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

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

3

Acos'eC

tanT22

2

radians in is here w

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

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'11kkRkx

eo

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

xD1N

1m1111 NN )( latitudefootprint at the calculated R and ,R T, C, are R and ,R ,T ,Cradiansin is here w

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

111

11

eeea

M642

2565

643

41

oo k

yMM

7

1

2223

11

cos6

21

5

120D

1111 24'832825 TeCTCDCTDo

1

111

tanN

RR

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

8

earth.) theof radius average the using approximated be alsocan factor scale UTM(The

factor Griddistance 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.

hRR

HRR

E

E

Elevation factor Approx. Elevation factor

2

2o

R2x1kk

valuesor true approximate either the usingcomputed 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

hHN

CentralMeridian

ko = 0.9996

k o = 1

.00

ko = 1.00

10

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°

11

Example

4.6.4 EXAMPLE A

4.6 Transformation Between Geographic and UTM Coordinates

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

eee

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.

13Meridian 3o West of Control Meridian

Equator

Meridian 3o East of Control 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

14

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

15

FIND:

b) Latitude, longitude and height of point A with respect to NAD 83 ellipsoida’ = 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

mRffRa

azyxh NN

sin1sinsincoscoscos 2

rad 0004874.0105059.8 6

m

e

eaRM 842.6372127

sin1

1

23

22

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

hRyx

N coscossin

RRN

N

hR

cossinf1-f1-

Rfa

cossineazcossinysincosxsin2

M

M

17

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 Coordinates4.7.1 EXAMPLE B con’t.

http://www.geod.nrcan.gc.ca/apps/ntv2/ntv2_utm_e.php

18

4.7 Application of UTM Coordinates4.7.1 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 Coordinates4.7.2 EXAMPLE C

20

4.7 Application of UTM Coordinates4.7.2 EXAMPLE C

21

4.7 Application of UTM Coordinates4.7.2 EXAMPLE C

720D

C3'e252T45C298T9061

24D'e9C4C10T35

2D

Rtan

N

0093924.0kNx

D

000,60)easting false(Xx0028879.0cos'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

6DCT21D

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 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 Coordinates4.7.2 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

orth

= 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

o 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

ATT24

4A

eCCTA

Ckk 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 UTMearth spherical aon based factors scale eApproximat

999513.0999884.099963.0factor grid eApproximat

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

26

MAP AZIMUTH AND SCALE FACTORS OF LINE A - B

4.8 Map Azimuth and Scale Factors of Line A

2

φφsin""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”