CONVERSIONS BETWEEN GEOGRAPHICAL AND
TRANSVERSE MERCATOR (UTM, GAUSS.KRUGER)
GRID COORDINATES
J.
VARGA
Institute of Geodesy, Technical University, Budapest, H-1521
(Received: March 2, 1981.)
Presented by Prof. Dr. F. Sarkozy, Director
To now, mutual converSIOns between Univel"sal Transverse 1Ylercato:r
(UTM) grid coordinates and geographical coordinates applied computers or
tables. Programmable desk and pocket calculators revolutionized calculations not only in the office but also on the terrain. A single magnetic card may
replace volumes of tahles. A programmable calculator, fed with correct
data input provides correct results as against the multitude of sources of error,
inherent in the use of tahles. Another significant advantage of programmed
calculation is a lower qualified operator needed to ohtain error-free results, at a
reduced running time.
The mutual conversion hetween UTM and geographical coordinates hy
means of a minicomputer will be presented helow.
The chosen computer methods require low data and program storage
capacity fit to a smaller programmable pocket calculator. The prepared programs with the indicated constants provide for a mean error of ± 1 mm in the
grid coordinate and : 0.0002" in the geographical coordinate, all along the ellipsoid.
Notations
a = semi-major axis of the ellipsoid;
b = semi-minor axis of the ellipsoid;
c = a 2jb, polar radius of curvature;
e-9 - a
"
e'2
=
a
2
2
-
a2
2
-
p
b , square 0 f t h e f'ITst eccentrICIty;
..
b2
2
e
= -l-~'
- . square
••
of the second eccentrICIty;
.
fP = latitude of point;
fPl = latitude of foot of perpendicular to central meridian from point;
A = longitude of point;
/10 = longitude of origin (the central meridian) of the projection;
9
J. VARGA
), = difference of longitude from the central meridian;
), = A - Ao in the Eastern Hemisphere;
Ao - A in the Western Hemisphere;
),
t
= tg q);
'rJ2
=
e'2 cos 2 q);
V = V1 + 'rJ2 = 1f 1 + e'2 cos 2 q) ;
M = c/T/"3, radius of curvature in the meridian;
M = cl V, radius of curvature in the prime vertical;
S = true meridional distance on the ellipsoid from the equator;
ko = central scale factor;
Xo= false northing;
Yo= false easting;
x = grid distance from the equator;
y = grid distance from the central meridian;
X = x + Xo = grid northing;
Y = Y
Y o = grid easting.
Relationships between grid and geographical coordinates in the transversal conformal Mercator projections of the ellipsoid are:
Geographical to projection:
x
= (S + A z ),2 + A4 ),4) ko,
Y = (AI)'
+ A3),3 + A 5).5) ko,
X= x +Xo,
Y=y
(1)
+ Yo'
Projection to geographical:
x=
X-Xo
y=
,
ko
Y- Y o
ko
,
(2)
After substitutions and transformations, (1) and (2) take the following,
efficiently programmable form: [2]
x = ko {S + N }.2 sin <P cos <P[1 +
2
~ cos 2 <P (5 _
12
+ ~ cos4 <P(61 360
58 t 2
+ t4 )]},
t2 + 9'YJ2 + 41]2) +
(3)
GRID COORDINATES
y = koi..N cos (P [1
+ ~ COS" (P (5 120
(P =
(PI -
18 t2
cos 2(P(1 -
+ t" + 147]2 -
y2 ft [1 -- ~ (5
2 MINI
12Nf
+
i.. =
+~
y4
360 Ni
y
(1 - L (1
NI COS (PI
6 Ni
(61
241
t2
+ 7]2) +
58 t27]2)] ,
+ 3tt + 7]t -
9ti7]t)
(4)
+
+ 90 tt + 45 ti)l '
+ 2 ti + 7]i) +
y4
120 Ni
(5
(5)
+ 28 ti + 24 tn] .
(6)
The angle values have to be substituted in radians into formulae (3) to
(6) to avoid calculation with powers ofthe analytical angle unit e. S and (P~ can
be determined as follows. Subscripts 1 in (5) and (6) indicate that for calculating
the given quantity (PI has to be substituted into the formula.
1. Determination of S (trne meridional distance on the ellipsoid)
For determining S and (PI' VINCENTY suggested a useful direct method in
[7]. For each ellipsoid 7 constant values have to be calculated and stored.
Constants will be derived in a similar way, some of them are, however, simple
to express from the others and therefore it is needless to store them separately.
LEE [3] calculates the plane coordinates by lJI using the isometric latitude
to obtain symmetrical terms for x andy, easy to program exempt from S. Disadvantage of this method is that series have to be taken into consideratian to
a high order number likely to inhibit use of a minicomputer.
S is given by integrals:
(7)
or
<1>
<1>
f
~
S=jMdtP=c
(1
o
0
dtP
+ e'2 cos2 tP)3/2
(8)
showing the arc length S of the meridian to be proportional to the area
between curves of functions
w = (1 9*
e2 sin 2 tP) -312,
J. VARGA
242
or
OJ
=
(1
and the coordinate axis OJ = o. It is also seen that for determining S, storage
of two constants for each ellipsoid may be enough; in one case these are a and
e2, in the other c and e'2. Functions OJ = f(tP) are continuous, mildly ascending
in the interval 0 < tP ::s;: n/2 fit to numerical integration by dividing interval
o to tP into relatively few parts to yield the area below the curves at the desired
accuracy.
Let us write Simpson's parabola formula for S divided to even n parts
according to (8):
where h = tP/n and OJi is the substitution value ofthe function at the i-th place.
n = f (tP) namely with increasing tP, the interval has to be divided into ever
more parts. This method of calculating S with a pocket calculator takes much
time, therefore the use of a faster desk computer is recommended. For n
30,
generally S is obtained with mm accuracy even near the poles.
In most practical works, calculations refer to the area of a single continent
or country. In this case it is expedient to calculate So once accurately for the
limit parallel circle if>o outside the boundary of the area, to store it in the memory as a constant and to divide only the interval tPu to tP into n parts.
n can be determined empirically in each case.
This latter method takes not much time even with a pocket calculator
because n ::s;: 10. The great advantage of this method is that only c and e'2 have
to be stored, these constants being also necessary for solving Eqs (3) to (6).
For determining S, a faster method but needing an increased storage
capacity will be shown below.
Table 1
Ellipsoid
Everest
Bessel
Qarke/66
Clarke/80
Hayford
Krassowsky
IUGG/1967
c(m)
6398547.993
6 398 786.849
6399902.551
6400057.735
6 399 936.608
6399698.902
6 399 617.430
e'!
·10-'
6.682201 989
6.719218798
6.814784832
6.850 116 152
6.768170197
6.738525415
6.739725198
6366680.291
6366742.521
6367399.690
6 367 386.645
6 367 654.500
6 367 558.497
6367471.748
GRID COORDINATES
243
Series expanding (8) and integrating term by term:
m - -B' c sm
. 2'"
C' . 4'"
D' C SIll
. 6'"
S = A ' c"...
~,- c SIn ~ - ~
2 4 6
I
A' = 1 - ~e'2
4
B'=
+
3 '2
,-e
I
4
+ ...
(9)
45 e'4 _ 175 e'6.L 11025 e'8_
64
256
I
16384
15e'4 -I -525'6
- e - -2205
- - e '8 -I ...
16
I
512
2048
I
C' =
15'4
105'6
e ---e
64
256
D'=
-35
- e '"6
2205 '8
,---e
I
4096
'8
-315
--e
,
2048
I
512
(10)
Reducing (9):
S
= atP -
fJsin 2 tf>
+ y sin 4 tf> -
<5
sin 6 tf>
... .,
(11)
where
et =
A' c,
fJ
B'
=-c,
2
C'
y=-c,
4
D'
o- =--c.
6
a, fJ, y, <5 are constants, to be determined separately for each ellipsoid.
The values for the most frequently used ellipsoids are found in Table 1.
2. Determination of tPl
Fig. 1 shows that coordinate x of foot point PIon the central meridian
at point P of latitude tf> equals SI' permitting (11) to he written as:
P
15900.693 22
15 988.638 53
16216.94389
16 300.700 72
16107.03468
16036.48028
16039.10744
y
16.5466
16.72995
17.2094
17.3876
16.9762
16.8281
16.8338
0.021
0.022
0.023
0.023
0.022
0.022
0.022
·10-'
PI::.
:c-/,:::
. 10-'
·10-'
2.497486
2.511275
2.546871
2.560030
2.529508
2.518466
2.518913
2.599
2.628
2.703
2.731
2.666
2.643
2.644
3.4
3.5
3.6
3.6
3.5
3.5
3.5
~/::.
J. VARGA
244
Fig. 1
Expressing lP 1 at a good approximation:
m = -X
'PI
cc
I
T
-
f3 srn
• 2m
r·SIn 4m'PI T - ~
'PI - I
cc
cc
cc
•
SIn
6m
'PI
+ ...
(12)
As lPl occurs on hoth sides of the sign of equality, it can he determined hy
iteration.
First a fair approximation value of lPl is determined from ($1) = ~.
cc
Substituting lPl into the right-hand side of (12) yields a closer approximation
for lP1 • The numher of re-substitutions depends on the needed accuracy hut it is
never more than three.
Disadvantage of the method is to need iteration, its advantage is, however, not to need storage of further constants that are easy to express from constants involved in calculating S.
In case of a computer with an increased storage capacity, also values of
f3/cc, r/cc, a/cc can he stored, of them those referring to ellipsoids most frequently
used are as well included in Table 1.
3. Programs
The programs were developed for a Hewlett-Packard 67/97 calculator.
For converting geographical to grid coordinates Eqs (3), (4) and (ll); for
converting grid to geographical coordinates Eqs (5), (6) and (12) were used.
The programs can he accommodated on a magnetic card, and the constants
helonging to an ellipsoid are stored on one side of a card.
The prepared programs are shown hy Program Listings 1 and 2, their use
hy Users' Instructions 1 and 2. The memory content is identical in hoth programs; this is illustrated in Table "Registers". a/cc was not separately stored
GRID COORDINATES
245
but built in into the program as a constant. From Table 1 it is seen that its value
may be taken for all the ellipsoids uniformly as 3.5 . 10-9 • Similarly also ~
might have been built into the program with a value of 0.022.
The elaborated examples refer to the Universal Transverse Mercator grid
system, with the following properties to be reckoned with in our calculations.
1. Ellipsoid: Hayford. The necessary constants are found in Table l.
2. False northing, Xo = 0 m (10000000 for Southern Hemisphere).
3. False easting, Y o = 500 000 m.
4. Scale factor at the central meridian, ko = 0.9996.
As in the former case, the stored matter of the memories is recorded in
the Data Sheet, supposing work on the Northern Hemisphere.
The programs can of course be used for any ellipsoid, even for tangential
or intersecting position of the image plane. Remind that in tangential position
ko = 1.
Data Sheet
6.367654500
1.610703468
1.697620000
2.200000000
2.529508000
2.666000000
9.996000000
0.000000000
0.000000000
0.000000000
5.000000000
0.000000000
0.000000000
6.399936608
6.768170197
0.000000000
+ 06
+ 04
+ 01
-
02
03
06
01
+ 00
+ 00
+ 00
+ 05
00
00
+ 06
- 03
00
+
+
+
0
1
2
3
4
5
6
7
8
9
A
B
C
D
E
I
Constants of ellipsoids missing from Table 1 are determined from Eqs (10)
and (ll).
The t'wo programs confined suit conversion between belts, using the following method. Grid coordinates for one belt are converted to geographical
coordinates, then, ·with the geographical longitude of the central meridian of
the adjacent belt as input, grid coordinates ofthe point on the adjacent projection belt are calculated with the other program.
246
J. VARGA
User Instructions
Geographical to projection
STEP
1
INSTRUCTIONS
Load sides 1 and 2
program
2
Load
3
KEYS
Ao
CJCJ
CJCJ
CJCJ
c=Jc=J
CJCJ
CJCJ
LDCJ
CJCJ
CJCJ
IENT.t1 CJ
CJ
CJr=:J
[TICJ
of
card
of d·ata
side 1
(Choose an
first only
of the central meridian.
Longitude
degree,
cord
at
ellipsoid "
if
needed: deg., min., sec.
4
5
Latitude, deg., min., sec.
q,
Longitude
A
6
Northern Hemisphere, East
-11-
7
UTM
grid
from Greenwich
For
0
new
-u-
u::J [:=J
-11-
West
-11-
CDCJ
C[JCJ
CJCJ
CJCJ
coordinates
or 3,
or
case
DATA/UNITS
East
~CJ
-11-
8
-
West
Southern Hemisphere,
-
OUTPUT
INPUT
DATA/UNITS
go
X (m)
Y (m)
CJCJ
CJCJ
CJCJ
to step 2,
4.
EXAMPLE
STEP
INSTRUCTION
1
Load program
2
·Load data
3
Longitude of
INPUT
9'
the cent. meridian
Latitude
Longitude
6
N. Hemisphere, East.
7
UTM\ X.
8
UTM, Y.
OUTPUT
OAT A fUNITS
CJCJ
CJCJ
CJCJ
CJCJ.
(Hayford 1
4
5
KEYS
DATA/UNITS
47° 15' 38,4257'
60 27' 49,7791 u
CD I
,
I
CJCJ
IENTlI CJ
I
CJ
I
o;::J CJ
CJCJ
CE:ill CJ
CJ
CJCJ
CJI
I
User Instructions 1
5237353\489 m
308121,657 m
!
GRID COORDINATES
247
User Instructions
Projection to geographical
1
INPUT
DATA/UNITS
INSTRUCTIONS
STEP
load
sides 1 and 2
of
program card
2
3
load side 1 data card
(Choose an ellipsoid), at first on(y
H,
needed: deg" min., sec.
if
ccord: :-:ates
UTM g,id
-h-
0
6
Ao
Longitude of the centraL meridian
degree,
Northern
Hemisp"r-,ere, =-'::15:
7rom G.. eeD,.... ic·r
Southern Hemisphere,
::.::15:
-Il-
West
-li-
7
latitude, deg" min./ sec.
8
Longitude,
c:::=J c:::=J
c:::=J c:::=J
c:::=J c:::=J
c::J c::J
c:::=J c:::=J
CJ c:::=J
CD c:::=J
c:::=J c=:::J
c:::=Ji
I
X (m)
lENT. t :
Y (m)
c:::=J
CJ c:::J
I
I
I
o::::J c:::J
-1.-
V/est
OUTPUT
DATA/UNITS
KEYS
I
CL] I
I
L:Q:J c:::J
j
c:::J c:::=J
c:::J c:::J
CB:ill c:::=J
c:::=J c:::=J
c:::J c:::=J
c:::=J c:::=J
I
-11-
For a new case go to step 2,
or 3, or 4,
~
i
<P
A
EXAMPLE
STEP
INSTRUCTION
1
load program
2
load data
3
lonqitude of the
4
UTM, X
5
UTM, Y
6
N. Hemisphere, Ea sI.
7
latitude
8
Longitu-ie
INPUT
I
DATA / UNITS
KEYS
OUTPUT
DATA/UNITS
c:::J c:::=J
(Hayford)
9°
cent. meridian.
5237353,489 m
308121
I
657 m
c:::J
COCJ
c:::=J c:::=J
lENT. t I c:::=J
c:::=J CJ
u:::::J CJ
c:::=J c:::=J
IB:TIJ c:::=J
CJ c:::=J
c:::=J CJ I
c:::=J c:::=J
c:::=J c:::J I
User Instructions 2
47° 15' 38,4257'
6° 27' 49,7791'
,
J. VARGA
248
Program Listing I
001
002
003
004
005
006
007
008
009
010
011
012
013
014
015
016
017
018
019
020
021
022
023
024
025
026
027
028
029
030
031
032
033
034
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036
037
038
039
040
041
042
043
044
045
046
047
048
049
050
051
052
053
054
055
056
*LBLB
SFO
GTOA
*LBLD
SFO
"LBLC
SFl
*LBLA
RAD
P:;tS
HMS--RCLC
FO?
CHS
D---R
STOO
X:;tY
HMS--D---R
ST02
SIN
ST03
LSTX
COS
ST04
X2
ST06
RCL4
X2
RCLE
x
ST05
, 1
,
yX
RCLD
X:;tY
RCL4
x
RCLO
x
ST07
1
RCL6
RCL5
+
RCL4
X2
RCLO
X2
x
ST08
057
058
059
060
061
062
063
064
065
066
067
068
069
070
071
072
073
074
075
076
077
078
079
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081
082
083
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085
086
087
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094
095
096
097
098
099
100
101
102
103
104
105
106
107
108
109
110
111
112
6
ST09
x
1
+
5
RCL6
1
8
x
RCL6
X2
+
RCLS
1
4
x
+
RCL5
RCL6
x
5
8
x
RCL8
X2
1
2
0
ST08
x
..L
I
RCL7
x
STOI
5
RCL6
RCL5
9
x
+
RCL5
x
X2
4
+
RCL9
2
x
1
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
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148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
,,
6
1
RCL6
5
8
x
RCL6
X2
+
RCL8
3
x
+
RCL7
RCLO
x
RCL3
x
2
x
RCL2
P:;tS
ST07
RCLO
X
+
RCL7
x
2
SIN
RCL1
x
RCL7
4
x
SIN
RCL2
x
+
RCL7
x
6
SIN
RCL3
x
RCL6
x
F1?
CHS
RCLB
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
+
CFO
CFl
DEG
RjS
RCLI
RCL6
x
RCLA
+
RTN
-LBLE
HMS--STOC
R~
RTN
RjS
GRID COORDINATES
249
Program Listing 2
001
002
003
004
005
006
007
008
009
010
011
012
013
014
015
016
017
018
019
020
021
022
023
024
025
026
027
028
029
030
031
032
033
034
035
036
037
038
039
040
041
042
043
044
045
046
047
048
049
050
051
052
053
054
055
056
*LBLB
SFO
GTOA
*LBLD
SFO
*LBLC
SFl
*LBLA
RAD
RCLA
RCL6
FO?
CHS
ST07
X=y
RCLB
RCL6
ABS
RCLO
ST08
ST09
3
STOI
*LBLO
RCL8
RCL9
2
x
SIN
RCL4
X
+
RCL9
4
x
SIN
RCL5
X
RCL9
6
><
SIN
3
5
EEX
9
CHS
x
+
057
058
059
060
061
062
063
064
065
066
067
068
069
070
071
072
073
074
075
076
077
078
079
080
081
082
083
084
085
086
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090
091
092
093
094
095
096
097
098
099
100
101
102
103
104
105
106
107
108
109
110
III
112
ST09
DSZI
GTOO
STOI
RCL7
p~S
STOO
RCLD
RCLI
COS
X2
RCLE
x
STOl
1
+yX
ST02
ST03
X2
ST04
X2
ST06
3
6
0
RCLI
TA1~
X2
ST05
9
0
><
6
I
+
RCL5
x
X2
4
5
+x
RCL5
3
x
,
5
...L
RCLl
+
RCLl
RCL5
x
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
x
9
RCL4
x
+
1
2
1
RCL5
y:x
x
RCLO
X2
x
RCL2
X2
X2
x
RCLD
X2
2
x
CHS
RCLI
I
T
R ..... D
..... ai\IS
CFO
CFl
RIS
RCL5
2
8
x
5
,,
RCL5
X2
2
4
,x,
RCL6
x
1
2
0
RCL5
2
x
RCLl
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
-L
I
1
-L
I
RCL4
x
6
1
I
T
RCLO
x
RCL3
RCLI
COS
R ..... D
RCLC
+
..... HMS
DEG
P=S
RTN
*LBLE
HMS .....
STOC
Rj.
RTN
RIS
J. VARGA.
250
Registers
0
,
d.
so
A
;3
P
a-
12
51
la
Ye
er
53
52
1
4
(3/«
$4
5
6
6!c£.
ka
56
SS
7
a
9
57
sa
59
1
0
le
x0
lE
c
.../,!..IJ
e
I'
Z
4. Transformation of the programs for lower accuracy requirements
The enclosed programs can also be used for the highest accuracy requirements on the entire ellipsoid; therefore they exceed that needed for lower accuracy requirements. If a mean error of =1 cm is made up with, steps 72 to 83
and 99 to 107 in program 1 and steps 108 to 115 in program 2 may be omitted.
Working ,·tithin a domain of 10° of the geographical latitude and having
So calculated up to a boundary parallel circle q>o, (3) to (5) may be replaced by
simpler expressions:
y = k o )' N cos q>
[1 + A;
cos 2 q> (1
where:
-
<PI = <Po
+ --Vo Xoo
No
3x60,
---'1}0 to
2 N5
-j-
xg
90
--'1}0 (to --I)! ... ,
<
2N~
or
<PI
= <P
.
},'"
+ -} ,V22sm
<P cos <P + -
2
24
.
2
sm <P cos 3 <P (5 - t )
<
!
...
These formulae can be composed to a program uniting the two programs
above so that a single input on magnetic card before start suffices and thereafter also conversion between the projectional belts to and from can be carried
out.
Similar simple formulae are those by FIELD [12]. Also nomograms al'e
made for non-programmable pocket calculators facilitating conversions such
as those by REITHOFER [10] for the domain 45°
q>
57°.
<
GRID COORDINATES
251
Summary
:Most of difficulties in mutual computer conversions between Universal Transverse
:Mercator (UTM) grid and geographical coordinates arise in determining S (the meridian arc
length from the equator) and cfJ 1 (the ellipsoidal geographical latitude of the foot point which
is on the central meridian).
Effective methods for such conversions are suggested, suiting pocket calculators \\ith
less storage capacities, just as are programs for calculators type HP 67/97 suitable for conversions on the entire surface of any ellipsoid, as indicated in the title.
References
1. HAZAY L-T.-iRczY-H. A.: A Gauss-Kriiger koordimitak szamltasa. (Akademiai Kiad6
Budapest 1951.).
2. JORDAN-EGGERT-KN;EISSL: Handbuch der Vermessungskunde, IV. (J. B. ::\Ietzlersche
Verlagsbuchhandlung, Stuttgart, 1958.).
3. LEE, L. P.: The Transverse Mercator Projection of the Entire Spheroid. Empire Survey
Review, XVI (123), 1962.
4. LEE, L. P.: The length of a meridian arc, and the Transverse :Mercator Projection of the
entire spheroid. Survey Review, XVII (133), 1964. pp. 343.
5. HAZAY 1.: Vetiilettan. (Tankonyvkiad6, Budapest, 1964.).
6. GDOWSKI, B.: The Transverse Mercator Projection of the entire spheroid. Survey Review,
XVIII (141). 1966. pp. 339-341.
7. VINCENTY. T.: The meridional distance problem for desk computers. Survey Review,
XXI (161), 1971. 136-140.
'
8. GERSTBACH, G.: Zur Losung ellipsoidischer Aufgaben mit elektronischen Taschenrechnern.
Zeitschrift fUr Vermessnngswesen. 5. 1974. pp. 205.
9. HUBENY, K.: Isotherme Koordinatensysteme und konforme Abbildungen des Rotationsellipsoides. :Mitteilungen der geodatischen Institute der Technischen Universitat. 27.
Graz. 1977.
10. REITHOFER, A.: Koeffiziententafeln und Rechenprogramme fUr die Gauss-Kriiger(UTM)-Koordinaten der Ellipsoide von Bessel. Hayford, Krassowsky und des Referenzellipsoids 1967. Mitteilungen der geodatischen Institute der Technischen Uuiversitat, 27, Graz. 1977.
11. JACKSOl'\', J. E.: Transverse ::\Iercator Projection. Survey Review, XXIY (188). 1978. pp.
278-285.
12. FIELD, N. J.: Conversions Between Geographical and Transverse :Mercator Co-ordinates.
Survey Review. XXV (195), 1980. pp. 228-230.
Dr. J6zsefYARGA, H-1521 Budapest