Gauss-Krueger Warrnambool Conference

7/29/2019 Gauss-Krueger Warrnambool Conference

1/20

THEGAUSSKRGER PROJECTION

R. E. Deakin1, M. N. Hunter

2and C. F. F. Karney

3

1School of Mathematical and Geospatial Sciences, RMIT University

2Maribyrnong, VIC, Australia.

3 Princeton, N.J., USA.

email: [email protected]

Presented at the Victorian Regional Survey Conference, Warrnambool, 10-12 September, 2010

ABSTRACT

The Gauss-Krger projection appears to have two

different forms. One is a set of equations capable ofmicrometre accuracy anywhere within 30 of a central

meridian of longitude. The other is equations limited tomillimetre accuracy within 3 6 of a central meridian.These latter equations are complicated but are in wide use

in the geospatial community. The former equations arerelatively simple (although they do contain hyperbolicfunctions) and are easily adapted to computers. But they

are not in wide use. This paper will give some insightinto both forms of the Gauss-Krger projection.

INTRODUCTION

The Gauss-Krger projection is a conformal mapping of areference ellipsoid of the earth onto a plane where the

equator and central meridian remain as straight lines andthe scale along the central meridian is constant; all other

meridians and parallels being complex curves (Figure 5).

The Gauss-Krger is one of a family of transverseMercator projections of which the spherical form was

originally developed by Johann Heinrich Lambert (1728-1777). This projection is also called the Gauss-Lambertprojection acknowledging the contribution of CarlFriedrich Gauss (17771855) to the development of thetransverse Mercator projection. Snyder (1993) and Lee(1976) have excellent summaries of the history of

development which we paraphrase below.

Gauss (c.1822) developed the ellipsoidal transverseMercator as one example of his investigations inconformal transformations using complex algebra andused it for the survey of Hannover in the same decade.

This projection had constant scale along the centralmeridian and was known as the Gauss conformal orGauss' Hannover projection. Also (c. 1843) Gaussdeveloped a 'double projection' consisting of a conformalmapping of the ellipsoid onto the sphere followed by amapping from the sphere to the plane using the spherical

transverse Mercator formula. This projection wasadapted by Oskar Schreiber and used for the PrussianLand Survey of 1876-1923. It is also called the Gauss-Schreiber projection and scale along the central meridianis not constant. Gauss left few details of his originaldevelopments and Schreiber (1866, 1897) published an

analysis of Gauss' methods, and Louis Krger (1912) re-evaluated both Gauss' and Schreiber's work, hence thename Gauss-Krger as a synonym for the transverseMercator projection.

The aim of this paper is to give a detailed derivation of aset of equations that we call the Gauss-Krger projection.These equations give micrometre accuracy anywherewithin 30 of a central meridian; and at their heart are

two important series linking conformal latitude and

rectifying latitude . We provide a development of

these series noting our extensive use of the computer

algebra systems MAPLE and Maxima in showing theseseries to high orders ofn; unlike Krger who only hadpatience. And without these computer tools it would beimpossible to realize the potential of his series.

Krger gave another set of equations that we wouldrecognise as Thomas's or Redfearn's equations (Thomas1952, Redfearn 1948). These other equations alsoknown as the Gauss-Krger projection are in wide usein the geospatial community; but they are complicated,

and only accurate within a narrow band (3 6) about acentral meridian. We outline the development of these

equations but do not give them explicitly, as we do notwish to promote their use. We also show that the use ofthese equations can lead to large errors in somecircumstances.

This paper supports the work of Engsager & Poder(2007) who also use Krger's series in their elegantalgorithms for a highly accurate transverse Mercatorprojection but provide no derivation of the formulae.Also, one of the authors (Karney 2010) has a detailedanalysis of the accuracy of our Gauss-Krger projection

equations and this paper may be regarded as backgroundreading.

The preliminary sections set out below contain

information that can be found in many geodesy and mapprojection texts and could probably be omitted but theyare included here for completeness. As is the extensiveAppendix that may be useful to the student following thedevelopment with pencil and paper at hand.

SOME PRELIMINARIES

The Gauss-Krger projection is a mapping of a referenceellipsoid of the earth onto a plane and some definition ofthe ellipsoid and various associated constants are useful.We then give a limited introduction to differentialgeometry including definitions and formulae for theGaussian fundamental quantities e, fand g, the

differential distance ds and scale factors m,h and k.

Next, we define and give equations forthe isometric

latitude , meridian distanceM, quadrant length Q, the

1
mailto:[email protected]:[email protected]:[email protected]


2/20

rectifying radiusA and the rectifying latitude . This

then provides the basic 'tools' to derive the conformal

latitude and show how the two important series

linking and are obtained.

The ellipsoid

In geodesy, the ellipsoid is a surface of revolution createdby rotating an ellipse (whose semi-axes lengths are a and

b and a ) about its minor axis. The ellipsoid is the

mathematical surface that idealizes the irregular shape ofthe earth and it has the following geometrical constants:

b

flatteninga b

fa

(1)

eccentricity2 2

2

a b

a

(2)

2nd eccentricity2 2

2

a b

b

(3)

3rd flatteninga b

na b

(4)

polar radius2

ac

b (5)

These geometric constants are inter-related as follows

2

2

1 11 1

11

b n a

a n c

f (6)

2

22 2421 1

nf fn

(7)

22

2 2

2 42

1 1

f f n

1 f n

(8)

2

2

1 1

2 1 1

fn

f

(9)

From equation (7) an absolutely convergent series for 2

is

(10)

2 2

8n n

3 4 5

12 16 20n n n

4nsince .0 1

The ellipsoid radii of curvature (meridian plane) and

(prime vertical plane) at a point whose latitude is

are

2 23 2 3 3

c

V2 2

1 1

1 sin

a a

W

(11)

1 22 21 sin

a a c

W V

(12)

where the latitude functions Vand Ware defined as

2 2 2 2 2 21 sin , 1 cosW V (13)

Some differential geometry: the differential rectangle andGaussian fundamental quantities e, f, g; , ,E F G andE,F, G

Curvilinear coordinates (latitude), (longitude) are

used to define the location of points on the ellipsoid (thedatum surface) and these points can also havex,y,z

Cartesian coordinates where the positive z-axis is therotational axis of the ellipsoid passing through the northpole, thex-y plane is the equatorial plane and thex-zplane is the Greenwich meridian plane. The positivex-axis passes through the intersection of the Greenwichmeridian and equator and the positivey-axis is advanced90 eastwards along the equator.

The curvilinear and Cartesian coordinates are related by

2

, cos cos

, cos sin

, 1 si

x x

y y

z z n

(14)

The differential arc-length ds of a curve on the ellipsoidis given by

(15) 2 2 2

ds dx dy dz 2

And the total differentials are

x xdx d d

y ydy d d

z zdz d d

(16)

Substituting equations (16) into (15) gathering terms andsimplifying gives

2 2

2ds e d f d d g d 2

(17)

where the coefficients e, fand g are known as theGaussian fundamental quantities and are given by

2 2 2

2 2 2

x y ze

x x y y z zf

x y zg

(18)

For the ellipsoid

2 , 0, cose f g 2 2 (19)

2


3/20

d

d

ed

g d

P

ds

Q

Figure 1. The differential rectangle

In Figure 1 the parametric curves and pass through

P and the curves d and d

ds

pass through Q, and

the length of the curve . The differential

rectangle formed by the curves may be regarded as aplane figure whose opposite sides are parallel straightlines enclosing a differentially small area da.

PQ

The differential distances along the parametric curves are

cos

ds e d d

ds g d d

(20)

and the angle between the parametric curves can be

found from

cosf

eg (21)

Thus, if the parametric curves on the surface intersect at

right angles (i.e., they form an orthogonal system of

curves) then1

2 and cos 0 , and so fromequation (21) . Conversely, if the

parametric curves form an orthogonal system.

0f 0f

Suppose there is another surface (projection surface) with

curvilinear coordinates andX,Y,ZCartesian

coordinates related by

,

(22) , , , , ,X X Y Y Z Z

Then, using equations (15) and (16) replacingx,y,z with

X,Y,Zand , with we have the differential

distance dSon this surface defined by

,

2 2

2dS E d F d d G d 2

(23)

with the Gaussian fundamental quantities

2 2

2 2 2

X Y ZE

2

X X Y Y Z ZF

X Y ZG

(24)

Alternatively, theX,Y,ZCartesian coordinates (projection

surface) can be expressed as functions of the ,

curvilinear coordinates (datum surface) as

, , , , ,X X Y Y Z Z

2

(25)

and

2 2

2dS E d F d d G d (26)

where

2 2

2 2

X Y ZE

2

2

X X Y Y Z ZF

X Y ZG

(27)

Scale factors m, h and k

The scale factorm is defined as the ratio of differentialdistances dS(projection surface) and ds (datum surface)and is usually given as a squared value

2 2

2

2 2

2 2

2 2

2

2

2

2

dS E d F d d G d m

ds e d f d d g d

E d F d d G d

e d f d d g d

2

2

(28)

WhenE G

e g and 0F or

E G

e g and the

scale factorm is the same in every direction and suchprojections are known as conformal. For the ellipsoid

(datum surface) where the parametric curves

0F

, are an

orthogonal system and 0f , this scale condition forconformal projection of the ellipsoid is often expressed as

and 0h k F F (29)

h is the scale factor along the meridian and kis the scalefactor along the parallel of latitude. Using equations (28)

andE d E G d G

h ke d e g d g

(30)

Isometric latitude

According to the SOED (1993) isometric means: ofequal measure or dimension and we may think of

isometric parameters (isometric latitude) and

0 (longitude difference) in the following way.Imagine you are standing on the earth at the equator andyou measure a metre north and a metre east; both of theseequal lengths would represent almost equal angular

changes in latitude d and longitude d . Now imagine

you are close to the north pole; a metre in the northdirection will represent (almost) the same angular change

d as it did at the equator. But a metre in the east

direction would represent a much greater change in

longitude, i.e., equal north and east linear measures nearthe pole do not correspond to equal angular measures.

3


4/20

What is required is isometric latitude , a variable

angular measure along a meridian that is defined byconsidering the differential rectangle of Figure 1 andequations (17) and (19) giving

2 22

2 22 2 2

2

22 2

2 22 2

cos

coscos

cos

ds e d g d

d d

dd

d d

(31)

where isometric latitude is defined by the relationship

cosd

d

(32)

Integrating this gives (Deakin & Hunter 2010b)

12

1 1

4 2

1 sinln tan

1 sin

(33)

Note that if the reference surface for the earth is a sphereof radiusR; then R , 0 and the isometriclatitude is

1 14 2ln tan (34)

Meridian distance M

Meridian distanceMis defined as the arc of the meridian

from the equator to the point of latitude

23

0 0 0

1a c3

M d dW V

d

(35)

This is an elliptic integral that cannot be expressed interms of elementary functions; instead, the integrand isexpanded by use of the binomial series then the integral isevaluated by term-by-term integration. The usual form of

the series formula forMis a function of and powers of2 ; but the German geodesist F.R. Helmert (1880) gave a

formula for meridian distance as a function of latitude

and powers of the ellipsoid constant n that required fewer

terms for the same accuracy than meridian distance

formula involving powers of 2 . Using Helmert's

method of development (Deakin & Hunter 2010a) a

formula forMcan be written as

0 2 4

2

6 8

10

sin 2 sin 4

1 1 sin 6 sin8

sin10

b b b

M a n n b b

b

(36)

where the coefficients nb have the following form

2 2 2

2 4 6

0

2

8

3 5

2

7

4

3 3 5 3 5 71

2 2 4 2 4 6

3 5 7 9

2 4 6 8

3 3 3 5 3 5 3 5 7

2 2 2 4 2 4 2 4 62

2 3 5 7 3 5 7 9

2 4 6 2 4 6 8

3

2

4

b n n n

n

n n

b

n

b

n

2 4

6

3 5

6

7

8

5 3 3 5 7

2 4 2 2 4 6

3 5 3 5 7 9

2 4 2 4 6 8

3 5 7 3 3 5 7 9

2 4 6 2 2 4 6 82

6 3 5 3 5 7 9 11

2 4 2 4 6 8 10

n n

n

n n

b

n

b

4 6

8

5

10

7

3 5 7 9 3 3 5 7 9 11

2 4 6 8 2 2 4 6 8 102

8 3 5 3 5 7 9 11 13

2 4 2 4 6 8 10 12

3 5 7 9 11

2 4 6 8 102

10 3 3 5 7 9 11 13

2 2 4 6 8 10 12

n n

n

n

b

n

Noting that

2 2

22 2 2

11 1 1 1

1

1 1 1

1 1

nn n n n

n

n n n

n n

then multiplying the coefficients in

equation

0 2 4, , ,b b b

(36) by 2

21 n gives

0 2 4

6 8

10 12

14 16

sin 2 sin 4

sin 6 sin 8

sin10 sin121

sin14 sin16

c c c

c caM

c cn

c c

(37)

where the coefficients nc are to order as follows8n

4


5/20

2 4 6 8

0

3 5 7

2

2 4 6 8

4

3 5 76

4 6 8

8

5 7

10

12

1 1 1 251

4 64 256 16384

3 3 3 15

2 16 128 2048

15 15 75 105

16 64 2048 8192

35 175 24548 768 6144

315 441 1323

512 2048 32768

693 2079

1280 10240

1

c n n n n

c n n n n

c n n n n

c n n n

c n n n

c n n

c

6 8

7

14

8

16

001 1573

2048 8192

6435

14336

109395

262144

n n

c n

c n

(38)

[This is Krger's equation forXshown in 5, p.12,

extended to order ]8n

Quadrant length Q

The quadrant length of the ellipsoid Q is the length of themeridian arc from the equator to the pole and is obtained

from equation (37) by setting 12

, and noting that

sin 2 , sin 4 , sin 6 , all equal zero, giving

02 1a

Qn

c (39)

[This is Krger's equation forM shown in 5, p.12.]

Rectifying radiusA

Dividing the quadrant length Q by 12

gives the

rectifying radiusA, which is the radius of a circle havingthe same circumference as the meridian ellipse andA to

order is8n

2 4 6 81 1 1 2511 4 64 256 16384a

A n n n nn

(40)

Rectifying latitude The rectifying latitude is defined in the following way

(Adams 1921):

If a sphere is determined such that the length of agreat circle upon it is equal in length to a meridian

upon the earth, we may calculate the latitudes uponthis sphere such that the arcs of the meridian upon it

are equal to the corresponding arcs of the meridianupon the earth.

If denotes this latitude on the sphere of radiusR then

meridian distanceMis given by

M R (41)

and since 12

when M Q then and the

rectifying latitude

R A

is defined as

M

A (42)

An expression for as a function of and powers ofn

is obtained by dividing equation (40) into equation (37)giving to order 4n

2 4 6

8

sin 2 sin 4 sin 6

sin8

d d d

d

(43)

where the coefficients nd are

3

2

2 4

4

36

4

8

3 9

2 16

15 15

16 32

3548

315

512

d n n

d n n

d n

d n

(44)

[This is Krger's equation (6), 5, p. 12.]

An expression for as a function of and powers ofn

is obtained by reversion of a series using Lagrange's

theorem (see Appendix) and to order 4n

2 4 6

8

sin 2 sin 4 sin 6

sin8

D D D

D

(45)

where the coefficients nD are

3

2

2 4

4

3

6

4

8

3 27

2 32

21 55

16 32

151

48

1097

512

D n n

D n n

D n

D n

(46)

[This is Krger's equation (7), 5, p. 13.]Conformal l atitude

Suppose we have a sphere of radius a with curvilinear

coordinates , (meridians and parallels) andX,Y,ZCartesian coordinates related by

, cos cos

, cos sin

, sin

X X a

Y Y a

Z Z a

(47)

Substituting equations (47) into equations (24), replacing

, with , gives the Gaussian FundamentalQuantities for the sphere as

2 , 0, cosE a F G a2 2 (48)

5


6/20

The conformal projection of the ellipsoid (datum surface)onto the sphere (projection surface) is obtained by

enforcing the condition ; and with equationsh k,

(19),

(30) and (48), replacing with , whereappropriate, we have,

cos

cos

a d a d

d d

(49)

This differential equation can be simplified by enforcingthe condition that the scale factor along the equator beunity, so that

0

0 0

cos1

cos

a d

d

and since 0 0 0 then 0 0cos cos 1 , 0 a

and d d . Substituting this result into equation (49)gives

cos cos

d d

(50)

Integrating both sides gives

12

1 1 1 14 2 4 2

1 sinln tan ln tan

1 sin

(51)

The conformal longitude d

is obtained from the

differential relationship d which is aconsequence of the scale factor along the equator being

unity, and longitude on the ellipsoid is identical tolongitude on the conformal sphere, which makes

(52)

Series involv ing conformal latitude and rectifying latitude

Two series that are crucial to the Gauss-Krger projectionwill be developed in this section; they are (i) a series for

conformal latitude as a function of the rectifying

latitude , and (ii) a series for as a function of .The method of development is not the same as employed

by Krger, but does give some insight into the power anduse of Taylor's theorem (see Appendix). Also, we shouldremember that Krger had only pencil, paper and

perseverance whilst we have the power of computeralgebra systems.

A series for conformal latitude as a function oflatitude may be developed using a method given by

Yang et al. (2000).

The right-hand-side of equation (51) is the isometriclatitude and we write

1 14 2exp tan (53)

where exp e is the exponential function.

Solving equation (53) gives the conformal latitude

1 122 tan exp

Let 1, tan expF so that equation (54)becomes

122 ,F (55)

and

12

1 1 14 2

1 sin, tan tan1 sin

F

Now since the eccentricity satisfies 0 1 , ,F

may be expanded into a power series of about 0 using Taylor's theorem [see Appendix, equation (139)]

00

2 2

2

0

3 3

3

0

, , ,

,2!

,3!

F F F

F

F

(56)

All odd-order partial derivatives evaluated at 0 arezero and the even-order partial derivatives evaluated at

0 are

2

2

0

4

4

0

6

6

0

8

8

0

1, sin 2

2

5 5, sin 2 sin 4

2 4

135 63sin 2 sin 4

4 2,

39sin6

4

1967 4879sin 2 sin 4

2 4,

1383 1237sin 6 sin8

2 8

F

F

F

F

[We note that the last of these derivatives is incorrectly

shown in Yang et al. (2000, p. 80).]Substituting these partial derivatives with

1 14 20,F into equation (56), and re-

arranging them into equation (55) gives

(54)

6


7/20

2 4 6

8

4 6 8

6 8

8

1 5 3

2 24 32sin2

281

5760

5 7 697sin4

48 80 11520

13 461 sin6480 13440

1237sin8

161280

(57)

[Note that equation (57) is incorrectly shown in Yang etal. (2000, eqn (3.5.8), p. 80) due to the error notedpreviously.]

Extending the process outlined above to higher even-

powers of and higher even-multiples of ; and then

using the series (10) to replace with powersofn gives a series for conformal latitude

2 4 6

, , , as a function

of latitude to order as4n

2 4 6

8

sin 2 sin 4 sin 6

sin8

g g g

g

(58)

where the coefficients ng are

2 3 4

2

2 3 4

4

3 4

6

4

8

2 4 822

3 3 45

5 16 13

3 15 926 34

15 21

1237

630

g n n n n

g n n n

g n n

g n

(59)

[This is Krger's equation (8), 5, p. 14.]

A series for conformal latitude as a function ofrectifying latitude can be obtained by using a method

set out in Krger (1912, p.14) that involves Taylor's

theorem (see Appendix); where

2

3 4

2 2sin 2 sin 2 2 2 cos 2 sin 2

2!

2 2 2 2cos 2 sin 2

3! 4!

Replacing 2 and 2 with 4 and 4 ; 6 and 6 ;

etc.; and with expressions for 2 2 , 4 4 , etc.from equation (45) we can obtain expressions for

sin 2 , sin sin tc.4 , 6 , e

sin 2 , sin sin etc.

6 ,

as functions ofn and

4 , Substituting these

expressions into equation (58) and simplifying gives a

series for conformal latitude as a function ofrectifying latitude to order 8n

2 4 6

8 10 12

14 16

sin 2 sin 4 sin 6

sin8 sin10 sin12

sin14 sin16

(60)

where the coefficients n are

2 3 4 5

2

6 7

8

2 3 4 5

4

6 7

8

3 4

6

1 2 37 1 81

2 3 96 360 512

96199 5406467

604800 38707200

7944359

67737600

1 1 437 46

48 15 1440 105

1118711 51841

3870720 1209600

24749483

348364800

17 37 209

480 840 4480

n n n n n

n n

n

n n n n

n n

n

n n

5 6

7 8

4 5 6

8

7 8

5 6

10

5569

90720

9261899 6457463

58060800 17740800

4397 11 830251

161280 504 7257600

466511 324154477

2494800 7664025600

4583 108847 8005831

161280 3991680 63866880

22894433

12454041

n n

n n

n n n

n n

n n

7n

8

6 7

12

8

7 8

14

8

16

6

20648693 16363163

638668800 518918400

2204645983

12915302400

219941297 497323811

5535129600 12454041600

191773887257

3719607091200

n

n n

n

n n

n

(61)

[This is Krger's equation (10), 5, p. 14 extended to

order ]

8

nAn expression for rectifying latitude as a function of

conformal latitude and powers ofn is obtained byreversion of a series using Lagrange's theorem (seeAppendix).

2 4 6 8

10 12 14

16

sin 2 sin 4 sin 6 sin8

sin10 sin12 sin14

sin16

(62)

where the coefficients n are

7


8/20

2 3 4 5

2

7 8

2 3 4 5 6

4

7 8

3 4 5

6

1 2 5 41 127 7891

2 3 16 180 288 37800

72161 18975107

387072 50803200

13 3 557 281 1983433

48 5 1440 630 1935360

13769 14800388328800 174182400

61 103 15061

240 140 26880

n n n n n n

n n

n n n n n

n n

n n n

6

6

7 8

4 5 6 7

8

8

5 6 7

10

167603

181440

67102379 79682431

29030400 79833600

49561 179 6601661 97445

161280 168 7257600 49896

40176129013

7664025600

34729 3418889 14644087

80640 1995840 91238402605413599

622

n

n n

n n n n

n

n n n

8

6 7

12

8

7 8

14

8

16

702080

212378941 30705481

319334400 10378368

175214326799

58118860800

1522256789 16759934899

1383782400 3113510400

1424729850961

743921418240

n

n n

n

n n

n

(63)

[This is Krger's equation (11), 5, p. 14 extended toorder ]8n

THE GAUSS-KRGER PROJECTION

The Gauss-Krger projection is the result of a triple-mapping in two parts (Bugayevskiy & Snyder 1995).The first part is a conformal mapping of the ellipsoid to asphere (the conformal sphere of radius a) followed by a

conformal mapping of this sphere to the plane using thespherical transverse Mercator projection equations with

spherical latitude replaced by conformal latitude .This two-step process is also known as the Gauss-

Schreiber projection (Snyder 1993) and the scale alongthe central meridian is not constant. [Note that theGauss-Schreiber projection commonly uses a conformal

sphere of radius0 0

R where0

and0

are

evaluated at a central latitude for the region of interest.]The second part is the conformal mapping from the

Gauss-Schreiber to the Gauss-Krger projection wherethe scale factor along the central meridian is madeconstant.

To understand this process we first discuss the sphericalMercator and transverse Mercator projections. We then

give the equations for the Gauss-Schreiber projection andshow that the scale factor along the central meridian ofthis projection is not constant. Finally, using complexfunctions and principles of conformal mapping developed

by Gauss, we show the conformal mapping from theGauss-Schreiber projection to the Gauss-Krgerprojection.

Having established the 'forward' mapping , ,X Y from the ellipsoid to the plane via the conformal sphereand the Gauss-Schreiber projection we show how the

'inverse' mapping , ,X Y from the plane to theellipsoid is achieved.

In addition to the equations for the forward and inversemappings we derive equations for scale factorm and grid

convergence .

Mercator projection of the sphere

X

Y

Figure 2 Mercator projection

graticule interval 15, central meridian0

120 E

The Mercator projection of the sphere is a conformalprojection with the datum surface a sphere of radiusRwith curvilinear coordinates , and Gaussian

fundamental quantities

2 2, 0, cose R f g R 2 (64)

The projection surface is a plane withX,YCartesian

coordinates and X X and Y Y and Gaussian

fundamental quantities

2 2

, 0,Y

E F GX

(65)

Enforcing the scale condition h and using equationsk(30), (64) and (65) gives the differential equation

1

cos

dY dX

d d (66)

This equation can be simplified by enforcing thecondition that the scale factor along the equator be unity

giving

1and

cosdX R d dY d

Integrating and evaluating constants of integration gives

the well known equations for Mercator's projection of thesphere

8


9/20

0

1 1

4 2ln tan

X R R

Y R

(67)

where0

is the longitude of the central meridian and

0 .

An alternative set of equations for Mercator's projection

may be derived as follows by using the half-angle

formula1 cos

tan2 1 cos

x x

x

and writing

121 1

4 2 12

1 cos 1 sintan

1 cos 1 sin

. Using

this result in equation (67) gives

0

1 sin 1 sinln ln

1 sin 2 1 sin

X R R

RY R

(68)

Transverse Mercator projection of the sphere(Gauss-Lambert projection)

The equations for the transverse Mercator projection ofthe sphere (also known as the Gauss-Lambert projection)can be derived by considering the schematic view of thesphere in Figure 3 that shows P having curvilinear

coordinates , that are angular quantities measured

along great circles (meridian and equator).

Now consider the great circleNBS(the oblique equator)with a poleA that lies on the equator and great circles

throughA, one of which passes through P making anangle with the equator and intersecting the oblique

equator at C.

N

S

Aequator

u

v

C

P

B

Figure 3 Oblique poleA on equator

and are oblique latitude and oblique longitude

respectively and equations linking , and , can be

obtained from spherical trigonometry and the right-

angled spherical triangle CNP having sides ,1

2 ,1

2 and an angle atNof 0

sin cos sin (69)

tantan

cos

(70)

Squaring both sides of equation (69) and using the

trigonometric identity 2 2sin cos 1x x gives, aftersome algebra

2 2cos cos tan cos (71)

From equations (71) and (69)

2 2

sintan

tan cos

(72)

Replacing with and with in equations (68);

then using equations (69), (70) and (144) give theequations for the transverse Mercator projection of the

sphere (Lauf 1983, Snyder 1987)

1

1

tantan

cos

1 sin 1 cos sinln ln

2 1 sin 2 1 cos sin

tanh cos sin

u R

R

R Rv

R

(73)

u

v

Figure 4 Transverse Mercator projection


120 E

9


10/20

The transverse Mercator projection of the sphere isconformal, which can be verified by analysis of the

Gaussian fundamental quantities e, f,g of the ,

spherical datum surface [see equations (64)] andE, F, Gof the u,v projection surface.

For the projection surface ,u u , ,v v

2 2

2 2

u vE

u u v vF

u vG

(74)

Differentiating equations (73) noting that

1

2

1tan

1

d dyy

dx dxy

and 12

1tanh

1

dy

dx dxy

dygives

2 2 2 2

2 2 2 2

cos sin cos cos,

1 cos sin 1 cos sin

sin sin cos cos,

1 cos sin 1 cos sin

u R u R

v R v R

(75)

and substituting these into equations (74) gives

2

2 2 2 2

cos, 0,

1 cos sin 1 cos sin

R RE F G

2 2

(76)

Now using equations (30), (64) and (76) the scale factors

h and kare equal and , then the projection is

conformal.

0f F

Gauss-Lambert scale factor

Since the projection is conformal, the scale factor

m h k E e and

2 2

2 2 4 4

1

1 cos sin

1 31 cos sin cos sin

2 8

m

(77)

Along the central meridian 0 and the central

meridian scale factor m 0 1

Gauss-Lambert grid convergence

The grid convergence is the angle between the

meridian and the grid-line parallel to the u-axis and isdefined as

tandv

du (78)

and the total differentials du and dv are

andu u v v

du d d dv d d

Along a meridian is constant and 0d , and thegrid convergence is obtained from

tanv u

(80)

and substituting partial derivatives from equations (75)

gives

1tan sin tan (81)

Transverse Mercator projection of the conformal sphere(Gauss-Schreiber projection)

The equations for the transverse Mercator projection ofthe conformal sphere are simply obtained by replacing

spherical latitude with conformal latitude inequations (73) and noting that the radius of the conformalsphere is a to give

1

1

tan

tan cos

tanh cos sin

u a

v a

(82)

[These are Krger's equations (36), 8, p. 20]

Alternatively, replacing with and with in

equations (67) then using equations (70), (72) and theidentity (147); and finally replacing spherical latitude

with conformal latitude gives (Karney 2010)

1

1

2 2

tantan

cos

sinsinh

tan cos

u a

v a

(83)

tan , which appears in both equations (83), can beevaluated by using hyperbolic and inverse hyperbolicfunctions (see Appendix) and equation (51) in thefollowing manner.

Using equation (147) of the Appendix we may write

11 14 2ln tan sinh tan (84)

And the right-hand-side of equation (51) can be written as

1 1 1

4 2 2

1 1

1 sinln tan ln

1 sin

sinh tan tanh sin

(85)

Equating (84) and (85) gives

1 1 1sinh tan sinh tan tanh sin (86)

With the substitution

(79)

1

2

tansinh tanh

1 tan

(87)

and some algebra, equation (86) can be rearranged as(Karney 2010)

10


11/20

2tan tan 1 1 tan2 (88)

The Gauss-Schreiber projection is also conformal since

can be replaced by in the previous analysis and

E e G g , and .0f F

Gauss-Schreiber scale factor

The scale factor is given by equation (28) as

2

2

2

dSm

ds (89)

For the datum surface (ellipsoid) equations (17) and (19)

give (noting that d d )

2 22 2 2cosds d d

2 (90)

For the projection plane

(91) 2 2

dS du dv 2

,u u and ,v v are given by equations (82);

and the total differentials are

andu u v v

du d d dv d d

(92)

Since the projection is conformal, scale is the same in alldirections around any point. It is sufficient then to

choose any one direction, say along a meridian where

is constant and 0d . Hence

2 2

2

21 u vm

(93)

The partial derivatives are evaluated using the chain rulefor differentiation and equations (33), (54) and (82)

andu u v v

(94)

with

2

2 2

2 2

2 2

1

1 sin cos

2expcos

1 exp 2

cos

1 cos sin

sin sin

1 cos sin

u a

v a

(95)

Substituting equations (95) into equations (94) and then

into equation (93) and simplifying gives the scale factorm for the Gauss-Schreiber projection as

2 2

2 21 tan 1 sin

tan cosm

Along the central meridian of the projection 0 andthe central meridian scale factor is

0m

2 2

0

2 2 4 4

cos1 sin

cos

cos 1 11 sin sin

cos 2 8

m

(97)

0m is not constant and varies slightly from unity, but a

final conformal mapping from the Gauss-Schreiberu,vplane to anX,Yplane may be made and this finalprojection (the Gauss-Krger projection) will have aconstant scale factor along the central meridian.

Gauss-Schreiber grid convergence

The grid convergence for the Gauss-Schreiber projectionis defined by equation (80) but the partial derivativesmust be evaluated using the chain rule for differentiation[equations (94)] and

tanv u v u

(98)

Using equations (95) the grid convergence for the Gauss-Schreiber projection is

1 12

tan tantan sin tan tan

1 tan

(99)

Conformal mapping from the Gauss-Schreiber to theGauss-Krger projection

Using conformal mapping and complex functions (seeAppendix), suppose that the mapping from the u,v planeof the Gauss-Schreiber projection (Figure 4) to theX,Yplane of the Gauss-Krger projection (Figure 5) is givenby

1

Y iX f u ivA

(100)

where the Y-axis is the central meridian, theX-axis is theequator andA is the rectifying radius.

Let the complex function f u iv be

2

1

sin 2 2r

r

u vf u iv i

a a

u vr i r

a a

(101)

where a is the radius of the conformal sphere and 2r are

as yet, unknown coefficients.

2

(96)

11


12/20

Expanding the complex trigonometric function inequation (101) gives

21

sin 2 cosh 2

cos 2 sinh 2r

r

u vf u iv i

r r

i r r

a a

u v

a a

u v

a a

(102)

and equating real and imaginary parts gives

2

1

2

1

sin 2 cosh 2

cos 2 sinh 2

r

r

r

r

Y u u vr r

a

u vr r

a a

A a

X v

A a

a

(103)

Now, along the central meridian and

and

0v

cosh 2 cosh 4 1v v Y

A

in equation (103)

becomes

2 4 6sin 2 sin 4 sin 6Y u u u u

A a a a a

(104)

Furthermore, along the central meridianu

ais an angular

quantity that is identical to the conformal latitude andequation (104) becomes

2 4 6sin 2 sin 4 sin 6

Y

A (105)

Now, if the central meridian scale factor is unity then the

Ycoordinate is the meridian distanceM, andY M

A A

is the rectifying latitude and equation (105) becomes

2 4 6sin 2 sin 4 sin 6 (106)

This equation is identical in form to equation (62) and we

may conclude that the coefficients are equal to the

coefficients

2r

2r in equation (62); and the Gauss-

Krger projection is given by

2

1

2

1

cos 2 sinh 2

sin 2 cosh 2

r

r

r

r

Av u

X ra a

u uY A r r

a a

vr

a

v

a

(107)

[These are Krger's equations (42), 8, p. 21.]

A is given by equation (40),u

aand

v

aare given by

equations (83) and we have elected to use coefficients

2r up to given by equations8r (63).

Figure 5 Gauss-Krger projection


120 E

[Note that the graticules of Figures 4 and 5 are fordifferent projections but are indistinguishable at theprinted scales and for the longitude extent shown. If a

larger eccentricity was chosen, say 110

1 199.5f

and the mappings scaled so that the distances from the

equator to the pole were identical, there would be somenoticeable differences between the graticules at large

distances from the central meridian. One of the authors(Karney 2010, Fig. 1) has examples of these graticuledifferences.]

Finally,Xand Yare scaled and shifted to giveE(east)andN(north) coordinates related to a false origin

0

0 0

E m X E

N m Y N

0

(108)

0m

0 ,E

is the central meridian scale factor and the quantities

are offsets that make theE,Ncoordinates positive

in the area of interest. The origin of theX,Ycoordinatesis at the intersection of the equator and the centralmeridian and is known as the true origin. The origin oftheE,Ncoordinates is known as the false origin and it is

located at

0N

0 0,X E Y N .

Gauss-Krger scale factor

The scale factor for the Gauss-Krger projection can be

derived in a similar way to the derivation of the scalefactor for the Gauss-Schreiber projection and we have

2 2 2

2 22 2 2cos

dS dX dY

ds d d 2

where , , ,X X u v Y Y u v and the total

differentials dXand dYare

X XdX du dv

u v

Y YdY du dv

u v

(109)

du and dv are given by equations (92) and substitutingthese into equations (109) gives

12


13/20

X u u X v vdX d d d d

u v

Y u u Y v vdY d d d d

u v

Choosing to evaluate the scale along a meridian where

is constant and 0d givesX u X v

dX du v

Y u Y vdY d

u v

(110)

and

2 2

2

2 2

dS dX dY m

ds d

2

2(111)

Differentiating equations (107) gives

, , ,X A X A Y X Y X

q pu a v a u v v u

(112)

where

2

1

2

1

2 sin 2 sinh 2

1 2 cos 2 cosh 2

r

r

r

r

u vq r r r

a a

u vp r r r

a a

(113)

Substituting equations (112) into (110) and then into theequation (111) and simplifying gives

2 22

2 2 2

2

1A um q p

a

v

(114)

The term in braces is the square of the scale factor

for the Gauss-Schreiber projection [see equation (93)]and so, using equation (96), we may write the scale factorfor the Gauss-Krger projection as

2 2 2

2 2

02 2

1 tan 1 sin

tan cos

Am m q p

a

(115)

where is the central meridian scale factor, q andp are

found from equations

0m

(113), tan from equation (88)andA from equation (40).

Gauss-Krger grid convergence

The grid convergence for the Gauss-Krger projection isdefined by

tandX

dY (116)

Using equations (110) and (112) we may write equation(116) as

tan

1

q v uu vq p p

u v q v up q

p

2

(117)

Let 1 , then using a trigonometric addition

formula write

11 21 2

tan tantan tan

1 tan tan

2

(118)

Noting the similarity between equations (117) and (118)we may define

1 2tan and tanq v u

p

(119)

and2 is the grid convergence on the Gauss-Schreiber

projection [see equations (98) and (99)]. So the gridconvergence on the Gauss-Krger projection is

1 1

2

tan tantan tan

1 tan

q

p

(120)

Conformal mapping from the Gauss-Krger pro jection tothe ellipsoid

The conformal mapping from the Gauss-Krgerprojection to the ellipsoid is achieved in three steps:

(i) A conformal mapping from the Gauss-Krger to theGauss-Schreiber projection giving u,v coordinates,then

(ii) Solving for tan and tan given the u,v Gauss-

Schreiber coordinates from which 0 , andfinally

(iii) Solving for tan by Newton-Raphson iteration and

then obtaining .

The development of the equations for these three steps isset out below.

Gauss-Schreiber coordinates from Gauss-Krgercoordinates

In a similar manner as outlined above, suppose that themapping from theX,Yplane of the Gauss-Krgerprojection to the u,v plane of the Gauss-Schreiberprojection is given by the complex function

1

u iv F Y iX a

(121)

IfE,Ncoordinates are given and , and are

known, then from equations0

E0

N0

m

(108)

0

0 0

andE E N N

X Ym

0

m(122)

13


14/20

Let the complex function F Y iX be

2

1

sin 2 2r

r

Y XF Y iX i

A A

Y XK r i r

A A

(123)

whereA is the rectifying radius and 2rK are as yet

unknown coefficients.

Expanding the complex trigonometric function inequation (123) and then equating real and imaginary partsgives

2

1

2

1

sin 2 cosh 2

cos 2 sinh 2

r

r

r

r

Y YK r r

X

A A

Y XK r r

A A

u

a A

v X

a A

(124)

Along the central meridianY M

A A the rectifying

latitude and and0X cosh 0 1 . Also,u

ais an

angular quantity that is identical to the conformal latitude

and we may write the first of equations (124) as

2 4 6sin 2 sin 4 sin 6K K K (125)

This equation is identical in form to equation (60) and we

may conclude that the coefficients are equal to the

coefficients

2rK

2r in equation (60) and the ratiosu

a

and

v

aare given by

2

1

2

1

sin 2 cosh 2

cos 2 sinh 2

r

r

r

r

Y Yr r

X

A A

Y Xr r

A A

u

a A

v X

a A

(126)

whereA is given by equation (40) and we have elected to

use coefficients2r

up to given by equations8r (61).

Conformal l atitude and longitude difference from Gauss-Schreiber coordinates

Equations (83) can be re-arranged and solved for tan

and tan as functions of the ratiosu

aand

v

agiving

2 2

sin

tan

sinh cos

tan sinh cos

u

a

v

a a

v u

a a

Solution for latitude by Newton-Raphson iteration

To evaluate tan after obtaining tan from equation(127) consider equations (87) and (88) with the

substitutions tat n and tat n

21 1t t t 2 (128)

and 12

sinh tanh1

t

t

(129)

tant can be evaluated using the Newton-Raphson

method for the real roots of the equation 0f t given

in the form of an iterative equation

1n

n n

n

f tt t

f t

(130)

where denotes the nth iterate andn

t f t is given by

2 21 1f t t t t (131)

The derivative d

f t fdt

t is given by

2 2

2 2

2 2

1 11 1

1 1

tf t t t

t

(132)

where tant is fixed.

An initial value for can be taken as1t

1 tant t and

the functions 1f t and 1f t evaluated from equations

(129), (131) and (132).2

t is now computed from

equation (130) and this process repeated to obtain

. This iterative process can be concluded when

the difference between1n

3 4, ,t t

t and nt reaches an acceptably

small value, and then the latitude is given by1tan 1nt .

This concludes the development of the Gauss-Krgerprojection.

TRANSFORMATIONS BETWEEN THE ELLIPSOID AND THEGAUSS-KRGER PLANE

Forward transformation: , ,X Y given 0 0, , ,a f m

1. Compute ellipsoid constants 2 , n and powers2 3 8, , ,n n n

u(127)

2. Compute the rectifying radiusA from equation (40)

3. Compute conformal latitude from equations (87)and (88)

4. Compute longitude difference0

5. Compute the u,v Gauss-Schreiber coordinates from

equations (83)

6. Compute the coefficients 2r from equations (63)

7. ComputeX,Ycoordinates from equations (107)8. Compute q andp from equations (113)

14


15/20

9. Compute scale factorm from equation (115)

10. Compute grid convergence from equation (120)

Inverse transformation: , ,X Y given0 0

, , ,a f m

1. Compute ellipsoid constants 2 , n and powers2 3 8, , ,n n n

2. Compute the rectifying radiusA from equation (40)3. Compute the coefficients 2r from equations (61)

4. Compute the ratios ,u v

a afrom equations (126)

5. Compute conformal latitude and longitudedifference from equations (127)

6. Compute tant by Newton-Raphson iterationusing equations (130), (131) and (132)

7. Compute latitude and longitude1tan t

0

8. Compute the coefficients n from equations (63)

9. Compute q andp from equations (113)

10. Compute scale factorm from equation (115)11. Compute grid convergence from equation (120)

ACCURACY OF THE TRANSFORMATIONS

One of the authors (Karney, 2010) has compared Krger's

series to order (set out above) with an exact transverse

Mercator projection defined by Lee (1976) and showsthat errors in positions computed from this series are less

than 5 nanometres anywhere within a distance of 4200

km of the central meridian (equivalent to at the

equator). So we can conclude that Kruger's series (toorder ) is easily capable of micrometre precision

within 30 of a central meridian.

8n

37.7

8n

THE 'OTHER' GAUSS-KRGER PROJECTION

In Krger's original work (Krger 1912) of 172 pages(plus vii pages), Krger develops the mapping equationsshown above in 22 pages, with a further 14 pages ofexamples of the forward and inverse transformations. Inthe next 38 pages Krger develops and explains analternative approach: direct transformations from the

ellipsoid to the plane and from the plane to the ellipsoid.

The remaining 100 pages are concerned with theintricacies of the geodesic projected on the transverseMercator plane, arc-to-chord, line scale factor, etc.

This alternative approach is outlined in the Appendix andfor the forward transformation [see equations (160)] theequations involve functions containing powers of the

longitude difference and derivatives2 3, , dM

d,

2

2

d M

d,

3

3,

d M

dFor the inverse transformation [see

equations (165) and (167)] the equations involve powers

of theXcoordinate and derivatives2 3 4

, ,X X X , 11

d

d

,

2

1

2

1

d

d

,

3

1

3

1

,d

d

and 1

d

dY

,

2

1

2

d

dY

,

3

1

3,

d

dY

For both

transformations, the higher order derivatives becomeexcessively complicated and are not generally known (orapproximated) beyond the eighth derivative.

Redfearn (1948) and Thomas (1952) both derive identical

formulae, extending (slightly) Kruger's equations, andupdating the notation and formulation. These formulaeare regarded as the standard for transformations betweenthe ellipsoid and the transverse Mercator projection. Forexample, GeoTrans (2010) uses Thomas' equations and

Geoscience Australia defines Redfearn's equations as themethod of transformation between the Geocentric Datumof Australia (ellipsoid) and Map Grid Australia(transverse Mercator) [GDAV2.3].

The apparent attractions of these formulae are:

(i) their wide-spread use and adoption by governmentmapping authorities, and

(ii) there are no hyperbolic functions.

The weakness of these formulae are:

(a) they are only accurate within relatively small bands

of longitude difference about the central meridian

(mm accuracy for 6 ) and

(b) at large longitude differences they can

give wildly inaccurate results (1-2 km errors).

30

The inaccuracies in Redfearn's (and Thomas's) equationsare most evident in the inverse transformation

, ,X Y . Table 1 shows a series of points each

having latitude but with increasing longitudedifferences

75

from a central meridian. TheX,Y

coordinates are computed using Krger's series and can

be regarded as exact (at mm accuracy) and the column

headed Readfearn , are the values obtained from

Redfearn's equations for the inverse transformation. The

error is the distance on the ellipsoid between the given

, in the first column and the Redfearn , in the

third column.

The values in the table have been computed for the

GRS80 ellipsoid ( 6378137 ma , 1 298.257222101f )

with 0 1m point Gauss-Krger Redfearn error

75 6

X 173137.521

Y 8335703.234

75 00' 00.0000" 5 59' 59.9999"

0.001

75 10

X 287748.837

Y 8351262.809

75 00' 00.0000" 9 59' 59.9966"

0.027

75 15

X 429237.683

Y 8381563.943

75 00' 00.0023" 14 59' 59.8608"

1.120

75 20

X 567859.299

Y 8423785.611

75 00' 00.0472" 19 59' 57.9044"

16.888

75 30

X 832650.961Y 8543094.338

75 00' 03.8591" 29 58' 03.5194"

942.737

75 35

X 956892.903

Y 8619555.491

75 00' 23.0237" 34 49' 57.6840"

4.9 km

Table 1

15


16/20

This problem is highlighted when considering a map ofGreenland (Figure 6), which is almost the ideal 'shape' fora transverse Mercator projection, having a small east-west extent (approx. 1650 km) and large north-south

extent (approx. 2600 km).

Y

A

B

Figure 6 Gauss-Krger projection of Greenland


45 W

PointsA andB represent two extremes if a central

meridian is chosen as . A ( ,

) is a point furthest from the central

meridian (approx. 850 km); andB (

75 W ) would have the greatest wes

0 45 W

70 N

78 N

t longit

22 30 W

ude.

Table 2 shows the errors at these points for the GRS80ellipsoid with 0 for the inverse transformation using

Redfearn's equations.

1m

point Gauss-Krger Redfearn error

A 70 22.5

X 842115.901

Y 7926858.314

75 00' 00.2049" 22 29' 53.9695"

64.282

B 78 -30

X -667590.239

Y 8837145.459

78 00' 03.1880"-29 57' 59.2860"

784.799

Table 2

CONCLUSION

We have provided here a reasonably complete derivationof the Gauss-Krger projection equations that allow

micrometre accuracy in the forward and inversemappings between the ellipsoid and plane. And we haveprovided some commentary on the 'other' Gauss-Krgerequations in wide use in the geospatial community.

These other equations offer only limited accuracy andshould be abandoned in favour of the equations (and

methods) we have outlined.

Our work is not original; indeed these equations weredeveloped by Krger almost a century ago. But with the

aid of computer algebra systems we have extendedKrger's series as others have done, e.g. Engsager &

Poder (2007) so that the method is capable of very highaccuracy at large distances from a central meridian. Thismakes the transverse Mercator projection a much moreuseful projection for the geospatial community.

We also hope that this paper may be useful to mappingorganisations wishing to 'upgrade' transformationsoftware that use formulae given by Redfearn (1948) or

Thomas (1952) they are unnecessarily inaccurate.

NOMENCLATURE

This paper Krger

k coefficients in series for

k coefficients in series for

oblique latitude

grid convergence c

eccentricity of ellipsoid e2 eccentricity of ellipsoid squared 2e2 2nd eccentricity of ellipsoid squared

oblique longitude

longitude L

0 longitude of central meridian

rectifying latitude

radius of curvature (meridian) R

radius of curvature (prime vertical) N

function of latitude

latitude B

conformal latitude b isometric latitude

longitude difference: 0 angle between parametric curves

A rectifying radius A

a semi-major axis of ellipsoid ab semi-minor axis of ellipsoid bc polar radius of curvatureds differential distance on datum surfacedS differential distance on projection surfacee Gaussian fundamental quantity

f flattening of ellipsoid

Gaussian fundamental quantityg Gaussian fundamental quantityh scale factor along meridiank scale factor along parallel

M meridian distance Xm scale factor m

0m central meridian scale factor 0mn 3rd flattening of ellipsoid n

Q quadrant length of ellipsoid M

s distance on ellipsoid st tant

t tant

u transverse Mercator coordinate V ellipsoid latitude function Q

v transverse Mercator coordinate W ellipsoid latitude function

16


17/20

APPENDIX

Reversion of a series

If we have an expression for a variablez as a series of

powers or functions of another variabley then we may,by a reversion of the series, find an expression fory as

series of functions ofz. Reversion of a series can be doneusing Lagrange's theorem, a proof of which can be foundin Bromwich (1991).

Suppose that

ory z xF y z y xF y (133)

then Lagrange's theorem states that for anyf

22

3 2 3

2

1

1

1!

2!

3!

!

n nn

n

xf y f z F z f z

x dF z f z

dz

x dF z f z

dz

x dF z f z

n dz

(134)

As an example, consider the series for rectifying latitude

2 4 6sin 2 sin 4 sin 6d d d (135)

And we wish to find an expression for as a function of

.

Comparing the variables in equations (135) and (133),

z , y and ; and if we choose1x f y y

then f z z and . So equation f z 1 (135) can be

expressed as

F (136)

and Lagrange's theorem gives

2

23

2

1

1

1

21

6

1

!

n nn

n

dF F

dd

Fd

dF

n d

(137)

where

2 4 6sin 2 sin 4 sin 6F d d d

and so

2 4 6sin 2 sin 4 sin 6F d d d

Taylor's theorem

This theorem, due to the English mathematicianBrook Taylor (16851731) enables a function

( )f x near a point x a to be expressed from the

values ( )f a and the successive derivatives of

( )f x evaluated at x a .

Taylor's polynomial may be expressed in thefollowing form

2

3

1

1

( ) ( ) ( ) ( ) ( )2!

( )3!

( )1 !

n

n

n

x af x f a x a f a f a

x af a

x af a R

n

(138)

where is the remainder aftern terms andn

R ( )f a ,

( ) etc.f a , are derivatives of the function f x

evaluated at x a .

Taylor's theorem can also be expressed as power series

0 !

kk

k

f af x

k

x a (139)

where

kk

k

x a

df a f x

dx

As an example of the use of Taylor's theorem, suppose

we have an expression for the difference between latitude and the rectifying latitude [see equation (45)]

2 4 6sin 2 sin 4 sin 6D D D (140)

and we wish to find expressions for sin 2 , sin4,

sin6 , etc. as functions of .

We can use Taylor's theorem to find an expression for

sinf about as

22

2

33

3

sin sin sin

1 sin2!

1sin

3!

d

d

d

d

d

d

giving

2

3 4

sin sin cos sin2

cos sin

6 24

(141)

Replacing with 2 and with 2 in equation (141)

and substituting from equation (140) gives an

17


18/20

expression for sin 2. Using similar replacements and

substitutions, expressions for sin 4, sin6, etc. can be

developed.

Hyperbolic funct ions

The basic functions are the hyperbolic sine ofx, denoted

by sinhx , and the hyperbolic cosine ofx denoted by

coshx ; they are defined as

exp

exp

exp

2

exp

2

sinh

cosh

x xx

x x

x

(142)

Other hyperbolic functions are in terms of these

sinh

cos

co

1tanh ,

h tanh

1sech , c

sh sinh

xcoth

1osech

x xx x

x x

x x

(143)

The inverse hyperbolic function of sinhx is 1sinh x and

is defined by sinh 1sinh x x . Similarly 1cosh x and1tanh x are defined by 1 coshcosh x x and

nh 1tanh ta x x ; both requiring and as a

consequence of the definitions

0x

1 2

1 2

1

sinh 1

cosh 1 1

1 1tanh 1 12 1

x x

x x

xx xx

ln

ln

ln

x x

x x (144)

A useful identity linking circular and hyperbolicfunctions used in conformal mapping is obtained byconsidering the following.

Using the trigonometric addition and double angleformula we have

1 12 21 1

4 2 1 12 2

21 12 2

2 21 12 2

cos sinln tan ln

cos sin

cos sin 1 sin

ln ln coscos sin

x xx

x x

x x x

xx x

(145)

Also, replacingx with tanx in the definition of the

inverse hyperbolic functions in equations (144) we have

1sinh tan ln tan 1 tan

1 sinln tan sec ln

cos

2x x x

xx x

x

(146)

And equating1 sin

lncos

x

x

from equations (145) and (146)

gives

11 14 2ln tan sinh tanx x (147)

Conformal mapping and complex functions

A theory due to Gauss states that a conformal mapping

from the , datum surface to theX,Yprojection

surface can be represented by the complex expression

Y iX f i (148)

Providing that and are isometric parameters andthe complex function f i is analytic. 1i

(the imaginary number), and the left-hand side ofequation (148) is a complex number consisting of a realand imaginary part. The right-hand-side of equation(148) is a complex function, i.e., a function of real and

imaginary parameters and respectively. The

complex function f i is analytic if it is

everywhere differentiable and we may think of ananalytic function as one that describes a smooth surfacehaving no holes, edges or discontinuities.

Part of a necessary and sufficient condition for f i to be analytic is that the Cauchy-Riemann

equations are satisfied, i.e., (Sokolnikoff & Redheffer1966)

andY X Y X

(149)

As an example, consider the Mercator projection of the

sphere shown in Figure 2 where the conformal mappingfrom the sphere (datum surface) to the plane is given byequations (67) and using the isometric latitude given byequation (34) the mapping equations are

0

1 1

4 2ln tan

X R R

Y R R

(150)

These equations can be expressed as the complexequation

Z Y iX R i (151)

whereZis a complex function defining the Mercatorprojection.

The transverse Mercator projection of the sphere shownin Figure 4 can also be expressed as a complex equation.

Using the identity (147) and equation (150) we maydefine the transverse Mercator projection by

1sinh tanZ Y iX R i (152)

Now suppose we have another complex function

(153)1tan sinhw u iv Z

representing a conformal transformation from theX,Yplane to the u,v plane.

18


19/20

What are the functions u and v?

It turns out, after some algebra, u and v are of the sameform as equations (83)

1

1

2 2

tantan

cos

sinsinhtan cos

u R

v R

(154)

and the transverse Mercator projection of the sphere isdefined by the complex function

1

1

2 2

tantan

cos

sinsinh

tan cos

w u iv

R

i

(155)

Other complex functions achieve the same result. Forexample Lauf (1983) shows that

1 142 tan expw u iv R i (156)

is also the transverse Mercator projection.

An alternative approach to developing a transverse

Mercator projection is to expand equation (148) as apower series.

Following Lauf (1983), consider a point P having

isometric coordinates , linked to an approximate

location0 0, by very small corrections , such

that0

and 0 ; equation (148)becomes

0 0

0 0

0

Y iX f i

f i

f i i

f z z f z

(157)

The complex function f z can be expanded by a

Taylor series [see equation (139)]

21 2

0 0

3

3

0

2!

3!

z0

f z f z z f z f z

zf z

(158)

where 1 20 0, , etc.f z f z are first, second and

higher order derivatives of the function f z can be

evaluated at . Choosing, as an approximate

location, a point on the central meridian having the same

isometric latitude as P, then

0z z

0 (since0

and 0 ) and (since 0 and0

0 ), hence 0 0 0z i z i i

and .

The complex function f z can then be written as

2

1 2

3

3

2!

3!

f z f i

if i f f

i

f

(159)

1f , 2f , etc. are first, second and higher order

derivatives of the function f .

Substituting equation (159) into equation (157) and

equating real and imaginary parts (noting that

and2 3 41, , 1, etc.i i i i f M ) gives

3 3 5 5 7 7

3 5 7

2 2 4 4 6 6

2 4 6

3! 5! 7!

2! 4! 6!

dM d M d M d M X

d d d d

d M d M d M Y M d d d

(160)

In this alternative approach, the transformation from the

plane to the ellipsoid is represented by the complexexpression

(161)i F Y iX

And similarly to before, the complex function F Y iX

can be expanded as a power series giving

2

1 2

3

3

2!

3!

iXi F Y iXF Y F Y

iXF Y

(162)

When 0X , 0 ; but when the point0X

,P becomes 1 1,0

1

P , a point on the central

meridian having latitude known as the foot-point

latitude. Now1

is the isometric latitude for the foot-

point latitude and we have 1F Y

Substituting equation (162) into equation (161) andequating real and imaginary parts gives

2 4 62 4 6

1 1 1

1 2 4 6

3 5 73 5 7

1 1 1 1

4 5 7

2! 4! 6!

3! 5! 7!

d d dX X X

dY dY dY

d d d d X X XX

dY dY dY dY

(163)

The first of equations (163) gives in terms of1

but

we require in terms of1

. Write the first of equations

(163) as

1 (164)

where

2 4 62 4 6

1 1 1

2 4 62! 4! 6!

d d dX X X

dY dY dY

(165)

19


20/20

And latitude 1g g can be expanded as

another power series

2

1 2

1 1

3

3

1

2!

3!

g g g

g

1

(166)

Krger, L., (1912), Konforme Abbildung desErdellipsoids in der Ebene,New Series 52, RoyalPrussian Geodetic Institute, Potsdam. DOI10.2312/GFZ.b103-krueger28

Lambert, J.H., (1772),Beytrge zum Gebrauche derMathematik und deren Anwendung III, Part VI:Anmerkungen und Zustze zur Entwerfung der Land-

und Himmelscharten, pp.105-199, URLhttp://books.google.com/books?id=sf82AAAAMAAJ

Noting that 1g 1 we may write the transformation

asLauf, G.B., (1983), Geodesy and Map Projections, Tafe

Publications, Collingwood, Australia.

2 42 4

1 1 1

1 2 4

1 1 1

3 5 73 5 7

1 1 1 1

4 5 7

2! 4!

3! 5! 7!

d d d

d d d

d d d d X X XX

dY dY dY dY

(167)

Lee, L.P., (1976), Conformal projections based onJacobian elliptic functions, Cartographica, Monograph

16, supplement 1 to Canadian Cartographer, Vol. 13,128 pages.

Redfearn, J.C.B., (1948), 'Transverse Mercator formulae',Empire Survey Review, Vol. 9, No. 69, pp. 318-322,July 1948

REFERENCESSchreiber, O., (1866), Theorie der Projektionsmethode

der hannoverschen Landsvermessung, Hannover:Hahnsche Hofbuchhandlung.

Adams, O.S., (1921),Latitude Developments Connectedwith Geodesy and Cartography, Special PublicationNo. 67, U.S. Coast and Geodetic Survey, Departmentof Commerce, Washington. Schreiber, O., (1897),Die konforme Doppelprojektion

der trigonometrischen Abteilung der Knigl,

Preussichen Landesaufnahme, Berlin.Bromwich, T.J.I'A., (1991),An Introduction to theTheory of Infinite Series, 3rd edition, ChelseaPublishing Company, New York. Snyder, J.P., (1987),Map ProjectionsA Working

Manual, Professional Paper 1395, U.S. GeologicalSurvey, Government Printing Office , Washington.Bugayevskiy, L.M. and Snyder, J.P., (1995),Map

Projections: A Reference Manual, Taylor & Francis,London. Snyder, J.P., (1993), Flattening The Earth - Two

Thousand Years of Map Projections, The University

of Chicago Press, Chicago.Deakin, R.E. and Hunter, M.N., (2010a),Geometric

Geodesy Part A, School of Mathematical and

Geospatial Sciences, RMIT University, Melbourne,Australia, January 2010.

SOED, (1993), The New Shorter Oxford English

Dictionary, Ed. Leslie Brown, Clarendon Press,Oxford.

Deakin, R.E. and Hunter, M.N., (2010b), GeometricGeodesy Part B, School of Mathematical andGeospatial Sciences, RMIT University, Melbourne,

Australia, January 2010.

Sokolnikoff, I.S. and Redheffer, R.M., (1966),Mathematics of Physics and Modern Engineering, 2ndedition, International Student Edition, McGraw-Hill

Kogakusha, Ltd. Tokyo.Engsager, K. and Poder, K., (2007), 'A highly accurate

world wide algorithm for the transverse Mercator

mapping (almost)', Proceedings of XXIII InternationalCartographic Conference (ICC2007), 4-10 August,

Moscow.

Thomas, P.D., (1952), Conformal Projections in Geodesyand Cartography, Special Publication No. 251, U.S.

Coast and Geodetic Survey, Department ofCommerce, Washington. URLhttp://docs.lib.noaa.gov/rescue/cgs_specpubs/QB27U35no2511952.pdfGDAV2.3 (2009), Geocentric Datum of Australia -

Technical Manual, Version 2.3, Amendment 1,Intergovernmental Committee on Surveying andMapping, July 2009. URLhttp://www.icsm.gov.au/gda/gdatm/index.html

Yang, Q., Snyder, J.P. and Tobler, W., (2000),Map

Projection Transformation, Taylor & Francis, London.

GeoTrans, (2010), The Geographic Translator, Edition 3,National Geospatial-Intelligence Agency, URLhttp://earth-info.nima.mil/GandG/geotrans

Helmert, F.R., (1880),Die mathematischen und

physikalischen Theorieen der hheren Geodsie, Vol.1,Die mathematischen Theorieen, Leipzig.

Karney, C.F.F., (2010), 'Transverse Mercator projection

with an accuracy of a few nanometres',Journal ofGeodesy (in press).

Gauss-Krueger Warrnambool Conference

Documents