UDC 528.236.001.1 (497.12) 
STATE SYSTEM OF PLAIN 
RECTANGULAR COORDINATES 
(Draft proposal f or Republic 
Slovenia) 
Dr. Miroslav Peterca 
Ljubljana 
Received far publication: May 24, 1993 
Abstract 
Far the state system of plain rectangular coordinates a draft 
proposal of Gauss-Krueger coordinates is given. 
Geometrically the Gauss-Krueger system is based on GKP 
widened meridian zones. To achieve greater use of coordinate 
records the change of coordinate origin is suggested. The 
UTM (Universal Transverse Mercator) should serve as a 
connection with Europe. 
Keywords: Gauss-Krueger projection, proposal, Slovenia, 
system of plain rectangular coordinates, UTM 
l!NTRODUCTION 
he technical term „system of plai.n rectangular coordinates" contains 
surveyin.g-cartographic projection. and system of plain rectangular coordinates, 
which is based on projection. as a geometrical basi.s. The article deals with: 
• basic projection and system of plain rectangular coordinates for the territory 
of Republic Slovenia and, 
• system for connecting the basic Slovene system with Europe. 
Geographical position of Republic Slovenia 
If for adjacent ?1eridians an~ parallels th~ external ma~gins of _s~eets TK 25 are taken, Slovema then occup1es the followmg geograph1cal pos1t1on: 
AW = 13°22'30" 
AE = 16°37'30" 
AE-AW = 3°15' 
Projection position ofRepublic Slovenia 
<f)N = 46°52'30" 
<ps = 45°22'30" 
<f)N-<f)S = l 0 30' 
• The position in the Gauss-Krueger projection 
Medium meridian of the system: 
Width of the meridian zone: 
Geodetski vestnik 37 (1993) 2 
lo = 15° 
3°15' 
Ao-AW = AE-Ao = 1°37'30" 
o The position in the UTM system (Universal Transverse Mercator) 
Medi um meridian of the system: ;\,0 = 15° 
Width of the meridian zone: 6° 
Ao-Al = .,1,z-Ao = 3° 
where ;\,1 = 12°, ;\,z = 18° are the adjoining meridians of the six-degree zone. 
Zone :number: 33 
BASI C PROJECTION OF REPUBLIC SLOVENIA AND THE STATE SYSTEM OF PLAIN 
RECTANGULAR COORDINATES 
The basic projection for all surveying calculations, cadastral and topographic survey and cartographing of the en tire territory of Republic Slovenia is the · 
Gauss-Krueger projection of the meridian zone 3°15' longitude. The fact is that 
Ao-AW = AE-Ao = 1 °37'30" still enables, that after scale reductio:n (mo = 0,9999) 
the mapping keeps 1:10 000 accuracy on the e:ntire territory. 
The state system of plain rectangular coordinates is a system of Gauss-Krueger coordinates, which is geometrically based on Gauss-Krueger projection of the 
meridian zone. It is meant for uniform defining the position of points on the plane on the 
entire territory of Slovenia. The system is formed by two coordinate axes, intersecting 
under right angle in the origin of the coordinate system. The vertical axis or axis of 
abscissae (X-axis) is a projection ofthe medium meridian of the zone in GPK. The 
horizontal axis or axis of ordinates (Y-axis) is a projection of equator in GPK or according 
to the given specification: the axis, which is at <po :;i= 0° longitude the parallel projection 
of equator. The geographical coordinates of the origin are therefore: 
Ao = 15°, A-Ao = 0°, <po = 0° 
or after the modification 
Ao = 15°, A-Ao = 0°, <po > 0°. 
The rectangular coordinates of the origin are: 
Yo = 0,0 m, conditionally 500 000 m 
Xo = O,Om. 
As opposed to geographical coordinates <p and ;t on ellipsoid the system of plain 
rectangular Gauss-Krueger coordinates is isometric and equals the Descartes system 
of coordinates. 
PROJECTION SYSTEM FOR CONNECTJ!NG WITH EUROPE 
For location connecting on the territory of Republic Slovenia with Europe the UTM system should be used. It contains: 
o Meridional Mercator (Gauss-Krueger) projection of 6° meridian zones, and 
o UTM rectangular coordinate net, which is based on Meridional Mercator 
projection as a geometrical basis. 
Note: Naturally, a direct connection with Europe is impossible without previous 
connection of astronomy-surveying nets or transformation by aid of mutual data. 
Geodetski vestnik 37 (1993) 2 
General specifications for UTM 
one margins: the UTM zones are Hmited by meridian n. 6° east or west of 
Greenwich, where is n = 0,1,2, ... 30. 
Origin of geographical coordinates: A-Ao = 0°, <po = 0°, 
where Ao is the longitude of the medi um meridian zone. 
origin ofrectangular coordinates: 
Yo = 0,0 m, conditionally 500 000 m 
Xo = 0,0 m, for northern hemisphere 
Parallel margins of the system: 84°N, 80°S, 
Zone marking: from 1-60, beginning by 180° W 
Scale module on medium meridian: mo = 0,9996 
GAUSS.-KRUEGER (MERIDIONAL MERCATOR) PROJECTION EQUATIONS 
he most practical form of computer oriented equations is still a series of 
polynome approximations. Although there is a tendency to come to a so-called 
„closed" equations, which could be most effective for computer processing, the results 
are still unsatisfactory. The so-called American form of equations enables the 
demanded degree of accuracy for the zone up to ±4° of the medi um meridian. By 
using narrower zones and at our latitudes some articles may be left out: 
x = mo {B - Bo + N tg si, [A2/2 + (5 - T + 9C + 4c2) A 4/24 + 
+ (61-58T + T2 + 600C-330e'2) A6/720)} (1) 
y = moN [A + (1-T + C) A3/6 + (5 -18T + T2 + 72C - 58 e'2) A5/120] (2) 
m = mo [1 + (1 + C) A2/2 + (5 -4T + 42C + 13C2 - 28e'2) A 4/24 + 
+ (61-148T + 16T2) A6/720] (3) 
in which there are: 
mo = scale module on the medium meridian (0,9999 for GKP, 0,9996 for UTM) 
e'2 = e2/(1-e2) (4) 
N = a/(1- e2 sin2 <p)112 (5) 
A = (A -Ao) cos <p, withl andAo in radians. (6) 
T = ~<fJ m 
C = ~~<p ~ 
B = a [(1- e2/4 - 3e4/64 - 5e6/256- ... ) <p - (3e2/8 + 3e4/32 + 
+ 45e6/1024 + ... ) sin 2 <p + (15e4/256 + 45 e6/1024 + ... ) sin4 <p-
- (35e6/3072 + ... ) sin 6 <p + ... ] (9) 
with <p in radians. 
B is the distance along the medium meridian from equator to <p latitude. 
Geodetski vestnik 37 (1993) 2 
Bo is B calculated for <po , width of the parallel, which intersects the medium 
meridian Ao at the origin of y and x coordinates. Namely, there isn't always the case 
that the origin of coordinates in the Meridional Mercator projection would 
necessarily lie on the equator, but it is adapted to the area of the mapping. In the GPK 
case in our present use and six-degree UTM zones, Bo = O. 
The equation (3) for the m scale can be written also asa function ofy and <p : 
m = mo [1 + (1 + e'2 cos2 <p) y2 / (Zrno 2 N2)] (3a) 
Equations for the reverse task: 
<p = <p1 - (N1 tg <p1/M1) [D2/2- (5 + 3T1 + 10C1 - 4Ci2- 9 e'2) D4/24 + 
+ (61 + 90T1 + 298C1 + 45T12 - 252e'2 - 3Ci2) D6/720] (10) 
;t = Ao + [D - (1 + 2T1 + C1) D3/6 + (5 - 2C1 + 28T1 - 3Ci2 + 8e'2 + 
+ 24Ti2) D5/120]/cos <p1 (11) 
where the <p1 is the point width on the medium meridian, which has the same value 
of abscissae x as the <p, A point. The equation for calculation is as follows: 
<p1 = µ + (3e1/2- 27 ei3/32 + ... ) sin 2 µ + (21ei2/16- 55e14/32 - ... ) sin 4 µ + 
+ (151ei3/96 + ... )sin 6 µ (12) 
in which where 
e1 = [1- (1- e2)112] / [1 + (1- e2)112] 
µ = B/[a(l - e2/4 - 3e4/64-5e6/256- ... )] 
where 
B = Bo+ x/mo 
with Bo calculated by equation (9) for the given <po . If the origin of rectangular 
coordinates is at the point <po = 0°, Bo = O. 
(13) 
(14) 
(15) 
e'2 = e2/(1- e2) (4) 
C1 = e'2 cos2 <p1 (16) 
T1 = tg2 <p1 (17) 
N1 = a/(1- e2 sin2 <p1)112 (18) 
M1 = a(l- e2)/(1- e2 sin2 <p1)312 (19) 
D = y/(N1mo) •. (20) 
Note: Except by the equation (12), by introducing the auxiliary, so-called rectified 
width <p1, the width µ, can be calculated also by the equation (9) for the given x. The 
interaction procedure is described in reference (Borčic 1976). 
Geodetski vestnik 37 (1993) 2 
ALLOWABLE GAUSS-KRUEGER ZONE WIDTH AND DEFORMATIONS 
U nder conditions mo = 0,9999 and d ::; 0,0001, the latitu.des of Republic Slovenia the following allowable widths A -Ao can be obtained: 
1 rp 45°22'30" 46°00'00" 46°52'30" 
1 
,l-,1,o 1°37'43" 1°38'49" 1°40'25" 
is table shows that in the zone 3°15' latitude the accuracy 1:10 000 is maintained. 
On new mar gin meridians A -;\.o = ± 1 °37'30" for the same latitudes the 
following values of m scale e.g. relative deformations 9. are obtained: 
rp 45°22'30" 46°00'00" 46°52'30" 
m 1,000099 1,000095 1,000088 
d 1: 10100 1:10500 1: 11400 
MODIFICATION OF RECTANGULAR COORDINATES 
To obtain greater expediency of plain rectangular coordinates records, one of the following modifications is possible: 
a) Since on the en tire territory of Republic Slovenia aH <p > 45° the origin of the 
system for RPK listings is moved from the pointAo, <po = 0° to pointA0 , <po = 45°. 
The origin of abscissa is not measured from the equator but from the point, in which 
the parallel <po = 45° intersects the projection of the mediurn meridian. Abscissae 
are calculated by equation (1 ), in which Bo is the length of the medi um meridian from 
equator up to <po = 45° latitude. The transition from old (previously calculated) into 
new abscissae is carried out by equation 
X = Xt - moBo, 
where Xt is the old abscissa and moBo the calculated constant. 
b) Due to the fact that in the present system all abscissae in Republic Slovenia are 
greater than 5 000 000 meters, the starting point of the system is moved to the point 
on the medi um meridian, for which 
Bo = 5 000 000/mo, 
which corresponds to the order of magnitude Bo for <po z 45°09'. With this order of 
magnitude Bo abscissae are calculated by equation (1) and the reverse task in the 
equation (15) is solved. The transition from old to new abscissae is done by equation 
(21), in which 
moBo = 5 000 000. 
The rnodification a) is used in 22 USA states. As the surveying and topographic projection they use TM (GKP) and on it based system of plain rectangular UTM 
coordinates. E.g. for Arizona <po = 31°00', Georgia 30°00', Idaho 41°40', etc. 
Although the modification a) - due to its very simple transition from the present into 
the modified system of Gauss-Krueger coordinates - has already been proposed also 
Geodetski vestnik 37 (1993) 2 
to be used for Republic Slovenia (Peterca 1989), the modification b) is more suitable 
for Republic Slovenia. Unless the value moBo in a) is a round number, in the case b) 
all already calculated abscissae are depreciated for 5 000 000 meten,. As opposed to a), 
this change does not alter the position of the existing trigonometric sections and 
division on sheets from 1:10 000 to 1:500, for which the frames of sheets are lines of 
the rectangular net. Besides, in 1950 in the USA they have begun introducing a 
completely new system and so instead of the Polyconic projection they introduced the 
use of TM and the Lambert conformal conical projection and on these projections 
based systems of plain coordinates. In Slovenia there is not a question of changing the 
system but only its modification. Since the en.tire territory of Republic Slovenia would 
fall in.to one coordinate system e.g. one Gauss-Krueger zone, in ordinates value 
listings the number 5, whkh has been so far used to mark the zone number, could be 
left out. 
Example of Marking Gauss-Krueger coordinates 
The old mode: y = 5 576 979,6 
X = 6 132 590,1 
New mode: 
Description of the rectangular net on topographic maps: 
Old mode: 55 77 51 33 
New mode: 5 77 1 33 
Reference;;: 
576979,6 
132 590,1 
Barčic, B., 1976, Gauss-Kruegerjeva projekcija meridijanskih cona, Zagreb, 38-39. 
Newton, G.D., 1985, Computer Programs for Common Map Projections, U.S. Geological Survey 
Bulletin 1642, 23-33. 
Peterca, M et al, 1974, Kartografija, Beograd, 213-223. 
Peterca, M, 1989, Transfer podatkov med kartami in načrti različnih kartografskih preslikav s 
posebnim oziram na SR Slovenijo, Informacijski sistem v geodetski službi; 21, 30-31, 38. 
Snyder, J.P., 1982, Map Projections Used by the U.S. Geological Survey, Geological Bulletin 1532, 
53-69. 
Review: Ana Kokalj 
profdr. Branko Rojc 
Geodetski vestnik 37 (1993) 2