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Gauss Conformal Projection Guide

The document describes the Gauss conformal projection (Transverse Mercator) and Krüger's formulas for converting between geodetic coordinates (latitude and longitude) and grid coordinates (northing and easting) in a transverse mercator map projection. It provides the mathematical formulas and definitions for converting in both directions, including computing scale factors, false northings/eastings, and coefficients for the ellipsoid parameters. An example conversion is also given for a location in Sweden using the GRS 1980 ellipsoid.

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Igli Myrta
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153 views5 pages

Gauss Conformal Projection Guide

The document describes the Gauss conformal projection (Transverse Mercator) and Krüger's formulas for converting between geodetic coordinates (latitude and longitude) and grid coordinates (northing and easting) in a transverse mercator map projection. It provides the mathematical formulas and definitions for converting in both directions, including computing scale factors, false northings/eastings, and coefficients for the ellipsoid parameters. An example conversion is also given for a location in Sweden using the GRS 1980 ellipsoid.

Uploaded by

Igli Myrta
Copyright
© Attribution Non-Commercial (BY-NC)
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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L A N T M T E R I E T

1 (5)

Geodesi

2005-08-31

Gauss Conformal Projection (Transverse Mercator)


Krgers Formulas

Symbols and Definitions


a f
P

semi-major axis of the ellipsoid flattening of the ellipsoid first eccentricity squared
P

e2

geodetic latitude, positive north geodetic longitude, positive east grid coordinate, positive north grid coordinate, positive east longitude of the central meridian scale factor along the central meridian difference 0 false northing false easting

x y
0 k0

FN FE

All angles are expressed in radians. Please note that the x-axis is directed to the north and the y-axis to the east. The following variables are defined out of the ellipsoidal parameters a and f: e 2 = f (2 f ) n=
=

f (2 f )
a 1 2 1 4 n + ... 1 + n + 64 (1 + n) 4

Lantmteriet I-Divisionen Geodesi Tel. vxel: 0771-63 63 63 E-post: geodesi@lm.se Internet: www.lantmateriet.se/geodesi

Lantmteriet

2005-08-31

Conversion from geodetic coordinates ( , ) to grid coordinates (x,y).


Compute the conformal 1 latitude *
TPF FPT

* = sin cos (A + B sin 2 + C sin 4 + D sin 6 + . . .)


The coefficients A, B, C, and D are computed using the following formulas: A = e2
B=
C= D=

1 (5e 4 e 6 ) 6
1 (104e 6 45e 8 + . . .) 120 1 (1237 e 8 + . . .) 1260

Let = 0 and

= arctan(tan * / cos ) = arctan h(cos * sin )


then

+ 1 sin 2 cosh 2 + 2 sin 4 cosh 4 + 3 sin 6 cosh 6 + + FN x = k 0 a + 4 sin 8 cosh 8 + K + 1 cos 2 sinh 2 + 2 cos 4 sinh 4 + 3 cos 6 sinh 6 + + FE y = k 0 a + 4 cos 8 sinh 8 + K

1
TP PT

Older Swedish literature refers to this quantity as the isometric latitude. Today the term isometric latitude is applied to the quantity

= ln{ tan( / 4 + / 2 )[(1 e sin ) /( 1 + e sin )]


to the conformal latitude by

e /2

}.

The isometric latitude is related

= ln tan( / 4 + * / 2 ) . Cf. John P. Snyder: Map Projections -

A Working Manual, U.S. Geological Survey Professional Paper 1395.

Lantmteriet

2005-08-31

where the coefficients 1 , 2 , 3 and 4 are computed by


1 2 5 3 41 4 n n2 + n + n + ... 2 3 16 180 13 2 3 3 557 4 n n + n + ... 48 5 1440 61 3 103 4 n n + ... 240 140 49561 4 n + ... 161280

1 = 2 = 3 = 4 =

Lantmteriet

2005-08-31

Conversion from grid coordinates (x,y) to geodetic coordinates ( , )


Introduce the variables and as
x FN k0 a y FE k0 a

= =

Let

= 1 sin 2 cosh 2 2 sin 4 cosh 4 3 sin 6 cosh 6 4 sin 8 cosh 8 K = 1 cos 2 sinh 2 2 cos 4 sinh 4 3 cos 6 sinh 6 4 cos 8 sinh 8 K
where
1 2 37 3 1 4 n n2 + n n +K 2 3 96 360 1 2 1 3 437 4 n n +K n + 15 1440 48 17 3 37 4 n n +K 480 840 4397 4 n +K 161280

1 = 2 = 3 = 4 =

The conformal latitude * and the difference in longitude are obtained by the formulas

* = arcsin(sin / cosh )

Lantmteriet

2005-08-31

= arctan(sinh / cos )
Finally, the latitude and the longitude are obtained by the formulas
= 0 +

= * + sin * cos * (A * + B * sin 2 * + C * sin 4 * + D * sin 6 * + . . .)

where
A * = ( e 2 + e 4 + e 6 + e 8 + . . .)

1 B * = (7 e 4 + 17 e 6 + 30e 8 + . . .) 6 C* = 1 ( 224e 6 + 889e 8 + . . .) 120 1 ( 4279e 8 + . . .) 1260

D* =

Worked example:
ELLIPSOID GRS 1980 Semi-major axis (a) Flattening (f) 6378137.0000 m. 1/298.257222101

TRANSVERSE MERCATOR PARAMETERS Longitude of the central meridian 13 35 7.692000 degr. min. sec. Scale factor on the central merdian 1.000002540000 False northing -6226307.8640 m. False easting 84182.8790 m. Latitude and longitude Grid coordinates 66 0 0.0000 24 0 0.0000 degr. min. sec. 1135809.413803 555304.016555 m

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