Theoretical Aspects on The Mathematical Basis
of the GAUSS-KRÜGER
and 1970 STEREOGRAPHIC Projections
Major. Eng. Pantazi Radu
Defence Research Agency
PO Box 51-16 Bucharest, Zip 765550, Romania
Fax: +40-1-4231030
In Romania the most used projections for the map products are Gauss-Krüger and 1970
Stereographic.
The paper deals with the two projections from mathematical cartography point of view, as
follows: definitions and main features, defining of the geodetic and rectangular coordinates
computation, convergence of meridians angle, linear distortion coefficient and of the distances
formulas.
The Gauss-Krüger projection is described from as belonging to the conformal transverse
cylindrical projections group. To get rectangular and geodetic coordinates computation formulas
within the paper two functions are defined. The first allows the point by point representation of an
ellipsoid on a plan (the direct problem) and, the latter, the obtaining of the curvilinear coordinates of
the points on the elipsoid (the inverse problem).
The convergence of meridians angle, linear distortion coefficient and the distances are
defined by solving these two problems.
Further on, the paper refers to map identifications for this type of projection.
1970 Stereographic projection is described as any azimuthal conformal projection with some
specific features for the Romanian territory.
Defining of the computation formulas of this projection is also made within the direct and
inverse problem solving.
I GAUSS-KRÜGER PROJECTION
1. The definition of the projection system
The GAUSS-KRÜGER projection belongs to the group of the cylindrical
transversal conformal projections, which give the reference ellipsoid representation
straight on a cylinder plan. The surface of ellipsoid is represented on a plane after being
divided into limited meridian zones (slices) with a 6, or 3 width.
The projection of the ellipsoid’s surface on a
plane is made in the following terms:
- representation must be conform (orthometric),
i.e. the angles are to be kept undistorted;
- the representation of the central meridian of a
slice is a straight line in report to which the
projection is symmetrical;
- the scale on the axial meridian direction is
Figure 1
K0=1, this means that the cylinder which
covers the ellipsoid is tangent at the axial
meridian.
The meridians and the parallels are represented by random curves. The
meridians are symmetrical opposite to the central meridian of the slice, which,
according to the first condition, are represented by straight lines. The parallels are
symmetrical opposite to the equator and are represented by straight lines.
The position of a point P in the plane of the projection can be determined in a
system of Cartesian coordinates X and Y. The X axis coincides with the axial meridian
of the slice pointed to the North, and the Y axis coincides with the equator parallel
pointed to the East
It is used the Krasovsky ellipsoid associated with the Geodetic Reference
System 1942 (S42).
semi-major axis a = 6 378 245,0 m; semi-minor axis b = 6 356 863,0188m.
One can cosider the following derived parameters of the ellipsoid:
- the first eccentricity:
e2 
a 2  b2
a2
- the second eccentricity: e 2 
a2  b2
b2
- the major normal for a given latitude  : N 
- the curvature radius of the  latitude parallel:
a
1 e 2 sin2 
P  N cos
- the curvature radius of the  latitude meridian: M 


N 1  e2
1  e 2 sin 2 
-the meridian arc () between 1 and 2 parallels:  
2
 Md
1
The position of a random point P on the surface of the reference ellipsoid can be
determinated by the ellipsoidal or geodetic coordinates.
2.Defining of the transformation functions
A point P on the ellipsoid is the affix of a complex number =u+iv, determined
by its components u and v or by the curvilinear coordinates in an ellipsoidal or geodetic
rectangular reference system. In the same way, a point P’ on the plane is the affix of a
complex number z=x+iy, the Cartesian coordinated in an axis system, or defined by
polar coordinates.
In this way a certain mathematical biunivocal function can be defined, by which
to the u and v values, which define a point P on the ellipsoid, must correspond the (x,y),
variables which determine a point P’ on the plane (P’ is the image of P).
The mathematical mapping has two problems to be solved:
- defining a differential function () which allows the point by point
representation of an ellipsoid on a plan (the direct problem);
- defining a differential function =F(z) which allows the obtaining of
the curvilinear coordinates (u,v) of the P point on the ellipsoid, if the
Cartesian coordinates (x,y) of the P point projection on the plan are
known (the inverse problem).
Further on, the method of defining the two transformation functions (direct f,
inverse F) in the case of Gauss projection for the practical solving of the two kinds of
problem, will be presented.
A complex function z=f() is a biunivoc correspondence between two complex
variables:
z=x+iy ;
=u+iv
which means defining x and y like real and continuous functions by , i.e.:
x  xu, v  
 z  f  
y  yu, v 
(2.1)
and hence
y=x+iy=f()=f(u+iv)=x(u,v)+iy(u,v)
(2.2)
Since x(u,v) and y(u,v) have continuous derivative in u and v, not in the same
time equal with 0, the necessary and sufficient condition for the derivative in =u+iv to
exist, hence for the function z=f() to be analytical, is the following relation being:
x
y y
x
f
i

i
u
u v
v
or the relations
 x y
 u  v

 y   x
 u
v
(2.3)
The relations are called D’Alembert or Cauchy-Riemann’s equations.The
functions (2.1) define the ellipsoid-plane transformation, in which u=, v= are
geodetic coordinates. The equation z=f() is an orthometric correspondence if the
function f() is analytical; then, (x,y) and (u,v) are called isometric coordinates.
In the same way one can define the inverse transformation:
=u+iv=F(z)=F(x+iy)=u(x,y)+iv(x,y)
(2.4)
i.e., can unique determinate:
u=u(x,y); v=v(x,y)
(2.5)
in which x,y are Cartesian coordinates.
These considerations give the possibility for defining the functions (2.2) and (2.4)
with a wished approximation, by a Taylor expanding in power series around the
elements of symmetrical axis (e.g., central meridian of a slice).
On the basis of the first defining condition for Gauss projection (to be
orthometric) the analytical correspondences (2.2), (2.4) one can obtain :
z  f    x  iy
(2.6)
(2.7)
  F  z   q  i
which define two functions by two series with constant coefficients, determinated by
the second condition (the longitude preserved by transformation in report to the central
meridian).
Hence, the function (2.6) solves the direct problem and the function (2.7) solves
the inverse problem).
3.The Gauss projection transformation function
In figure 2:O - the origin of the Gauss Cartesian system; 0 - the central meridian
longitude; P(,) - a random point by  and  coordinates;X0
- the ordinate of 0
latitude parallel;  - the ordinate of  latitude parallel;
R
- the intersection point the
of the central meridian and the 
parallel; F
- a point on the
central meridian with y ordinate;
Let us consider :
=-0; =-0 ;
F ig u r e 2
q=q-q0; x=x-x0;
z=z-z0; =-0
then:
z=f()=x+iy
(2.8)
=F(z)=q+i
(2.8’)
Expanding in MacLaurin power series we obtain:
z  x  iy  f q  i   a1 q  i   a2 q  i   a3 q  i   ...
2
3
(2.9)
(2.10)
  q  i  F x  iy   b1 x  iy   b2 x  iy   b3 x  iy   ...
2
in which:
an 
1
n!
d n x  iy 
n
d q  i 
3
bn 
(2.11)
1
n!
d n q  i 
n
d x  iy 
(2.12)
The characteristics of the analytical functions allow to obtain the derivative
independent of the direction. Hence, we choose the central meridian direction (=0=0), uniquely defined by (x=), and then we can obtain the expression of the partial
derivate.
Accordingly, the coefficients an and bn will be:
an 
1
n!
d nz

d n
1
n!
d n
dq n
(2.13)
bn 
Taking into account that
d=Md
2
N
M
 1  e ' cos 2   1   2
dq 
d
M
N cos 
d n
the an coefficients for dq n
will be:
1
n!
d n

dz n
1
n!
d nq
d n
(2.14)

a1  N cos  ; a 2   12 N cos 2 tg ; a3   16 N cos 3  1  tg 2   2
a4 
1
24
a5 
1
120

N cos 4 tg 5  tg 2  9 2  4 4


;

N cos  5  18tg   tg   14  58 2 tg 2  ... ;
5
2

4
2

;

1
a 6   720
N cos 6 tg 61  58tg 2  tg 4  270 2  320 2 tg 2  ... ;
d 
will be:
dq n
n
and bn coefficients for

1
1
1
; b2 
tg ; b3 
1  2tg 2   2
2
3
N cos 
2 N cos 
6 N cos 
1
b4 
tg 5  5tg 2   2  4 4 ;
24 N 4 cos 
b1 

;


1


5  20tg 2  24tg 4  6 2  8 4tg 2  ... ;
120 N 5 cos
1
b6 
tg 61  180tg 2  120tg 4  46 2  48 4tg 2  ...
720 N 6 cos
b5 


;
4. The evaluation of the Gauss Cartesian coordinates from the geodetic coordinates
(direct problem)
Shifting origin O in R (in figure 2) and having given  and  of point P, one
may evaluate theirs Cartesian coordinates (x,y) in Gauss projection by (2.9) and (2.13):
 x  a 0  a 2 p 2  a 4 p 4  A6

 y 
b1 p  b3 p 3  B5
(3.1)
Taking into account that:
p    0  10 4
''
e, e’, a
A,B,C,D,E

-the reference ellipsoid parameters
-the coefficients which determine the meridian arc on the ellipsoid

A  a  1  e 2  1  34 e 2 

45
64

11025 8
e 4  350
e 6  16384
e
512


; B

a  1  e 2 15 4 210 6 2205 8
a  1  e2
 64 e  512 e  4096 e
; D

4
6
  A    B sin 2  C sin 4  D sin 6  E sin 8
C
1
sin 1" 
N
"
a
1e2 sin2 

35
512


a  1  e2

2
e6 
315
2048
e8

;
3
4
525 6
e 2  60
e 4  512
e 
64
E
e

a  1  e 2 315 8
 16384 e
8
(3.2)
K0 – scale factor; K0=1
;   e  cos
One can define the coefficients a0, a2, a4, A6, b1, b3, B5 thus:
a0  K 0  

2205 8
2048
; a 2  12 N  cos 2   tg  sin 2 1  K 0 10 8
; a4 

1
24

1
A6 
p 6 N cos 6   tg  sin 6 1 61  58tg 2  270 2  330 2 tg 2  K 0 10 24
720
b1  N  cos   sin 1  K 0 10 4 ; b3  16 N cos 3   sin 3 1  1  tg 2   2  K 0 1012




N  cos 4   tg  sin 4 1  5  tg 2   9 2  4 4  K 0 10 16


1 5
B5 
p  N cos 5  sin 5 1  5  18tg 2   18tg 4  14 2  58 2 tg 2   K 0 10 20
120

By agreement adopted, the meridian abscissa is set to y0 = 500 000 m, for
avoiding negative values for points placed in the west side of central meridian of the
slice. The same, by agreement adopted, the Equator ordinate is set to x0 = 10 000 000 m,
for avoiding negative values on southern hemisphere.
Thus the final coordinates are:
y1=500 000 + y; x1=x (northern hemisphere); x1=10 000 000 –x
(3.3)
(southern misphere)
For homogenizing the coordinates (x1,y1),
y2= y1+Nf * 1 000 000
(Nf is 4 or 5 for 6o slices on Romanian territory).
5. The evaluation of the geodetic coordinate from the Gauss Cartesian coordinates
(inverse problem)
Shifting the origin O in F (in figure 2) and having given (x,y) Gauss coordinates
of a point P, one may evaluate the geodetic coordinates (,) with the following
relations:
     a1 q 2  a3 q 4  D6

   0  b2 q  b4 q 3  E5
N 
(4.1)
a
1  e sin 2  
2
Taking into account that:
q= (y-500 000)10-6;
A, B, C, D, E - coefficients of meridian arc on ellipsoid;
  e cos  '
K0=1- scale factor;
0 , 0 – geodetic coordinates of central meridian;
e,e’,a - the reference ellipsoid parameters
x0= 500 000;
Latitude ’ is iterative evaluated
  x  x0  / A;  0   ;
  B sin 2  C sin 4  D sin 6  E sin 8  / A
;
    0  
(4.2)
(4.3)
     10
If then =’ and restarts from (4.2).
5
If      10
then the value ’ obtained by (4.3) will be used to
evaluate the coefficients a1, a3, D6, b2, E5 and the values N’ and ’.
The coefficients are:
5
tg 
1
 1    2  2 10 12
2 N  2 sin 1
K 0
1
a3 
 tg 5  3tg 2    6  2  6tg 2     2  3  4  9tg 2     4 K14 10 24
0
24 N  4 sin 1
tg 
6
2
4
2
2
2
4





D6 
q
61

90
tg


45
tg


107


162
tg



45
tg
    2  k16 10 36
0
720 N  6 sin 1
1
1
6
2
2
18
1
1
b2 
 10 ; b4 
 1  2tg       K 3 10
0
N  cos   sin 1 K 0
N  cos   sin 1
a1 







E5 



1
 q 5 5  28tg 2    24tg 4    6  2  8tg 2   2  K15 10 30
0
6 N  3 cos   sin 1
6. The meridians convergence angle ()
The meridians convergence angle is the angle  between the meridian projection
and a parallel to ordinate in the point P(x,y). This angle can be expressed in report with
geodetic coordinates or Gauss Cartesian coordinates.
dx
dx dy
From figure 3 we deduce that tg  dy  d : d
. Doing the differentiation of (3.1) in report
with , we obtain the meridian convergence :
    1 p   2 p 3  C5
in which:
”- the meridian convergence
expressed in arc seconds;
Figure 3


p  0.0001     10 4    0 ;  1  sin  10 4 ;  2  12  sin   cos 2   1  3 2  2 4  sin 2 1 1012


C5  151  sin   cos   2  tg   p  sin 1 10 ;   e  cos 
4
2
5
4
20
0= central meridian longitude of the slice.
The meridian convergence in report with Gauss Cartesian coordinates (x,y), is
expressed thus:
   g1  q  g 2  q  F5 (5.2)
in which:
q=(x-500 000)10-6
15 N  5 sin 1
K 0
 2  5tg 2   3tg 4  q 5 5  10 30
1
tg 
3 N  3 sin 1
K 0
N  sin 1 K 0
g1 

 10 6 ; g 2 
 1  tg 2     2  2  4  3  1018
1
1
tg 
tg 

F5 



’- the meridian arc on the ellipsoid, iterative evaluated as shown in II.4;
N
a
1  e sin 2  
2
; (5.3)   e cos  ; (5.4) K 0  1 -scale factor.
7.The coefficient of linear distortion (K)
The coefficient of linear distortion of a point P(,), in direct problem is
evaluated by: 2
K  K 0 1  p 2  1 cos 2   sin 2 11   2   10 8 
(6.1)
in which p  10 4   0 
;  e cos  ; 0- the longitude of the central meridian;
K0=1 – scale factor
K  K 0 1  k1  q 2  3 10 5 q 4
The coefficient of linear distortion of a point P(x,y), in inverse problem is evaluated by

q  10 6  y  500 000;

k1 


1
1
1012
2
12


1




10

2 N 2
K 02
2R2
(6.2)
in which N’,’- see II.6 ((5.3),(5.4)); K0=1 scale factor; R- mean curvature ray for the
latitude .
8. The evaluation of the relative distance to the ground
The linear Gauss values are distorsioned with a certain value. This value is given
by the projection distortion. To determine a certain distance to the ground with the
Gauss Cartesian coordinates we must take into account the linear distortion.
Using:
The distance value on the ground.
x  y
D
S 
2

2
;
K
K
1
1  1
4
1
  


K
6  K1 K m K 2



The mean inverse coefficient of linear distortion
K1 , K 2 , K m


1
1
K i  K 0 K i  K 0 1 
1  i2  2  1012  q 2  0.00003  q 4 ;
2

2
N
K
i
0




K0  1
q  x  10 6
N 
1
1  e 2 sin 2  i
The distortion coefficients in end points and mean point.
One can compute for Km
qm 
 y1  y 2 
2
10  6 ;  m 
 1   2
2
;
N i 
a
1  e sin  i
2
2
;  i  e  cos  ;
e 2 
a2 b2
b2
a,b- semi-axis of the reference ellipsoid
’- the latitude of the meridian arc X=j evaluated in Chap. 4 ((4.2)(4.3)).
II. STEREOGRAPHIC PROJECTION 1970
1.The definition of the projection system
The stereographic projection is an azimuthal orthometric projection. The
reference ellipsoid is represented on a tangent plane in central zone point of the
interested territory or to a secant plane using a secant circle (Romanian territory).
The geometrical representation is
- The Earth is a sphere with R0 radius (mean
curvature radius);
- P0 is central zone point for mapping projection
(projection pole);
- From P’0 (P0 antipode) on represent all the
sphere points on a tangent projection plane in
Figure 4
P0 or on a secant plane.
The principal elements of stereographic projection
plane are:
X  X0
  2R0 tg
The distance PP0= (defined by Roussille) is given
by: 2R0
In which X is the parallel of the point P and X0 is the
parallel of the point P0. Thus the Cartesian stereographic
coordinates of the point P are:
x=cosA; y=sinA
Now in Romania the Stereographic 1970 projection is
Figure 5
used, which has Făgăraş town as the pole of projection. Its
geographic coordinates are:
0=46 (North); 0=25 (East)
The scale factor is V = 0,99975.
The projection is applied on a secant plan with a minor circle (zenithal distance
1,4843).It is used the Krasovsky ellipsoid (see II.1).
2.The evaluation of the cartesian stereographical coordinates from the
geodetic coordinates
Let us consider a point P(,) on the ellipsoid, p(x,y) its projection on a plan
and q the isometric latitude of the point P. Denote 0, 0, q0 the pole projection
coordinates. On the basis of Cauchy-Riemann (chap II.2) conditions, x and y must be
an analytical function in report with q+ il: x+iy=f(q+il).
If denote: =-0, l=-0,
q=q-q0 and (x0,y0) are the pole projection stereographic coordinates, then:
x+iy=f[(q0+i0)+(q+il)] (7.1)
If the pole projection is the origin of the rectangular coordinates, then Taylor
expanding in power series (7.1) around the pole projection, one obtain:
x  iy  a1 q  il   a2 q  il   a3 q  il   ...
2
3
and separating the real side from the imaginary side, then:
 y  a1 Computing
2a 2 ql  3a3 q 2 l  a3l 3  4a 4 q 3l  4a 4 l 3  5a5 q 4 l  10a5 q 2 l 3  ...

 x  a1 q  a 2 q  a 2 l  a3 q  3a3 ql  a 4 q  6a 4 q l  a 4 l  a5 q  ...
2
3
2
4
2 2
4
5
in which:
an 
1  dnx 


n 
n! 
 dq 
One replace q by , through:
in which:
cn 
(7.2)
1  d nq

n
n! 
 d
q  c1  c2 2  c3 3  c4 4  ...
(7.3)




Introducing (7.3) in (7.2) one obtain the evaluation relations for the sterographic
coordinates of a point P(,):
2
2
2
2
4

 x  x0  a10   a 20   a 02 l  a30 l  a12 l  a 40   ...

2
3
3
3
4
2 3
5

 y  y 0  b01l  b11l  b21 l  b03l  b31 l  b13 l  b41 l  b23  l  b51 l  ...
(7.4)
Denote:
 0  e cos  0 ; N 0 
a
1  e sin 2  0
2
; t 0  tg 0
The first coefficients (aij) and (bij) from (7.4) have the following forms:

a10  N 0 1   02   04   06


b01  N 0 cos 

3
N 0 t 0 02 1   02
2
1
a 02  N 0 t 0 cos 2  0
2
1
a30  N 0 1  402  6t02  904  42t0202 
(7.5)
12
1
a12  N 0 cos 2  0 1  2t02  2t0202  2t0204 
4
1
a40 
N 0t0 cos 4  0 2  t02  602 
24
a 20 

b11  N 0t0 cos  0 1  02  04
b21  


1
N 0 cos  0 1  02  6t0202  04  12t0204
4



1
N 0 cos3  0 1  502  6t0202
12
1
b13   N 0t0 cos3  0 2  t02  402  t0202
6
.............................................................................
b03 


The coefficients (7.5) are constants computed by the means of the pole
projection latitude 0. The stereographic Cartesian coordinate (x,y) are evaluated using
(7.4) with (7.5).
For Romanian territory are enough till six order terms in (7.4).
By agreement adopted, the pole projection coordinate is set to x 0=500 000 m;
y0=500 000 m, for avoiding negative values (in the relation (7.4)).
The coordinates for P are:
x1=500 000+x;
y1=500 000+y
3.The evaluation of geodetic coordinates from stereographical coordinates
The inverse problem, to determinate the geodetic coordinates (,) from the
stereographical coordinates is solved similarly with the direct problem (III.2).
Thus one can be write the following expanded in power series:
q  il  A1 x  iy   A2 x  iy   A3 x  iy   A4 x  iy   ...
2
3
4
(8.1)
which became after the computation and separation of the real side from the imaginary
side:
q  A1 x  A2 x 2  A2 y 2  A3 x 3  3 A3 xy 2  A4 x 4  6 A4 x 2 y 2  A4 y 4  A5 x 5  10 A5 x 3 y 2  ...

l  A1 y  2 A2 xy  A3 x 2 y  A3 y 3  4 A4 xy3  5 A5 x 4 y  10 A5 x 2 y 3  ...
An 
The coefficients Ai are:
Replacing q by , through:
in which:
cn 
1  d n

n
n! 
 dq
1  d nq

n
n! 
 dx
(8.2)




  c1q  c2 q 2  c3 q 3  c4 q 4  ...
(8.3)




and introducing the first relation (8.2) in (8.3) and =l+0, =+0 then (8.2) became:
   0  A10 x  A20 x 2  A02 y 2  A30 x 3  A12 xy2  A40 x 4  A22 x 2 y 2  A04 y 4  A50 x 5  A32 x 3 y 2  ...

  0  B01 y  B11xy  A21x 2 y  B03 y 3  B31x 3 y  B13 xy3  B41x 4 y  B23 x 2 y 3  B05 y 5  B51x 5 y  ...
(8.4)
Denote:   e cos  0 ; N 0 
a
1  e 2 sin 2  0
t0  tg 0
;
The first coefficients Aij and Bij have the following forms:
A10 

1
1   02
N0

3 t 0 02
1   02
2 N 02
1 t0
A02  
cos 2  0
2 N 02

A20  


1 1
A30  
1  802  6t0202  1304  36t0204
12 N 03

1 1
A12  
1  2t02  202  4t0202  6t0204
4 N 03


3 t0 2
A40 
0
4 N 04

1 t0
A22  
1  2t02  802  t0202
4 N0
A04 


1 t0
3  3t02  202  6t0202
24 N 04
B01 
1
N 0 cos  0
B11 
t0
.
N 02 cos  0
B21 
1
1  4t02  02
4 N cos  0
B03 
1
1  4t02  02
12 N cos  0
B31 
t0
3  6t02  02
6 N 04 cos  0
B41 
1
1  12t02  16t04
16 N 05 cos  0

3
0


2
0




(8.5)


The coefficients (8.5) are computed constants in report with the pole projection
latitude, 0. The geodetic coordinates (,), are evaluated using (7.4), with (7.5).
4. The meridians convergence angle 
The meridians convergence angle , in report with
geodetic coordinates (,), in
   01l   11l   21 2 l   03l 3  ...
the stereographic projection can be evaluate by:
; in which:
l=-0 ; =-0
  e cos  0 ;
N0 
a
1  e sin 2  0
2
t0  tg 0


1
1
t 0 cos 3  0 1  9 02 ;  11  cos  0 ;
2
2
1
5
1

cos  0 ;  21    02 sin  0 ;  13  
cos 3  0 1  2t 02
24
4
24
 01  sin  0 ;  03 
 31
;


The meridian convergence in report with the stereographic coordinates can be evaluate
by:
   01 y   11 xy   21 x 2 y   03 y 3   31 x 3 y   13 xy3  ...
In which:
 
 01
t0
;
N0
 
 21
t0
4N
3
0
 
 03
3  4t
2
0




1
1
 
3  4t 02  5 02 ;  11
1  2t 02 02 0 ;
2
12 N 0
2 N 02

 
 5 02 ;  13


1
1  8t 02  8t 04 .
8 N 04
 
 31


1
1  8t 02  8t 04 ;
8 N 04
5. Distance reduction in the stereographic projection
The representation scale for direct problem is given by:
m  1  14 1   02  04  2  14 1  02 l 2  34 t 002  3  14 cos 2  0 t 0 1  8 2 l 2  241  4  481 cos 4  0 2  t 02 l 4
and for the inverse problem:
2t 0
x2  y2
m  1

xy 2 R0- the mean curvature radius of the Earth.
4 R02
R03
The distance d between two points P1(x1,y1) and P2(x2,y2) in report with the
distances on the ellipsoid can be evaluated by
2
x 2  ym
d
x 2  y 2
 1 m

2
s
4 R0
48 R02
,
in which:
x x
y y
xm  1 2 ;
ym  1 2 ; x  x2  x1;
y  y2  y1.
2
2
R0- the mean curvature ray of the Earth
III. MAP IDENTIFICATIONS AND SHEETS NUMBERING
IN THE GAUSS-KRÜGER AND 1970 STEREOGRAPHIC
PROJECTIONS
The numbering sheets system is named the identification (nomenclature) of the
map. This is usually printed on the north edge of each map sheet, by letters and numbers.
Romania has adopted, as starting point for all scale topographic maps identification the
projection used for the world international chart at 1:1,000000 scale namely the
projection of each hemisphere on a normal cone which can be easily unfolded, since
1952. According to this unfolding, the earth surface up to the proximity of the poles,
has been divided onto 4 width slices by means of parallels and onto 6 width columns
by means of meridians. The slices are parallel with the Equator, and the columns are
placed among the meridians which are straight lines pointed to the poles.
That's why, nowdays, the Romanian topographic maps, both those in GaussKrüger projection and 1970 Stereographic ones, have their identifications defined
within this system. The identification of the 1:1,000 000 scale map sheet consist of the
Latin alphabet capital letters (from A to V) corresponding to 4 slices beginning from
the Equator to the poles and of the numbers from 1 to 60, corresponding to the 6
longitude columns, beginning from the 180 meridian.
The Romanian teritory is mostly covered by the L-34 and L-35 and partly by the
M-34, M-35, K-34 and K-35 map sheets at 1:1,000000 scale.The map sheet at 1:1,000
000 scale consists of 4 map sheets at 1:500,000 scale, which are printed with the Latin
alphabet capital letters (A,B,C,D), 36 map sheets at 1:200,000 scale, printed with the
roman numbers (I to XXXVI) and of 144 map sheets at 1: 100,000 scale, printed with
the Arabian numbers (1 to 144).
The identifications of the 1:500,000;1:200,000 and 1:100,000 scale map sheets
consist of the identification of the 1:1,000 000 scale map sheet at which their proper
notations are added. For the other scales, one can detail as follows: the map sheet at
1:100,000 scale consists of 4 map sheets at 1:50,000 scale, printed with the first four
capital letters of the Latin alphabet (A,B,C,D); the map sheet at 1:50,000 scale consists
of 4 map sheets at 1:25,000 scale, printed by the first four small letters of the Latin
alphabet (a,b,c,d); the map sheet at 1:25,000 scale consists of 4 sheets of the
topographic plan at 1:10,000 scale, printed with the Arabian numbers 1,2,3,4; the
1:10,000 scale topographic plan consists of 4 sheets of the topographic plan at 1:5,000
scale, printed with the first four Roman numbers (I,II,III,IV).
IV.REFERENCES
[1] Câmpeanu, V. (1970), Practising Mathematical Cartography Handbook. Military Academy, Bucharest.
[2] Dragomir, V., Ghiţău, D., Mihăilescu, M., Rotaru, M. (1977) Theory of the Earth's Shape. Technical
Press,
Bucharest.
[3] Niţu, C.(1995), Mathematical Cartography. Military Technical Academy Press, Bucharest.
[4] Soloviev, M.D.(1956) Cartographical Projections. Military Press, Bucharest.
[5] * * * ,
(1973) Geodetic Engineer Handbook. Technical Press, Bucharest.