A Fresh Look at the UTM Projection - The Karney-Krueger Equations

8/10/2019 A Fresh Look at the UTM Projection - The Karney-Krueger Equations

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A FRESH LOOK AT THE UTM PROJECTION:Karney-Krueger equations

R. E. Deakin 1, M. N. Hunter 2 and C. F. F. Karney 3 1 School of Mathematical and Geospatial Sciences, RMIT University

2 Maribyrnong, VIC, Australia.3 Princeton, N.J., USA.

email: [email protected]

Presented at the Surveying and Spatial Sciences Institute (SSSI) Land Surveying Commission National Conference,Melbourne, 18-21 April, 2012

ABSTRACT

The Universal Transverse Mercator (UTM) coordinatesystem has at its core, equations that enable thetransformations from geographic to grid coordinates andvice versa. These equations have a long historystretching back to Gauss and his survey of Hannover inthe early 1800's but their modern formulation is due to L.Krueger (1912) who synthesised Gauss and other workon the transverse Mercator (TM) projection of theellipsoid. Krueger developed two methods oftransformation and the equations of his second methodhave become the de facto standard for the UTM and otherTM projection systems. Unfortunately these arecomplicated and have limited accuracy. But theequations for Kruegers first method which do not sufferfrom these accuracy limitations are undergoing arenaissance and a recent study by Charles Karney (2011)

has given these equations a make over. This papergives some insight into the development and use of theKarney-Krueger equations.

INTRODUCTION

The transverse Mercator (TM) projection is a conformalmapping of a reference ellipsoid of the earth onto a planewhere the equator and central meridian remain as straightlines and the scale along the central meridian is constant;all other meridians and parallels being complex curves(Figure 5). The UTM is a mapping system that uses theTM projection with certain restrictions (such as standard6 -wide longitude zones, central meridian scale factor of0.9996, defined false origin offsets, etc.) and the Gauss-Krueger system is similar. And both are members of afamily of TM projections of which the spherical form wasoriginally developed by Johann Heinrich Lambert (1728-1777). This (spherical) projection is also called theGauss-Lambert projection acknowledging thecontribution of Carl Friedrich Gauss (1777 1855) to thedevelopment of the TM projection. Snyder (1993) andLee (1976) have excellent summaries of the history ofdevelopment which we paraphrase below.

Gauss (c.1822) developed the ellipsoidal TM as oneexample of his investigations in conformal

transformations using complex algebra and used it for thesurvey of Hannover in the same decade. This projectionhad constant scale along the central meridian and wasknown as the Gauss conformal or Gauss' Hannover

projection. Also (c. 1843) Gauss developed a 'doubleprojection' consisting of a conformal mapping of theellipsoid onto the sphere followed by a mapping from thesphere to the plane using the spherical TM formula. Thisprojection was adapted by Oskar Schreiber and used for

the Prussian Land Survey of 1876-1923. It is also calledthe Gauss-Schreiber projection and scale along the centralmeridian is not constant. Gauss left few details of hisoriginal developments and Schreiber (1866, 1897)published an analysis of Gauss' methods, and LouisKrueger (1912) re-evaluated both Gauss' and Schreiber'swork, hence the name Gauss-Krueger as a synonym forthe TM projection.

The aim of this paper is to give a detailed derivation of aset of equations that we call the Karney-Kruegerequations. These equations give micrometre accuracyanywhere within 30 of a central meridian; and at their

heart are two important series linking conformal latitude and rectifying latitude . We provide a developmentof these series noting our extensive use of the computeralgebra system Maxima in showing these series to highorders of n; unlike Krueger who only had patience. Andwithout this computer tool it would be impossible torealize the potential of his series.

Krueger gave another set of equations that we wouldrecognise as Thomas's or Redfearn's equations (Thomas1952, Redfearn 1948). These other equations alsoknown as the TM projection are in wide use in thegeospatial community; but they are complicated, and onlyaccurate within a narrow band (3 6) about a centralmeridian. We outline the development of these equationsbut do not give them explicitly, as we do not wish topromote their use. We also show that the use of theseequations can lead to large errors in some circumstances.

This paper supports the work of Engsager & Poder(2007) who also use Krueger's series in their elegantalgorithms for a highly accurate transverse Mercatorprojection but provide no derivation of the formulae.Also, one of the authors (Karney 2011) has a detailedanalysis of the accuracy of our projection equations (theKarney-Krueger equations) and this paper may beregarded as background reading. [An earlier version ofthis paper was presented at The Institution of SurveyorsVictoria, 25th Regional Survey Conference,Warrnambool, 10-12 September 2010]
mailto:[email protected]:[email protected]:[email protected]:[email protected]


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The preliminary sections set out below containinformation 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 PRELIMINARIESThe transverse Mercator (TM) projection is a mapping ofa reference ellipsoid of the earth onto a plane and somedefinition of the ellipsoid and various associatedconstants are useful. We then give a limited introductionto differential geometry including definitions andformulae for the Gaussian fundamental quantities e, f andg, the differential distance ds and scale factors m, h and k.

Next, we define and give equations for the isometriclatitude , meridian distance M, quadrant length Q, therectifying radius A and the rectifying latitude . Thisthen provides the basic 'tools' to derive the conformallatitude and show how the two important serieslinking and are obtained.

The ellipsoid

In geodesy, the ellipsoid is a surface of revolution createdby rotating an ellipse (whose semi-axes lengths are a andb and a b ) about its minor axis. The ellipsoid is themathematical surface that idealizes the irregular shape ofthe earth and it has the following geometrical constants:

flatteninga b

f a

(1)

eccentricity2 2

2

a ba

(2)

2nd eccentricity2 2

2

a bb

(3)

3rd flatteninga b

na b

(4)

polar radius2a

cb

(5)

These geometric constants are inter-related as follows

2

2

1 11 1

11

b n af

a n c

(6)

22

2 2

42

1 1

nf f

n

(7)

22

2 2 2

2 41 1 1

f f n

f n

(8)

2

2

1 12 1 1

f n f

(9)

From equation (7) an absolutely convergent series for 2 is

2 2 3 4 54 8 12 16 20n n n n n (10)since 0 1n .The ellipsoid radii of curvature (meridian plane) and

(prime vertical plane) at a point whose latitude is are

2 23 2 3 32 2

1 1

1 sin

a a cW V

(11)

1 22 21 sina a c

W V

(12)

where the latitude functions V and W are defined as

2 2 2

22 2 2

2

1 sin

1 2 cos 21 cos1

W

n nVn

(13)

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

Curvilinear coordinates (latitude), (longitude) areused to define the location of points on the ellipsoid (thedatum surface) and these points can also have x,y,zCartesian coordinates where the positive z-axis is therotational axis of the ellipsoid passing through the north

pole, the x-y plane is the equatorial plane and the x-zplane is the Greenwich meridian plane. The positive x-axis passes through the intersection of the Greenwichmeridian and equator and the positive y-axis is advanced90 eastwards along the equator.

The curvilinear and Cartesian coordinates are related by

2

, cos cos

, cos sin

, 1 sin

x x

y y

z z

(14)

The differential arc-length ds of a curve on the ellipsoid

is given by

2 2 2 2ds dx dy dz (15)

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


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2 2 22ds e d f d d g d (17)

where the coefficients e, f and 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 2 2, 0, cose f g (19)

d

d

e d

g d

P

d s

Q

Figure 1. The differential rectangle

In Figure 1 the parametric curves and pass through

P and the curves d and d pass through Q, and

the length of the curve PQ ds . The differentialrectangle formed by the curves may be regarded as aplane figure whose opposite sides are parallel straightlines enclosing a differentially small area da.

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 befound from

cos f eg

(21)

Thus, if the parametric curves on the surface intersect atright angles (i.e., they form an orthogonal system ofcurves) then 12 and cos 0 , and so fromequation (21) 0f . Conversely, if 0f theparametric curves form an orthogonal system.

Suppose there is another surface (projection surface) with, curvilinear coordinates and X,Y,Z Cartesian

coordinates related by

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

Then, using equations (15) and (16) replacing x,y,z withX,Y,Z and , with , we have the differentialdistance dS on this surface defined by

2 2 22dS E d F d d G d (23)

with the Gaussian fundamental quantities

2 2 2

2 2 2

X Y ZE

X X Y Y Z ZF

X Y ZG

(24)

Alternatively, the X,Y,Z Cartesian coordinates (projectionsurface) can be expressed as functions of the , curvilinear coordinates (datum surface) as

, , , , ,X X Y Y Z Z (25)

and

2 2 22dS E d F d d G d (26)

where

2 2 2

2 2 2

X Y ZE

X X Y Y Z ZF

X Y ZG

(27)

Scale factors m, h and k

The scale factor m 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

22 2 2

2 2

2 2

2

2

2

2

dS E d F d d G dm

ds e d f d d g d

E d F d d G d

e d f d d g d

(28)

WhenE Ge g

and 0F or E Ge g

and 0F the

scale factor m is the same in every direction and suchprojections are known as conformal. For the ellipsoid(datum surface) where the parametric curves , are anorthogonal 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 k is the scale

factor along the parallel of latitude. Using equations (28)

andE d E G d G

h k e d e g d g

(30)


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Isometric latitude

According to the SOED (1993) i sometric means: ofequal measure or dimension and we may think ofisometric parameters (isometric latitude) and

0 (longitude difference) in the following way.Imagine you are standing on the earth at the equator and

you measure a metre north and a metre east; both of theseequal lengths would represent almost equal angularchanges in latitude d and longitude d . Now imagineyou are close to the north pole; a metre in the northdirection will represent (almost) the same angular changed as it did at the equator. But a metre in the eastdirection would represent a much greater change inlongitude, i.e., equal north and east linear measures nearthe pole do not correspond to equal angular measures.

What is required is isometric latitude , a variableangular 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

222 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 gives (Deakin & Hunter 2010b)

12

1 14 2

1 sinln tan

1 sin

(33)

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

1 1

4 2ln tan (34)

Meridian distance M

Meridian distance M is defined as the arc of the meridianfrom the equator to the point of latitude

23 3

0 0 0

1a cM d d d

W V

(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 ofthe series formula for M is a function of and powers of

2 ; but the German geodesist F.R. Helmert (1880) gave aformula for meridian distance as a function of latitude

and powers of the ellipsoid constant n that required fewerterms for the same accuracy than meridian distanceformula involving powers of 2 . Helmert's method ofdevelopment (Deakin & Hunter 2010a) and with somealgebra we may write

2 3 22 2

0

1 1 2 cos 21

aM n n n d

n

(36)

The computer algebra system Maxima can be used toevaluate the integral and M can be written as

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 8n as follows

2 4 6 8

0

3 5 7

2

2 4 6 8

4

3 5 7

6

4 6 8

8

5 7

10

12

1 1 1 251 4 64 256 16384

3 3 3 15

2 16 128 204815 15 75 105

16 64 2048 819235 175 245

48 768 6144315 441 1323

512 2048 32768693 2079

1280 102401

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 81926435

14336109395

262144

n n

c n

c n

(38)

[This is Krueger's equation for X shown 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 obtainedfrom equation (37) by setting 12 , and noting that

sin2 , sin4 , sin6 , all equal zero, giving

02 1a

Q cn

(39)

[This is Krueger's equation for M shown in 5, p.12.]

Rectifying latitude and rectifying radius A

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 meridianupon the earth, we may calculate the latitudes upon


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this sphere such that the arcs of the meridian upon itare equal to the corresponding arcs of the meridianupon the earth.

If denotes this latitude on the (rectifying) sphere ofradius A then meridian distance M is given by

M A (40)

An expression for A is obtained by considering the casewhen 12 , M Q and (40) may be re-arranged to

give 2A Q then using (39) and (38) to give A to

order 8n as

2 4 6 81 1 1 2511 4 64 256 16384aA n n n nn (41)Re-arranging (40) and using (37) and (41) gives the rectifying latitude to order 4n as

2 4 6

8

sin 2 sin 4 sin 6

sin8

MA

d d d

d

(42)

where the coefficients nd are

3 2 42 4

3 46 8

3 9 15 152 16 16 3235 31548 512

d n n d n n

d n d n

(43)

[This is Krueger's equation (6), 5, p. 12, and Maxima

has been used for the algebra by using a seriesrepresentation of 10c ]

An expression for as a function of and powers of nis obtained by reversion of a series using Lagrange'stheorem (see Appendix) and to order 4n

2 4 6

8

sin 2 sin 4 sin 6

sin8

D D D

D

(44)

where the coefficients nD are

3 2 42 4

3 46 8

3 27 21 552 32 16 32151 109748 512

D n n D n n

D n D n

(45)

[This is Krueger's equation (7), 5, p. 13.]

Conformal latitude

Suppose we have a sphere of radius a with curvilinearcoordinates , (meridians and parallels) and X,Y,ZCartesian coordinates related by

, cos cos

, cos sin

, sin

X X a

Y Y a

Z Z a

(46)

Substituting equations (46) into equations (24), replacing, with , gives the Gaussian Fundamental

Quantities for the sphere as

2 2 2, 0, cosE a F G a (47)

The conformal projection of the ellipsoid (datum surface)onto the sphere (projection surface) is obtained byenforcing the condition h k ; and with equations (19),(30) and (47), replacing , with , whereappropriate, we have,

coscos

a d a dd d

(48)

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

0

0 0

cos1

cosa d

d

and since 0 0 0 then 0 0cos cos 1 , 0 a and d d . Substituting this result into equation (48)gives

cos cosd d

(49)

An expression linking and can be obtained byintegrating both sides of (49) using the standard integral

result 1 14 2sec ln tanx dx x and the previous resultfor equation (33) to give

12

1 1 1 14 2 4 2

1 sinln tan ln tan

1 sin

(50)

The right-hand-side of (50) is the isometric latitude [see equation (33)] and we may write

1 14 2e tan (51)

Solving equation (51) gives the conformal latitude1 1

22tan e (52)

The conformal longitude is obtained from thedifferential relationship d d which is aconsequence of the scale factor along the equator beingunity, and longitude on the ellipsoid is identical tolongitude on the conformal sphere, which makes

(53)

Another expression linking and can be obtained byusing (11), (12) and (13) in the right-hand-side of (49)giving

2

2 2

1

cos cos 1 2 cos 2 cos

n dd d

V n n

(54)

and separating the right-hand-side of (54) into partialfractions gives


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2

4 coscos cos 1 2 cos 2d d n d

n n

(55)

Using the standard integral result

1 14 2sec ln tanx dx x and the useful identitylinking circular and hyperbolic functions

11 14 2

ln tan sinh tanx x [see (148) in Appendix]we may write

1 12

0

4 cossinh tan sinh tan

1 2 cos 2n d

n n

(56)

We now have a function linking conformal latitude with latitude and the integral on the right-hand-side of(56) can be evaluated using Maxima and then combinedwith 1sinh tan to give . The conformal latitude is then

1tan sinh (57)

And the series for to order 4n is

2 4 6

8

sin 2 sin 4 sin 6

sin8

g g g

g

(58)

where the coefficients ng are

2 3 42

2 3 44

3 46

48

2 4 822

3 3 455 16 133 15 9

26 3415 21

1237630

g n n n n

g n n n

g n n

g n

(59)

[This is Krueger's equation (8), 5, p. 14, although wehave not used his method of development.]

The series connecting rectifying latitude with conformallatitude

A method of obtaining as a function of ; and asa function of is set out in Deakin et al. (2010) that

follows the methods in Krueger (1912). A betteralternative, suggested by Charles Karney (2010), utilizesthe power of Maxima and is set out below:

[a] The series (58) gives conformal latitude as afunction of latitude

[b] Using Lagranges theorem (see Appendix) the series(58) may be reversed giving as a function of

2 4 6sin 2 sin 4 sin 6G G G (60)

where the coefficients nG are function of n

[c] Substituting (60) into (42) gives as a function

2 4 6

8 10

12 14

16

sin 2 sin 4 sin 6

sin8 sin10

sin12 sin14

sin16

(61)

where the coefficients n are

2 3 4 5

2

6 7

8

2 3 4 5

4

6 7

8

3 4 56

1 2 5 41 127

2 3 16 180 2887891 72161

37800 38707218975107

5080320013 3 557 281

48 5 1440 6301983433 13769

1935360 28800148003883

17418240061 103 15061

240 140 26880

n n n n n

n n

n

n n n n

n n

n

n n n

6 7

8

4 5 68

7 8

5 610

7

167603 67102379181440 290304007968243179833600

49561 179 6601661161280 168 7257600

97445 4017612901349896 7664025600

34729 341888980640 1995840

14644087 26054135999123840 622

n n

n

n n n

n n

n n

n

8

6 712

8

7 814

816

702080212378941 30705481319334400 10378368

17521432679958118860800

1522256789 167599348991383782400 31135104001424729850961743921418240

n

n n

n

n n

n

(62)

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

[d] Reversing series (61) gives conformal latitude as

a function of rectifying latitude to order 8n

2 4 6

8 10 12

14 16

sin 2 sin 4 sin 6

sin8 sin10 sin12

sin14 sin16

(63)

where the coefficients n are


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2 3 4 52

6 7

8

2 3 4 54

6 7

8

3 46

1 2 37 1 81

2 3 96 360 51296199 5406467

604800 387072007944359

67737600

1 1 437 4648 15 1440 1051118711 51841

3870720 120960024749483

34836480017 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 68

7 8

5 6 710

5569

907209261899 6457463

58060800 177408004397 11 830251

161280 504 7257600466511 324154477

2494800 76640256004583 108847 8005831

161280 3991680 6386688022894433

12454041

n n

n n

n n n

n n

n n n

8

6 712

8

7 814

816

620648693 16363163

638668800 5189184002204645983

12915302400

219941297 4973238115535129600 124540416001917738872573719607091200

n

n n

n

n n

n

(64)

[This is Krueger equation (10), 5, p. 14 extended toorder 8n ]

The series (61) and (63) are the key series in the Karney-Krueger equations for the TM projection.

THE KARNEY-KRUEGER EQUATIONS FOR THETRANSVERSE MERCATOR PROJECTION

The Karney-Krueger equations for the transverseMercator (TM) projection are 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 aconformal mapping of this sphere to the plane using thespherical TM projection equations with spherical latitude

replaced by conformal latitude . This two-stepprocess is also known as the Gauss-Schreiber projection(Snyder 1993) and the scale along the central meridian isnot constant. [Note that the Gauss-Schreiber projectioncommonly uses a conformal sphere of radius 0 0R

where 0 and 0 are evaluated at a central latitude forthe region of interest.] The second part is the conformalmapping from the Gauss-Schreiber to the TM projection

where the scale factor along the central meridian is madeconstant.

To understand this process we first discuss the sphericalMercator and transverse Mercator projections. We thengive the equations for the Gauss-Schreiber projection andshow that the scale factor along the central meridian ofthis projection is not constant. Finally, using complex

functions and principles of conformal mapping developedby Gauss, we show the conformal mapping from theGauss-Schreiber projection to the TM projection.

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 factor m and gridconvergence .

Mercator projection of the sphere

X

Y

Figure 2 Mercator projectiongraticule interval 15, central meridian 0 120 E

The Mercator projection of the sphere is a conformalprojection with the datum surface a sphere of radius Rwith curvilinear coordinates , and Gaussianfundamental quantities

2 2 2, 0, cose R f g R (65)

The projection surface is a plane with X,Y Cartesiancoordinates and X X and Y Y and Gaussianfundamental quantities

2 2

, 0,Y X

E F G

(66)

Enforcing the scale condition h k and using equations(30), (65) and (66) gives the differential equation

1cos

dY dXd d

(67)

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


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1 and

cosdX R d dY d

Integrating and evaluating constants of integration givesthe well known equations for Mercator's projection of thesphere

0

1 14 2ln tan

X R R

Y R

(68)

where 0 is the longitude of the central meridian and

0 .

An alternative set of equations for Mercator's projectionmay be derived as follows by using the half-angle

formula1 cos

tan2 1 cosx x

x and writing

121 1

4 2 12

1 cos 1 sintan

1 cos 1 sin

.

Using this result in equation (68) gives

01 sin 1 sin

ln ln1 sin 2 1 sin

X R R

RY R

(69)

TM projection of the sphere (Gauss-Lambert projection)

The equations for the TM projection of the sphere (alsoknown as the Gauss-Lambert projection) can be derivedby considering the schematic view of the sphere in Figure

3 that shows P having curvilinear coordinates , thatare angular quantities measured along great circles(meridian and equator).

Now consider the great circle NBS (the oblique equator)with a pole A that lies on the equator and great circlesthrough A, one of which passes through P making anangle with the equator and intersecting the obliqueequator at C.

N

S

Ae q u a to r

u

v

CP

B

Figure 3 Oblique pole A on equator

and are oblique latitude and oblique longituderespectively and equations linking , and , can beobtained from spherical trigonometry and the right-angled spherical triangle CNP having sides , 12 ,

12 and an angle at N of 0

sin cos sin (70)

tantan

cos

(71)

Squaring both sides of equation (70) and using thetrigonometric identity 2 2sin cos 1x x gives, aftersome algebra

2 2cos cos tan cos (72)

From equations (72) and (70)

2 2

sintan

tan cos

(73)

Replacing with and with in equations (69);then using equations (70), (71) and (145) give theequations for the TM 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

(74)

u

v Figure 4 Transverse Mercator (TM) projection

graticule interval 15, central meridian 0 120 E


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The TM projection of the sphere is conformal, which canbe verified by analysis of the Gaussian fundamentalquantities e, f, g of the , spherical datum surface [seeequations (65)] and E, F, G of 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

(75)

Differentiating equations (74) noting that

12

1tan

1d dy

ydx dxy

and 1 21

tanh1

d dyy

dx dxy gives

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

(76)

and substituting these into equations (75) gives

2 2 2

2 2 2 2

cos, 0,

1 cos sin 1 cos sinR R

E F G

(77)

Now using equations (30), (65) and (77) the scale factorsh and k are equal and 0f F , then the projection isconformal.

Gauss-Lambert scale factor

Since the projection is conformal, the scale factorm h k E e and

2 2

2 2 4 4

1

1 cos sin

1 31 cos sin cos sin

2 8

m

(78)

Along the central meridian 0 and the centralmeridian scale factor 0 1m

Gauss-Lambert grid convergence

The grid convergence is the angle between themeridian and the grid-line parallel to the u-axis and isdefined as

tan dv

du (79)

and the total differentials du and dv are

andu u v v

du d d dv d d

(80)

Along a meridian is constant and 0d , and the gridconvergence is obtained from

tan v u

(81)

and substituting partial derivatives from equations (76)gives

1tan sin tan (82)

TM projection of the conformal sphere (Gauss-Schreiberprojection)

The equations for the TM projection of the conformalsphere are simply obtained by replacing spherical latitude

with conformal latitude in equations (74) andnoting that the radius of the conformal sphere is a to give

1

1

tantan

cos

tanh cos sin

u a

v a

(83)

[These are Krueger s equations (36), 8, p. 20]Alternatively, replacing with and with inequations (68) then using equations (71), (73) and theidentity (148); and finally replacing spherical latitude

with conformal latitude gives (Karney 2011)

1

1

2 2

tantan

cos

sinsinh

tan cos

u a

v a

(84)

tan , which appears in both equations (84), can beevaluated by using hyperbolic and inverse hyperbolicfunctions (see Appendix) and equation (50) in thefollowing manner.

Using equation (148) of the Appendix we may write

11 14 2ln tan sinh tan (85)

And the right-hand-side of equation (50) can be written as

1 1 14 2 2

1 1

1 sinln tan ln

1 sin

sinh tan tanh sin

(86)Equating (85) and (86) gives

1 1 1sinh tan sinh tan tanh sin (87)

With the substitution

1

2

tansinh tanh

1 tan

(88)

and some algebra, equation (87) can be rearranged as(Karney 2011)

2 2tan tan 1 1 tan (89)


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10

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

22

2

dSm

ds (90)

For the datum surface (ellipsoid) equations (17) and (19)give (noting that d d )

2 2 22 2 2cosds d d (91)

For the projection plane

2 2 2dS du dv (92)

,u u and ,v v are given by equations (83);and the total differentials are

andu u v v

du d d dv d d

(93)

Since the projection is conformal, scale is the same in alldirections around any point. It is sufficient then tochoose any one direction, say along a meridian where is constant and 0d . Hence

2 2

22

1 u vm

(94)

The partial derivatives are evaluated using the chain rulefor differentiation and equations (33), (52) and (84)

andu u v v

(95)

with

2

2 2

2 2

2 2

1

1 sin cos

2expcos

1 exp 2

cos1 cos sin

sin sin1 cos sin

u a

v a

(96)

Substituting equations (96) into equations (95) and theninto equation (94) and simplifying gives the scale factorm for the Gauss-Schreiber projection as

2 2 2

2 2

1 tan 1 sin

tan cosm

(97)

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

2 20

2 2 4 4

cos1 sin

cos

cos 1 11 sin sin

cos 2 8

m

(98)

0m is not constant and varies slightly from unity, but afinal conformal mapping from the Gauss-Schreiber u,vplane to an X,Y plane may be made and this finalprojection (the TM projection) will have a constant scalefactor along the central meridian.

Gauss-Schreiber grid convergence

The grid convergence for the Gauss-Schreiber projectionis defined by equation (81) but the partial derivativesmust be evaluated using the chain rule for differentiation[equations (95)] and

tan v u v u

(99)

Using equations (96) the grid convergence for the Gauss-Schreiber projection is

1 12

tan tantan sin tan tan

1 tan

(100)

Conformal mapping from the Gauss-Schreiber to the TMprojection

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

1 Y iX f u ivA

(101)

where the Y-axis is the central meridian, the X-axis is theequator and A is the rectifying radius.

Let the complex function f u iv be

21

sin 2 2rr

u vf u iv i

a au v

r i ra a

(102)

where a is the radius of the conformal sphere and 2r areas yet, unknown coefficients.

Expanding the complex trigonometric function inequation (102) gives

21

sin 2 cosh 2

cos 2 sinh 2

rr

u vf u iv i

r r

i r r

a au va a

u va a

(103)

and equating real and imaginary parts gives


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11

21

21

sin 2 cosh 2

cos 2 sinh 2

rr

rr

Y u u vr r

a a

u vr r

a a

A aX vA a

(104)

Now, along the central meridian 0v and

cosh 2 cosh 4 1v v andYA in equation (104)

becomes

2 4 6sin 2 sin 4 sin 6Y u u u uA a a a a

(105)

Furthermore, along the central meridianua

is an angular

quantity that is identical to the conformal latitude andequation (105) becomes

2 4 6sin 2 sin 4 sin 6Y

A (106)

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

Y coordinate is the meridian distance M, andY MA A

is the rectifying latitude and equation (106) becomes

2 4 6sin 2 sin 4 sin 6 (107)

This equation is identical in form to equation (61) and wemay conclude that the coefficients 2r are equal to thecoefficients 2r in equations (62); and the TMprojection is given by

21

21

cos 2 sinh 2

sin 2 cosh 2

rr

rr

Av u v

X r ra a a

u u vY A r r

a a a

(108)

[These are Krueger equations (42), 8, p. 21.]

A is given by equation (41),ua and

va are given by

equations (84) and we have elected to use coefficients

2r up to 8r given by equations (62).

[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 alarger eccentricity was chosen, say 110 1 199.5f and the mappings scaled so that the distances from theequator to the pole were identical, there would be somenoticeable differences between the graticules at largedistances from the central meridian. One of the authors(Karney 2011, Fig. 1) has examples of these graticuledifferences.]

Figure 5 Transverse Mercator projectiongraticule interval 15, central meridian 0 120 E

Finally, X and Y are scaled and shifted to give E (east)and N (north) coordinates related to a false origin

0 0

0 0

E m X EN m Y N

(109)

0m is the central meridian scale factor and the quantities

0 0,E N are offsets that make the E,N coordinates positivein the area of interest. The origin of the X,Y coordinatesis at the intersection of the equator and the centralmeridian and is known as the true origin. The origin ofthe E,N coordinates is known as the false origin and it islocated at 0 0,X E Y N .

TM scale factor

The scale factor for the TM projection can be derived in asimilar way to the derivation of the scale factor for theGauss-Schreiber projection and we have

2 2 2

2 2 22 2 2cos

dS dX dY

ds d d

where , , ,X X u v Y Y u v and the totaldifferentials dX and dY are

X XdX du dv

u vY Y

dY du dvu v

(110)

du and dv are given by equations (93) and substitutingthese into equations (110) gives

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 gives


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12

X u X vdX d

u v

Y u Y vdY d

u v

(111)

and

2 2 22

2 22

dS dX dYm

ds d (112)

Differentiating equations (108) gives

, , ,X A X A Y X Y X

q pu a v a u v v u

(113)

where

21

21

2 sin 2 sinh 2

1 2 cos 2 cosh 2

rr

rr

u vq r r r

a a

u vp r r ra a

(114)

Substituting equations (113) into (111) and then into theequation (112) and simplifying gives

2 22

2 2 22

1A u vm q p

a

(115)

The term in braces is the square of the scale factorfor the Gauss-Schreiber projection [see equation (94)]and so, using equation (97), we may write the scale factorfor the TM projection as

2 2 22 2

0 2 2

1 tan 1 sin

tan cos

Am m q p

a

(116)

where 0m is the central meridian scale factor, q and p are

found from equations (114), tan from equation (89)and A from equation (41).

TM grid convergence

The grid convergence for the TM projection is defined by

tan dXdY

(117)

Using equations (111) and (113) we may write equation(117) as

tan1

q v uu vq p p

u v q v up qp

(118)

Let 1 2 , then using a trigonometric additionformula write

1 21 21 2

tan tantan tan

1 tan tan

(119)

Noting the similarity between equations (118) and (119)we may define

1 2tan and tanq v up

(120)

and 2 is the grid convergence on the Gauss-Schreiber

projection [see equations (99) and (100)]. So the gridconvergence on the TM projection is

1 1

2

tan tantan tan

1 tan

qp

(121)

Conformal mapping from the TM projection to the ellipsoid

The conformal mapping from the TM projection to theellipsoid is achieved in three steps:

(i) A conformal mapping from the TM to the Gauss-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 andthen obtaining .

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

Gauss-Schreiber coordinates from TM coordinates

In a similar manner as outlined above, suppose that themapping from the X,Y plane of the TM projection to theu,v plane of the Gauss-Schreiber projection is given bythe complex function

1 u iv F Y iXa

(122)

If E,N coordinates are given and 0E , 0N and 0m areknown, then from equations (109)

0 0

0 0

andE E N N

X Ym m

(123)

Let the complex function F Y iX be

21

sin 2 2rr

Y XF Y iX i

A AY X

K r i rA A

(124)

where A is the rectifying radius and 2rK are as yetunknown coefficients.

Expanding the complex trigonometric function inequation (124) and then equating real and imaginary partsgives


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13

21

21

sin 2 cosh 2

cos 2 sinh 2

rr

rr

Y Y XK r r

A A

Y XK r r

A A

ua Av Xa A

(125)

Along the central meridianY MA A

the rectifying

latitude and 0X and cosh 0 1 . Also, ua

is an

angular quantity that is identical to the conformal latitude and we may write the first of equations (125) as

2 4 6sin 2 sin 4 sin 6K K K (126)

This equation is identical in form to equation (63) and wemay conclude that the coefficients 2rK are equal to the

coefficients 2r in equation (63) and the ratiosua

and

va

are given by

21

21

sin 2 cosh 2

cos 2 sinh 2

rr

rr

Y Y Xr r

A A

Y Xr r

A A

ua Av Xa A

(127)

where A is given by equation (41) and we have elected touse coefficients 2r up to 8r given by equations (64).

Conformal latitude and longitude difference from Gauss-

Schreiber coordinatesEquations (84) can be re-arranged and solved for tan

and tan as functions of the ratiosua

andva

giving

2 2

sintan

sinh cos

tan sinh cos

ua

v ua a

v ua a

(128)

Solution for latitude by Newton-Raphson iteration

To evaluate tan after obtaining tan from equation(128) consider equations (88) and (89) with thesubstitutions tant and tant

2 21 1t t t (129)

and 12

sinh tanh1

t

t

(130)

tant can be evaluated using the Newton-Raphsonmethod for the real roots of the equation 0f t givenin the form of an iterative equation

1

nn n

n

f tt t

f t (131)

where nt denotes the nth iterate and f t is given by

2 21 1f t t t t (132)

The derivative df t f tdt

is given by

2 2

2 2

2 2

1 11 1

1 1

tf t t t

t

(133)

where tant is fixed.

An initial value for 1t can be taken as 1 tant t andthe functions 1f t and 1f t evaluated from equations(130), (132) and (133). 2t is now computed from

equation (131) and this process repeated to obtain3 4, ,t t . This iterative process can be concluded when

the difference between 1nt and nt reaches an acceptablysmall value, and then the latitude is given by

11tan nt

.

This concludes the development of the TM projection.

TRANSFORMATIONS BETWEEN THE ELLIPSOID AND THETRANSVERSE MERCATOR (TM) PLANE

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

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

2. Compute the rectifying radius A from equation (41)3. Compute conformal latitude from equations (88)

and (89)4. Compute longitude difference 0 5. Compute the u,v Gauss-Schreiber coordinates from

equations (84)6. Compute the coefficients 2r from equations (62)7. Compute X,Y coordinates from equations (108)8. Compute q and p from equations (114)9. Compute scale factor m from equation (116)10. Compute grid convergence from equation (121)

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

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

2. Compute the rectifying radius A from equation (41)3. Compute the coefficients 2r from equations (64)

4. Compute the ratios ,u va a from equations (127)

5. Compute conformal latitude and longitude

difference from equations (128)6. Compute tant by Newton-Raphson iteration

using equations (131), (132) and (133)


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14

7. Compute latitude 1tan t and longitude

0 8. Compute the coefficients n from equations (62)9. Compute q and p from equations (114)10. Compute scale factor m from equation (116)11. Compute grid convergence from equation (121)

ACCURACY OF THE TRANSFORMATIONS

One of the authors (Karney, 2011) has comparedKrueger s series to order 8n (set out above) with an exactTM projection defined by Lee (1976) and shows thaterrors in positions computed from this series are less than5 nanometres anywhere within a distance of 4200 km ofthe central meridian (equivalent to 37.7 at theequator). So we can conclude that Kruger's series (toorder 8n ) is easily capable of micrometre precisionwithin 30 of a central meridian.

THE 'OTHER' TM PROJECTION

In Krueger s original work (Krueger 1912) of 172 pages(plus vii pages), Krueger develops the mapping equationsshown above in 22 pages, with a further 14 pages ofexamples of the forward and inverse transformations. Inthe next 38 pages Krueger develops and explains analternative approach: direct transformations from theellipsoid to the plane and from the plane to the ellipsoid.The remaining 100 pages are concerned with theintricacies of the geodesic projected on the TM plane,arc-to-chord, line scale factor, etc.

This alternative approach is outlined in the Appendix andfor the forward transformation [see equations (161)] theequations involve functions containing powers of the

longitude difference 2 3, , and derivativesdMd

,

2

2

d Md

,3

3 ,d Md

For the inverse transformation [see

equations (166) and (168)] the equations involve powers

of the X coordinate 2 3 4, , ,X X X and derivatives 11

dd

,

212

1

d

d

,3

13

1

,d

d

and 1d

dY

,

21

2

d

dY

,

31

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 identicalformulae, extending (slightly) Kruger's equations, andupdating the notation and formulation. These formulaeare regarded as the standard for transformations betweenthe ellipsoid and the TM projection. For example,GeoTrans (2010) uses Thomas' equations and GeoscienceAustralia defines Redfearn's equations as the method oftransformation between the Geocentric Datum ofAustralia (ellipsoid) and Map Grid Australia (transverseMercator) [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 bandsof longitude difference about the central meridian

(mm accuracy for 6 ) and(b) at large longitude differences 30 they can

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

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 75 but with increasing longitudedifferences from a central meridian. The X,Ycoordinates are computed using Kruegers series and canbe regarded as exact (at mm accuracy) and the columnheaded Readfearn , are the values obtained from

Redfearn's equations for the inverse transformation. Theerror is the distance on the ellipsoid between the given

, in the first column and the Redfearn , in thethird column.

The values in the table have been computed for theGRS80 ellipsoid ( 6378137 ma , 1 298.257222101f )with 0 1m

point Gauss-Krueger Redfearn error 75 6

X 173137.521Y 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 429237.683Y 8381563.943

75 00' 00.0023" 14 59' 59.8608" 1.120

75 20

X 567859.299Y 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 956892.903Y 8619555.491

75 00' 23.0237" 34 49' 57.6840" 4.9 km

Table 1

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-southextent (approx. 2600 km).

Points A and B represent two extremes if a centralmeridian is chosen as 0 45 W . A ( 70 N ,

22 30 W ) is a point furthest from the centralmeridian (approx. 850 km); and B ( 78 N 75 W )would have the greatest west longitude.

Table 2 shows the errors at these points for the GRS80ellipsoid with 0 1m for the inverse transformation usingRedfearn's equations.


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15

point Gauss-Krueger Redfearn errorA 70

22. 5X 842115.901Y 7926858.314

75 00' 00. 2049" 22 29' 53.9695" 64.282

B 78 -30

X -667590.239Y 8837145.459

78 00' 03.1880"-29 57' 59.2860" 784.799

Table 2

Y

A

B

Figure 6 Transverse Mercator projection of Greenlandgraticule interval 15, central meridian 0 45 W

CONCLUSION

We have provided here a reasonably complete derivationof the Karney-Krueger equations for the TM projectionthat allow micrometre accuracy in the forward andinverse mappings between the ellipsoid and plane. Andwe have provided some commentary on the 'other' TMequations in wide use in the geospatial community.These other equations offer only limited accuracy andshould be abandoned in favour of the equations (andmethods) we have outlined.

Our work is not original; indeed these equations weredeveloped by Krueger almost a century ago. But with theaid of computer algebra systems we have extendedKrueger 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 (TM) projection a muchmore useful 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) orThomas (1952) they are unnecessarily inaccurate.

NOMENCLATURE

This paper Krueger

k coefficients in series for

k coefficients in series for oblique latitude

grid convergence c eccentricity of ellipsoid e2 eccentricity of ellipsoid squared 2e 2 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 Aa 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 quantityf flattening of ellipsoid

Gaussian fundamental quantityg Gaussian fundamental quantityh scale factor along meridiank scale factor along parallelM meridian distance Xm scale factor m

0m central meridian scale factor 0m n 3rd flattening of ellipsoid nQ quadrant length of ellipsoid M s distance on ellipsoid s t tant

t tant u transverse Mercator coordinate V ellipsoid latitude function Qv transverse Mercator coordinate W ellipsoid latitude function


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APPENDIX

Reversion of a series

If we have an expression for a variable z as a series ofpowers or functions of another variable y then we may,by a reversion of the series, find an expression for y as

series of functions of z. 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 (134)

then Lagrange's theorem states that for any f

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 d F z f zdz

x dF z f z

n dz

(135)

As an example, consider the series for rectifying latitude

2 4 6sin 2 sin 4 sin 6d d d (136)

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

Comparing the variables in equations (136) and (134),z , y and 1x ; and if we choose f y y then f z z and 1f z . So equation (136) can beexpressed as

F (137)

and Lagrange's theorem gives

2

23

2

1

1

1

216

1

!

n nn

n

dF F

dd

Fd

dF

n d

(138)

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 (1685 1731) enables a function

( )f x near a point x a to be expressed from thevalues ( )f a and the successive derivatives of

( )f x evaluated at x a .

Taylor's polynomial may be expressed in thefollowing form

2

3

11

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

( )3!

( )1 !

nn

n

x af x f a x a f a f a

x af a

x af a R

n

(139)

where nR is the remainder after n terms and ( )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 !

k k

k

f af x x a

k (140)

where k

k 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 (44)]

2 4 6sin 2 sin 4 sin 6D D D (141)

and we wish to find expressions for sin2 , 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!

dd

dd

dd

giving

2

3 4

sin sin cos sin2

cos sin6 24

(142)

Replacing with 2 and with 2 in equation (142)and substituting from equation (141) gives an


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17

expression for sin2 . Using similar replacements andsubstitutions, expressions for sin4 , sin6 , etc. can bedeveloped.

Hyperbolic functions

The basic functions are the hyperbolic sine of x, denotedby sinh x , and the hyperbolic cosine of x denoted bycosh x ; they are defined as

e esinh

2e e

cosh2

x x

x x

x

x

(143)

Other hyperbolic functions are in terms of these

sinh 1tanh , coth

cosh tanh1 1

sech , cosechcosh sinh

xx x

x x

x xx x

(144)

The inverse hyperbolic function of sinh x is 1sinh x and

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

1tanh tanh x x ; both requiring 0x and as aconsequence of the definitions

1 2

1 2

1

sinh ln 1

cosh ln 1 1

1 1tanh ln 1 12 1

x x x x

x x x x

xx xx

(145)

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 sinln ln coscos sin

x xx

x x

x x xxx x

(146)

Also, replacing x with tan x in the definition of theinverse hyperbolic functions in equations (145) we have

1 2sinh tan ln tan 1 tan

1 sinln tan sec ln

cos

x x x

xx x

x

(147)

And equating1 sin

lncos

xx

from equations (146) and (147)

gives

11 14 2ln tan sinh tanx x (148)

Conformal mapping and complex functions

A theory due to Gauss states that a conformal mappingfrom the , datum surface to the X,Y projectionsurface can be represented by the complex expression

Y iX f i (149)

Providing that and are isometric parameters andthe complex function f i is analytic. 1i (the imaginary number), and the left-hand side ofequation (149) is a complex number consisting of a realand imaginary part. The right-hand-side of equation(149) is a complex function, i.e., a function of real andimaginary parameters and respectively. The

complex function f i is analytic if it iseverywhere 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

(150)

As an example, consider the Mercator projection of thesphere shown in Figure 2 where the conformal mappingfrom the sphere (datum surface) to the plane is given byequations (68) and using the isometric latitude given byequation (34) the mapping equations are

0

1 14 2ln tan

X R R

Y R R

(151)

These equations can be expressed as the complexequation

Z Y iX R i (152)

where Z is 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 (148) and equation (151) we maydefine the transverse Mercator projection by

1sinh tanZ Y iX R i (153)Now suppose we have another complex function

1tan sinhw u iv Z (154)representing a conformal transformation from the X,Yplane to the u,v plane.


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18

What are the functions u and v?

It turns out, after some algebra, u and v are of the sameform as equations (84)

1

12 2

tantan

cos

sinsinhtan cos

u R

v R

(155)

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

(156)

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

1 142 tan expw u iv R i (157)is also the transverse Mercator projection.

An alternative approach to developing a transverseMercator projection is to expand equation (149) as apower series.

Following Lauf (1983), consider a point P havingisometric coordinates , linked to an approximatelocation 0 0, by very small corrections , suchthat 0 and 0 ; equation (149)becomes

0 0

0 0

0

Y iX f i

f i

f i i

f z z f z

(158)

The complex function f z can be expanded by aTaylor series [see equation (140)]

21 2

0 0 0

33

0

2!

3!

zf z f z z f z f z

zf z

(159)

where 1 20 0, , etc.f z f z are first, second andhigher order derivatives of the function f z can beevaluated at 0z z . Choosing, as an approximatelocation, a point on the central meridian having the sameisometric latitude as P, then 0 (since 0

and 0 ) and (since 0 and0 0 ), hence 0 0 0z i andz i i .

The complex function f z can then be written as

21 2

33

2!

3!

f z f i

if i f f

i

f

(160) 1f , 2f , etc. are first, second and higher order

derivatives of the function f .

Substituting equation (160) into equation (158) andequating real and imaginary parts (noting that

2 3 41, , 1, etc.i i i i and f M ) gives3 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 MX

d d d d

d M d M d MY M d d d

(161)

In this alternative approach, the transformation from theplane to the ellipsoid is represented by the complexexpression

i F Y iX (162)

And similarly to before, the complex function F Y iX can be expanded as a power series giving

21 2

33

2!

3!

iXi F Y iXF Y F Y

iXF Y

(163)

When 0X , 0 ; but when 0X the point ,P becomes 1 1 , 0P , a point on the central

meridian having latitude 1 known as the foot-point

latitude. Now 1 is the isometric latitude for the foot-

point latitude and we have 1F Y

Substituting equation (163) into equation (162) andequating real and imaginary parts gives

2 4 62 4 61 1 1

1 2 4 6

3 5 73 5 71 1 1 1

4 5 7

2! 4! 6!

3! 5! 7!

d d dX X XdY dY dY

d d d dX X XX

dY dY dY dY

(164)

The first of equations (164) gives in terms of 1 but

we require in terms of 1 . Write the first of equations(164) as

1 (165)

where

2 4 62 4 61 1 1

2 4 62! 4! 6!d d dX X XdY dY dY

(166)


19/19

And latitude 1g g can be expanded asanother power series

21 2

1 1 1

33

1

2!

3!

g g g

g

(167)

Noting that 1 1g we may write the transformationas

2 42 41 1 11 2 4

1 1 1

3 5 73 5 71 1 1 1

4 5 7

2! 4!

3! 5! 7!

d d dd d d

d d d dX X XX

dY dY dY dY

(168)

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