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

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


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



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



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


4.6.2 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



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



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


4.6.3 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


4.6.3 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


4.6.4 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


12


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


4.6.4 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


4.7.1 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


4.7.1 EXAMPLE B con’t.

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

18


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°


4.7.2 EXAMPLE C

20

4.7 Application of UTM Coordinates4.7.2 EXAMPLE C

21


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


4.7.2 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



= 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



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

26



2

φφsin""θ BA

More Precise Spherical Conversion

2038

2

38166667.4949.25sin"2688"θ


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

Documents

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