24 720 

UTM Scale Factor

 
A2
   
4 6
  
k k o 1 1 C      2

5 4T 42C 13C 28e' 2 A
  
61 148T 16T 2 A

 2 24 720 
Or in terms of Latitude and Longitude
 x2 
 
k  k o 1  1  e' cos   2 2 
2 2

 2 k o RN 

113
 4.6 Transformation Between Geographic
and UTM Coordinates
4.6.2 Conversion from UTM to Geographic Coordinates

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

Where 1 = footprint latitude which is the latitude at the

central meridian which has the same y
coordinate of the point
= the rectifying latitude

114
 4.6 Transformation Between Geographic
and UTM Coordinates
4.6.2 Conversion from UTM to Geographic Coordinates
y
M  Mo 
ko
M

 e2 e 4
e 6 
a1  3 5  
 4 64 256 
1 1 e2
e1 
1 1 e2
 e1 3   e12 4 
1     3  27  sin 2    21  55  sin 4  
e1 e1

 2 32   16 32 
 e 3   e 4 
 151  sin 6   1097
1 1   sin 8  
 96   512 
where  is in radians
C1 , T1 , RN1 , and R 1 are C, T, RN, and R m calculated at the footprint latitude (1 )
x
D
RN1k o
115
 4.6 Transformation Between Geographic
and UTM Coordinates
4.6.2 Conversion from UTM to Geographic Coordinates

D2 
 
4
2 D
 5  3T1  10C1  4C1  9e'
2

 tan 1   2 24
  1   RN1   
6 
 R1  
 
 61  90T1  298C1  45T1  252e ' 3C1
2 2 2 D

720 

 
3 5
D  1  2T1  C1   5 2C1  28T1  3C1  8e '  24T 1
D 2 2 2 D

  o  6 120
cos1 

116
 Elevation factor (Scale factor to sea level)

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.
R RE
Elevation factor  Approx. Elevation factor 
R  h RE  H
(The UTM scale factor can also be approximated using
the average radius of the earth.)
 x 2 
Approximate UTM Scale factor k  k o  1  
 2R 2 
The grid scale factor for UTM maps can then be computed
using either the approximate or true values
Grid Factor  (scale factor) X (Elevation factor)

Grid distance
Ground Distance 
Grid factor 117
 4.6 Transformation Between Geographic
and UTM Coordinates
4.6.3 UTM Map Scale Factor
[Review] Central
Meridian Ground surface
H
h N Mean
sea level

Projection
Ellipsoid surface
surface

ko = 0.9996

118
 4.6 Transformation Between Geographic
and UTM Coordinates
4.6.4 EXAMPLE A
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

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°

119
 4.6 Transformation Between Geographic
and UTM Coordinates
4.6.4 EXAMPLE A

Example

120
 4.6 Transformation Between Geographic
and UTM Coordinates
4.6.4 EXAMPLE A
e  2 f  f  0.00676865
2 2

2
e
e' 
2
 0.00681478

1 e
2

1 e
2
RM  a  6,372,127.842 m
1  e2 sin 2  2
3

a
RN  N   6,390,630.874 m N used in this equation is
 2
1 e sin
2
  not to be confused with
T  tan 2   1.34689285 geiodal height.
C  e'2 cos2   0.00290374
A    o cos  0.0075952
 e 2 e
4
e
6
  e2 e
4
e
6
 
 1   3  5  
    3  3  45   sin 2  
  4 64 256   8 32 1024  
M  a  5457211.606
 e4 e 6
  e 6

15  45   sin 4   35   sin 6   
 256 1024   3072  
Mo  0
121
 4.6 Transformation Between Geographic
and UTM Coordinates
4.6.4 EXAMPLE A

5

 
3
   
x ko N  A 1 T C  A
   2
 
5 18T T 72C 58e '
2 A
  48,518.5439 m
 6 120 
add a false easting of 500,000m
E = 548,518.544 m
  A2 6 
2 A 
y  ko M  M o  N tan    5  T  9C  4C   61  58T  T  600C  330e'   
4
2 A 2

 2 24 720  

N = 5,455,242.563 m 6o Zone UTM Coordinates

x
Meridian 3o East of
O m North Y Control Meridian

Central Meridian
500,000 m East
Equator

Meridian 3o West of
Control Meridian 122
 4.7 Application of UTM Coordinates
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

GSRUG UTM coordinates: Output- Calculated UTM coordinates:

UTM Zone: 13 UTM Zone: 13
Easting: 548518.573 meters EAST Easting: 548518.544 meters EAST
Northing: 5455242.533 meters NORTH Northing: 5455242.563 meters NORTH
123
 4.7 Application of UTM Coordinates

4.7.1 EXAMPLE B

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

Note: dx = x

124
4.7 Application of UTM Coordinates
a  a ' a  6378137  6378206 .4  69.4 m EXAMPLE B
f  f ' f  1 1  3.72587  10 5
298 .257 294 .979
a
RN   6390630 .874 m
1  e sin  
2 2

1 e
2

RM  a  6372127 .842 m
1  e 2
sin 2  
3
2

 RNe 2 sin cos    RM 

   xsin  cos    ysin  sin    zcos    a   f   RN 1- f sin cos 
 1- f  
   
a
RM h
  7.5149  10  7 rad  4.306  10 5 deg. = 0.155”
 x sin   y cos 
 
RN  h cos 
  8.5059  10  6 rad  0.0004874  = 1.75”

 f 1  f RN sin 2  m
a
 h  x cos  cos   y cos  sin   z sin   a
RN
 h = - 25.704m
 '      4915'0.155" = 49o 15’ 0.16” N
 '      104 20' ( 1.75" ) = 104o 20’ 1.75 W
h '  h  h  740 .66 m  ( 25.704 m ) = 714.956m
125
 4.7 Application of UTM Coordinates
http://www.geod.nrcan.gc.ca/apps/ntv2/ntv2_utm_e.php

4.7.1 EXAMPLE B con’t.

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: Calculated Output data:

LATITUDE: 49o 15’ 0.11403“ N LATITUDE: 49o 15’ 0.155“ N
Shift: 0.11403 seconds
Standard deviation: 0.078m
LONGITUDE: 104o 20’ 1.87927” W LONGITUDE: 104o 20’ 1.75” W
Shift: 1.87927 seconds
Standard Deviation: 0.208m
126
 4.7 Application of UTM Coordinates
4.7.1 EXAMPLE B con’t.

Works up to 50o N
In Western Canada

National Geodetic Survey

http://www.ngs.noaa.gov/cgi-bin/nadcon.prl
127
 4.7 Application of UTM Coordinates
4.7.2 EXAMPLE C
FIND:
c) Latitude and longitude of point B with respect to NAD 27
E = 560,000m N = 5,470,000m
a = 6,378,206.4 1/f = 294.979
o= 105°W ko= 0.9996 Mo= 0°

128
4.7 Application of UTM Coordinates
4.7.2 EXAMPLE C
y
M  Mo   5472188.876
ko
e 2  2 f  f 2  0.00676865
e2
e' 
2
 0.00681478
1  e2
M
  0.85940716 rad
 e2 e4 e6 
a1   3  5  
 4 64 256 
1  1  e2
e1   0.00169791
1 1 e 2

 e1 e13   e12 e14 

1     3  27  sin 2     21  55  sin 4 
  
 2 32   16 32 
 e13   e14 
 151  sin 6   1097   sin 8     0.8619251 rad
 96   512 
1  e2
R1  a  6372278.77
1  e 
3
2
sin 1 2 2

a
RN1   6390681.33
1  e sin 1 
2 2
129
 4.7 Application of UTM Coordinates
4.7.2 EXAMPLE C
T1  tan 2 1  1.35976
C1  e'2 cos 2 1  0.0028879
x  X  ( false easting )  60,000
x
D  0.0093924
N1k o
 D2 
 
4
2 D
    2

 tan1   2 
5 3 T 1 10 C1 4C1 9 e'

  1   N1   24 

 R1  
  
6
2 D
  61  90 T  298 C  45 T 2
 252 e ' 2
 3C 
720 
1 1 1 1

  0.8618735 rad  49.381713   49  22 ' 54.168 " N

 
5
D3
   
D 1 2T1 C1   5  2C1  28T1  3C12  8e'2 24T12
D
  o  6 120
cos1 
  1.832596 rad  0.0144274  1.818168 rad  -104.17337   104 10'24.134" W

130
 4.7 Application of UTM Coordinates
4.7.2 EXAMPLE C

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: Calculated geographic coordinates:

LATITUDE: 49o 22’ 54.168061” N LATITUDE: 49o 22’ 54.168” N
LONGITUDE: 104o 10’ 24.134352” W LONGITUDE: 104o 10’ 24.134” W
131
 4.8 Map Azimuth and Scale Factors of Line A

Given:  = 49o 22’ 54” N

A-B has a calculated grid Azimuth  = 37o 52’ 59.5” =104o 10’ 24” W

Find:
“True” Azimuth of line from A to B” (seconds)


Grid North


Central Meridian
Δα  θ   sin  m sec  ( ) 3  F  B
2

Grid North
Δα  θ  2688" sin 49.3158333  1  ( )
3
 F m
1 
F sin m cos  m sin 1"
2 2

12
A
 = 2038 “ = 0o 33’ 58”  = 49o 15’ N

=105o W
=104o 20” W
Corrected (Astronomic) Azimuth A to B
 = 37o 52’ 59.5 + 0o 33’ 58” = 38o 26’ 57.5”

132
 4.8 Map Azimuth and Scale Factors of Line A
MAP AZIMUTH AND SCALE FACTORS OF LINE A - B
Corrected (Astronomic) Azimuth A to B
 = 37o 52’ 59.5 + 0o 33’ 58” = 38o 26’ 57.5”
Precise scale Factor (S.F.)
UTM Scale factor
 A4
    A6 
2
k  k o 1  1  C   5  4T  42C  13C  28e'
A 2 2  61  148T  16T 2
24 
 2 720 
k  0.99963
Precise Elevation Factor (E.F.)
RM RN
R   R  6379250.925
RM sin 2 ( )  RN cos 2  
From Geodetics Canada GPS - H v.2 software :
“A”
H  740.66m h  722.71m N  -17.95m
R
Precise Elevation Factor (E.F.)   0.999887
R  h
True grid factor = S.F. X E.F.

0.99963  0.999887  0.999517

133
 4.8 Map Azimuth and Scale Factors of Line A
MAP AZIMUTH AND SCALE FACTORS OF LINE A - B

Approximate scale factors based on a spherical earth

UTM Scale Factor (S.F.)
 x 2
  48 ,548.5 2

M p  ko 1  2   0.99961    0.99963
2 
 2R   2  6,367,272 
Elevation Factor (E.F.)
6,367,272
 0.999884
6,367,272  740
Approximate grid factor
0.99963  0.999884  0.999513
Rough Conversion
 
  52.13d tan   52.13  48518.5 3.2808 tan4915'
5280
  52.13  30.15 1.1606  1824.1"  030'24.1"
134
 4.8 Map Azimuth and Scale Factors of Line A
MAP AZIMUTH AND SCALE FACTORS OF LINE A - B

More Precise Spherical Conversion

θ"  " sin

φ A  φ B 
2

θ"  2688" sin

49.25  49.38166667   2038
2

Corrected (Astronomic) Azimuth A to B

 = 37o 52’ 59.5 + 0o 33’ 58” = 38o 26’ 57.5”

135