Karsten Engsager
Senior advisor, Danish National Spacecenter, Danish Technical
University (ke@space.dtu.dk).
Working with classical geodesy in the areas: Reference systems,
Reference frames, Adjustment theory, Mappings,
Transformations, Interpolation.
A HIGHLY ACCURATE WORLD WIDE ALGORITHM FOR
THE TRANSVERSE MERCATOR MAPPING (ALMOST)
K. ENGSAGER1, K. PODER2
1
Danish National Space Center, Geodetic Department, Danish Technical University,
Juliane Mariesvej 30, DK-2100 Copenhagen E, Denmark. 2 DK-3650 Ølstykke, Denmark
Introduction
This presentation is in no way new or
genius. It solely presents the ideas and
developments from colleagues: König
und Weise (1951). “Map Projections”
Grafarend and Krumm (2006) appeared
shortly before this paper was written, but
some comments are included. The single
steps in performing the mapping from
geographical (geodetic) coordinates to
Transverse Mercator coordinates (and
reverse) are outlined but not proved. The
formulas given are valid at high/low
altitudes with only singularity at the
equator 90 degrees from the central
meridian.
It is our hope that this presentation will
convince the reader that the Transverse
Mercator projection is a useful tool for
world wide presentations.
The Transverse Mercator mapping is a
conformal mapping of the ellipsoid
coordinates to a plane. Any mapping
function satisfying the Cauchy-Riemann
differential formula is conformal. It is
therefore nearby to use series of complex
functions to describe the Transverse
Mercator projection. Some of the
complex functions are given as Clenshaw
sine summation where the coefficients are
elaborated on the third flattening n which
gives very efficient series. Our
contribution is to extend these series from
degree 4 to 7. Using series to degree 4 the
accuracy is 0.03 mm up to 4400 km from
the central meridian. Using the series to
higher degree will increase this limit.
a 1  n   a  b /2
Coordinate designation
for the scale furthermore
improves the efficiency of the series (b is
the minor axis).
-1


 Geodetic latitude
 Geodetic longitude
p
c
  2    Geodetic co - latitude
  r  ii  Complex geodetic latitude
pc
  2   r  ii  Complex geodetic co - latitude



c


e

 1  e cos p  2 

p




 ln  tan 

  2  1  e cos p  


e

 1  e cos p  2 





ln  tan

4
2  1  e cos p  


 

Isometricl atitude
  i  Complex Isometric coordinate s


 y  ix
 Complex (normalize d) mapping coordinate s
N  iE  Complex (metric) mapping coordinate s
u
Help functions
hypot( x, y )  x 2  y 2
atan 2r sin p, r cos p   p ; in proper quadrant
Clenshaw sine accumulation (simple or
complex) is used in calculation of the
series:
N
CS z    A2 sin  2 z 
 1
Details may be found in Poder and
Engsager (1998).
Derivatives of a mapping function
Ellipsoidal definition
a
 Equatorial radius
f
n
 Flattening
 f 2  f 
; Third flattening
F    1  2ncos 2   n
2
M    a1  n  1  n  F  
2
1
3
2
; Meridian curvature radius
a
1

1  14 n 2  641 n 4  256
n6
1 n
; Meridian arc unit

Q

The size of some coefficients to the series
expansions have been calculated using the
derived value of n in the Geodetic
Reference System 1980 (GRS80).
The derivatives of a mapping function can be
found as the complex differential with
modulus m and argument g of the mapping
function to the complex differential of
isometric latitude and longitude. The
simplification is possible because both systems
are orthogonal and isometric.
du  dy  ix
ds  r  d  id 
du
  m exp( ig ) 
ds
hypot y  , x  
m
r  
g  atan 2x  , y  
(Mapping fundamenta l form)
(Geodetic fundamenta l form)
(Complex scale)
(Scale)
(Meridian Convergenc e)
The Transverse Mercator mapping
The meridian arc unit Q is the mean length
of one radian of the meridian thus the
length of the meridian quadrant is:
 2    2 1 a n 1 
G
1
4
1
n 2  641 n 4  256
n6

The last term being 0.1 10-18 shows the
superiority of series expansion in powers
of the third flattening n against using
2
given
in
e 2  f 2  f   4n1  n  as
Grafarend and Krumm (2006) (8.40).
Using the mean axis
The Transverse Mercator is basically a
generalisation of the meridian to the plane:
Point P with geodetic coordinate s  ,   c ,
c  longitude of central meridian
Genuine origo : N, E   0,0  at   0 and   0
ELLIPSOID
Complex latitude : c  r  ii
dc cosc 

1  n 2  2 cos2c 
dc 1  n 2
Normalized transv. crd : u  y  ix

du M c 

d c
Q
SPHERE
 c   r  i i

U  Y  iX
Mapping sequence for the Transverse
Mercator mapping
The mapping for the Transverse Mercator
is split up in several conformal mappings
during the development of the mapping.
This is done to ensure as efficient
expansion of series and a simple
development strategy. Finally some of the
mappings are united again ending up with
only three mappings:
1. Ellipsoidal coordinates to the
Soldner Sphere
2. Soldner Sphere to Complex
Gaussian coordinates
3. Complex Gaussian coordinates to
Transverse Mercator coordinates
Mapping Ellipsoid  Soldner Sphere
The Soldner Sphere is a pure parameter
sphere of unity size. The Geodetic
coordinates of the ellipsoid is mapped on
the sphere with the simple and virtually
unique mapping equations as shown below.
Strictly speaking we should account for the
Riemann leaves arising from the
periodicity of the complex functions.
Mapping equations :
 

cos d  id  
cos 


m ;
N  cos d  id   N  cos 
Cauchy - Riemann :
 

1
 
g0



0


Function :
Direct mapping
┌────────────────────────┐
│
φ,λ: Geodetic coord
│
└────────────┬───────────┘
(1) 
│
 (2)
┌────────────┴───────────┐
│
Φ,Λ: Soldner Sphere
│
└────────────┬───────────┘
(3) 
│
 (4)
┌────────────┴───────────┐
│ Y,X: Complex Gaussian coord. │
└────────────┬───────────┘
(5) 
│
 (6)
┌────────────┴───────────┐
│
N,E: Transversal coord.
│
└────────────────────────┘
Inverse mapping
 1  e cos p  

exp     tan  p 2  
 1  e cos p  
e2
 tan P 2 
This may be reformulated into a Clenshaw
sine summation:
7
     e2 sin 2 
1

Array e7  e2 , e4 , e6 , e8 , e10 , e12 , e14 
e2  2n  23 n 2  43 n3  82
n 4  32
n5  4642
n 6  8384
n7
45
45
4725
4725
e4 
2288 7
 53 n 2  16
n3  139 n 4  904
n5  1522
n 6  1575
n
15
315
945
e6 
26 3
44644 7
 15
n  34
n 4  85 n5  12686
n 6  14175
n
21
2835
e8 
24832 6
 1237
n 4  125 n5  14175
n  1077964
n7
630
155925
e10 
 734
n5  109598
n 6  1040
n7
315
31185
567
e12 
444337 6
941912 7
 155925
n  184275
n
e14 
 2405834
n7
675675
(1)
Geodetic coordinates  Soldner
Sphere
-19
The last coefficient is e14 = -1.3 10
Limiting the series to n4 gives the last
coefficient in e8 = 1.6 10-11
7
     G2 sin 2 
1

Array G7  G2 , G4 , G6 , G8 , G10 , G12 , G14 
G2  2n  23 n 2  2n3  116
n4
45

G4 
2704
315
G6 
 73 n 2  85 n3 
227
45
n4 
26
45
n5 
n5 
56 3
 15
n  136
n 4  1262
n5 
35
105
G8 
G10 

4279
630
n 
4

2854
675
2323
945
 31256
n7
1575
n6
73814
2835
n6
98738 7
 14175
n
n 
11763988
155925
332
35
n 
4174
315
n5  144838
n6 
6237
G12 
5

399572
14175
601676
22275
6
n 
G14 
(2) Soldner
coordinates
 16822
n7
4725
n6
6
n
2046082
31185
7
n7
115444544
2027025
n
7
 38341552
n7
675675
Sphere

Geodetic
The last coefficient is G14 = 2.1 10-18
Limiting the series to n4 gives the last
coefficient in G8 = 5.4 10-11
Soldner Sphere 
Complex Gaussian coordinates
The Complex Gaussian coordinates gives a
transverse mapping, where the central
meridian is mapped with unity scale i.e. it
behaves as if the central meridian is the
“virtual equator” and the normal to the
central meridian plane passing through the
centre of the sphere is the “virtual rotation
axis”. A point with the Gaussian
coordinates (Φ, Λ) should be mapped to
the Complex Gaussian coordinates Φc = Φr
+ iΦi.
The figure shows the latitudes and the
longitude and the auxiliary parameter t,
which in fact is the virtual latitude, while
Φr is the virtual longitude in a Mercator
mapping, where the central meridian acts
as the virtual equator and the virtual poles
are found in the real equator 90 degrees
from the central meridian.
The singularity at the (true) Poles is
handled by using the atan2 function.
Input : ,  ; on the soldner sphere
Y   r  atan 2sin  , cos cos 
t  atan 2cos sin  , hypot sin  , cos cos 
X   i  ln tan  4  t 2
U  Y  iX   r   i ; Complex Gaussian Coordinate s
(3) Soldner Sphere  Complex Gaussian
coordinates
The parameter t (atan2(…)) is crucial for
the accuracy of the entire Transverse
Mercator mapping. For geodetic purposes
up with series expantion to degree 4 t up to
40° is acceptable giving 0.03 mm in
accuracy. For cartographic purposes it is
possible to go close 90° and still getting an
accuracy of one meter.
U  Y  iX   r  i i
; Complex Gaussian Coordinate s
t  2 atan exp X    2
  atan 2sin t , cost cosY 
  atan 2sin Y cost , hypot sin t , cost cosY 
(4) Complex Gaussian coordinates
Soldner Sphere

Complex Gaussian coordinates

Transverse Coordinates
The Complex Gaussian coordinates are
mapped to Transverse Coordinates in two
steps via the Normalized Transverse
Coordinates.
The
two
Clenshaw
summations have been reformulated to one
which is presented below. Interested
readers may find details in Poder and
Engsager
(1998).
The
normalized
transversal coordinates are then scaled to
Transversal Mercator coordinates.
input : Gaussian complex coordinate s :
U  Y  iX   r  i i
Implementation details
normalized transvers al coordinate s :
All coordinates except the Transversal
Mercator coordinates may be considered be
angular units in radians. The Meridian arc
unit Q is used to transform the Transverse
Mercator coordinates into angular units
called Normalized Transverse Mercator
coordinates.
7
u  U   U 2 κ sin 2κU 
1
output : metric scaled transvers al coordinate s :
N  iE  uQ
Array U [7]  U 2 , U 4 , U 6 , U 8 , U10 , U12 , U14 
7891 6
72161 7
41
U 2   12 n  23 n 2  165 n3  180
n 4  127
n5  37800
n  387072
n
288
U4 
557
281 5
1983433 6
 13
n 2  53 n3  1440
n 4  630
n  1935360
n  13769
n7
48
28800
U6 
61
103 4
167603 6
 240
n3  140
n  15061
n5  181440
n  67102379
n7
26880
29030400
U8 
49561 4
6601661 6
 161280
n  179
n5  7257600
n  97445
n7
168
49896
U10 
34729 5
3418889 6
 80640
n  1995840
n  14644087
n7
9123840
U12 
212378941 6
30705481 7
 319334400
n  10378368
n
U14 
 1522256789
n7
1383782400
(5) Complex Gaussian coordinates
Transversal Mercator coordinates

The last coefficient is U14 = 4.1 10-20
Limiting the series to n4 gives the last
coefficient in U8 = 2.4 10-12
NB.: Clenshaw Complex sine summation
is used in (5) and (6).
input : metric scaled transvers al coordinate s :
N, E
normalized transvers al coordinate s :
u  N  iE  / Q
output : Gaussian complex coordinate s :
7
U  u   u2 κ sin 2κu 
1
Array u[7]  u2 , u4 , u6 , u8 , u10 , u12 , u14 
81 5
96199 6
5406467 7
1
u2   12 n  23 n 2  37
n3  360
n 4  512
n  604800
n  38707200
n
96
u4 
u6 
u8 
u10 
437 4
46 5
51841
- 481 n 2  151 n3  1440
n  105
n  1118711
n 6  1209600
n7
3870720
17
37 4
- 480
n3  840
n 
209
4480
5569 6
9261899 7
n5  90720
n  58060800
n
4397
830251 6
466511 7
11 5
- 161280
n 4  504
n  7257600
n  294800
n
4583
108847 6
8005831 7
 161280
n5  3991680
n  63866880
n
u12 
20648693 6
16363163 7
 638668800
n  518918400
n
u14 
219941297 7
 5535129600
n
(6) Transversal Mercator coordinates 
Complex Gaussian coordinates
The last coefficient is u14 = -1.5 10-21
Limiting the series to n4 gives the last
coefficient in u8 = -2.2 10-13
Using the scaled meridian arc unit Qm
instead introduce the scale in a simple way:
Scale on central meridian : m0
Qm  Q  m 0
Accuracy check
Taking the difference between the input
coordinates and the result of the
backward_trf(forward_trf(input
coordinates)) the accuracy may be checked
against the accuracy required for
transformation. In the figure below is the
limit 0.03 mm shown at series to degree 4
and 5. This limit may be controlled in (3)
by fabs  0.810 and in (4) by
fabsU   0.810 as a very coarse limit
(i.e. E  5100 km ). The limit is in fact a
function of the latitude and longitude. For
degree 5 the limit will be 1.144 (i.e.
E  7300 km ).
An accuracy limit of 30 m will give a
limiting
factor
of
2.290
(i.e.
E  14500 km ).
The singularity points may be controlled
by the limiting factor above or a coarser
one.
It has been demonstrated that it is possible
in a fairly simple sequence of very efficient
mappings to transform the geographical
coordinates to Transversal Mercator
Coordinates.
It has been pointed out that there is a
substantial difference in expanding series
in powers of the third flattening n against
in powers of the eccentricity e2. Due to the
relation between the two parameters any
series expansion may be reformulated to
the other.
UTM zone 32 coordinates : E coordinate is
given +E0 (=500km)
Performance
The mapping implemented to 4th degree
run on a PC with Intel Pentium 4 cpu 2.66
GHz
makes
more
than
224000
transformations pr. second in accuracy
checking
mode:
backward_trf(forward_trf((input
coordinates)).
An implementation to 5th degree decreases
the throughput to 222000. When the
mapping is run trough our general
transformation manager the throughput
decreases by further 10%.
More convolutions
The formulas presented are valid for the
latitude      
and for the


longitude  2       2   , where δ
may be related to the control setup above.
When the input latitude goes beyond the
limits it is forced to the interval by
adding/subtracting κ2π to the latitude and
the northing N is either increased or
decreased by κ4Qm . The reverse situation
is handled in a similar way.
Conclusion
To illustrate the accuracy the size of some
coefficients depending on the third
flattening n have been calculated to the
derived value of n in the Geodetic
Reference System 1980 (GRS80).
Implementation details have also been
given.
The C-code for the Transversal Mercator
mapping is free and may be found on the
homepage
spacecenter.dk
under
Research/Geodesy/Mapping (the URL is
not ready at writing time) or will be
forwarded
on
request
to
ke@spacecenter.dk
Appendix
Scaled meridian arc length from equator
to the latitude
7


G    Qm     p 2 sin 2 
 1


4
6
57
15 2
135
p 2   32 n  169 n 3  323 n 5  2048
n 7 p 4   16
n  15
32 n  2048 n
p6 
3
7
105 5
105
 35
p8 
48 n  256 n  2048 n
6
315 4
 512
n  189
512 n
p10 
693 5
693 7
 1280
n  2048
n p12 
6
 1001
2048 n
p14 
6435 7
 14336
n
The last coefficient is p14 = 1.7 10-20
Limiting the series to n4 gives the last
coefficient in p8 = 4.9 10-12
Normalized Transversa l coordinate s :
u  ( N  iE )Qm1  y  ix
  latitude of the point (N, E)
A  G  Qm1 ; Normalized scaled meridian length
Latitude at the normalized arc length
,    coordinate s on the Soldner sphere of (N, E)
A  G / Qm
MERIDIAN CONVERGENC E
7
  A  A   q2 sin 2A

 1
q2   n 
3
2
q6 
27
32
n 
3
269
512
n 
5
6607
24576
n
7
q4   n 
417 5
 151
n3  128
n  87963
n 7 q8 
96
2048
q10 
q14 

8011
2560
n5  69119
n 7 q12 
6144
21
16
2
55
32
n 
4
6759
4096
n
6
 1097
n 4  15543
n6
512
2560

293393
61440
n6
 6459601
n7
860160

4
  atan 2sin  y  tanh x , cos y      C2 cos2u 
  1

4


 atan 2sin  sin  , cos      C2 cos2u 
  1

LOCAL SCALE
  m0 hypot cos y , sinh x  cos A
  4

 4
 exp    C2 cos2u    cos2A


1


1




The last coefficient is q14 = 2.8 10-19
Limiting the series to n4 gives the last
coefficient in q8 = 1.7 10-11
 m0 1  cos 2  sin 2  
Meridian convergence and local scale
  4

 4
 exp    C2 cos2u    cos2A
  1
   1



1
4
3 2
3 3
55
C 2  + 12 n  8 n  32 n  1152 n
3
5 2
C 4  + 16 n  13 n 
C6 
C8 
83 3
173 4
+ 480
n  899 n
2421
772
n
4
4
+ 1531
336 n
The last coefficient is C8 = 3.6 10-11 and in
the Clenshaw summation is used both
cosine and complex cosine.
References
Bomford, G (1962): Geodesy, second edition, Oxford
Bougayevskiy Lev M, Snyder John P (1995) Map Projections. A reference Manual. Taylor
and Francis, London
Grarafend E.W., Krumm F.W. (2006): Map Projections. Springer, Berlin/Heidelberg/New
York
Krüger L (1912): Konforme Abbildung des Erdellipsoids in der Ebene. Veröffentlichung
des Königlisch Preuszischen Geodätischen Institutes. Neue Folge No. 52, Potsdam
König R, Weise K.H. (1951): Mathematische Grundlagen der Höheren Geodäsie und
Kartographie, Erster Band, Springer, Berlin/Göttingen/Heidelberg
Poder Knud, Engsager Karsten (1998): Some Conformal Mappings and Transformations
for Geodesy and Topographic Cartography. National Survey and Cadastre, Denmark,
Publications 4. series vol. 6, Copenhagen
Richardus Peter, Adler Ron K (1972): Map projections, North Holland, Amsterdam
Journal of Geodesy (2004): The Geodesist’s Handbook 2004, Springer