| 214 |
| 64/2 |
V
G 2020
GEODETSKI VESTNIK | letn. / Vol. 64 | št. / No. 2 |
RECENZIRANI ČLANKI | PEER-REVIEWED ARTICLES
KLJUČNE BESEDE KEY WORDS
ABSTRACT IZVLEČEK 
cartographic projection, CAMPREL, Chebyshev-Grave's 
theorem, Meshcheryakov's theorem, Laplace's partial 
differential equations, pan-European mapping, maximum 
absolute linear distortion
kartografska projekcija, CAMPREL, Chebyshev-Grave teorem, 
Meshcheryakov teorem, Laplaceove parcialne diferencialne 
enačbe, vseevropsko kartiranje, linearno popačenje
UDK: 528.235:528.721:528.9(4) 
Klasifikacija prispevka po COBISS.SI: 1.01
Prispelo: 11. 4. 2018
Sprejeto: 10. 4. 2020
DOI: 10.15292/geodetski-vestnik.2020.02.214-226
SCIENTIFIC ARTICLE
Received: 11. 4. 2018
Accepted: 10. 4. 2020
Ivan Nestorov, Milan Kilibarda, Dragutin Protić
OPTIMALNA KONFORMNA 
PROJEKCIJA ZA VSEEVROPSKO 
KARTIRANJE
THE OPTIMAL CONFORMAL 
PROJECTION FOR PAN-
EUROPEAN MAPPING
There is an increasing need for pan-European spatial datasets, 
mainly to support the common European Union policies. This 
has inevitably raised demands for adopting pan-European 
cartographic projections to visualize the spatial data. The Map 
Projection Workshop organized by EuroGeographics in 2001 
provided a recommendation to the European Commission to 
adopt the Lambert conformal conic projection for conformal 
pan-European mapping at the scales smaller or equal to 
1:500,000. This paper discusses if the projection is an optimal 
solution in terms of linear deformations over the mapping 
region. An optimized CAMPREL (conformal adaptive 
mapping projection of rotation ellipsoid) projection for the 
area of interest is proposed as an alternative solution. The 
projection quality criteria were calculated and compared with 
those of the Lambert conformal conic projection. The maximal 
possible absolute linear distortion for conformal mapping of the 
pan-European area is also given. It has been shown that the 
CAMPREL projection designed for pan-European mapping 
better meets the projection selection criteria.
V Evropi je predvsem v povezavi s podporo različnim 
politikam opazna vse večja potreba po vseevropskih prostorskih 
podatkovnih nizih. Za vizualizacijo teh podatkov bi bilo treba 
sprejeti ustrezne kartografske projekcije. Na delavnici na temo 
kartografskih projekcij združenja EuroGeographic leta 2001 
so bila oblikovana priporočila Evropski komisiji, in sicer da se 
v namene kartiranja Evrope v merilih, manjših ali enakih 1 : 
500.000, uporabi Lambertova konformna stožčna projekcija. 
V članku razpravljamo, ali je predlagana kartografska 
projekcija optimalna rešitev z vidika linearnih deformacij na 
območju kartiranja. Kot alternativna rešitev je predlagana 
optimizirana prilagojena konformna projekcija rotacijskega 
elipsoida CAMPREL (angl. Conformal Adaptive Mapping 
Projection of Rotation Ellipsoid ). Izračunali smo vrednosti 
cenilk za oceno kakovosti predlagane projekcije in jih primerjali 
z vrednostmi za Lambertovo konformno stožčno projekcijo. 
Dodatno je izračunana največja mogoča absolutna linearna 
popačenost pri konformnem kartiranju Evrope. Izkazalo se je, 
da predlagana projekcija CAMPREL bolje ustreza izbranim 
kriterijem glede primernosti za vseevropsko kartiranje.
Ivan Nestorov, Milan Kilibarda, Dragutin Protić | OPTIMALNA KONFORMNA PROJEKCIJA ZA VSEEVROPSKO KARTIRANJE | THE OPTIMAL CONFORMAL PROJECTION FOR PAN-EUROPEAN 
MAPPING | 214-226 |
SI  |  EN

| 215 |
GEODETSKI VESTNIK | 64/2 |
RECENZIRANI ČLANKI | PEER-REVIEWED ARTICLES SI | EN
Ivan Nestorov, Milan Kilibarda, Dragutin Protić | OPTIMALNA KONFORMNA PROJEKCIJA ZA VSEEVROPSKO KARTIRANJE | THE OPTIMAL CONFORMAL PROJECTION FOR PAN-EUROPEAN 
MAPPING | 214-226 |
1 INTRODUCTION
There are increasing demands for seamless pan-European spatial datasets (e.g. GMES land monitoring 
data, LUCAS, LPIS.), mainly aimed to support definition and monitoring of the European Union poli -
cies, e.g. Common Agriculture Policy, Environment (Annoni et al., 2001; Annoni and Smits, 2003). 
T o cartographically represent and store such spatial data, there is a need for a common spatial reference 
system for Europe. According to ISO19111:2019 (ISO, 2019), a spatial reference system consists of a 
datum and a coordinate system that includes the definition of a map projection. 
The European Commission Interservices Committee for Geographic Information-COGI adopted 
ETRS89 with the underlying GRS80 ellipsoid, like the geodetic datum (Annoni et al., 2000). The 
Map Projection Workshop, organized in 2001 by EuroGeographics following a request from the 
JRC (Joint Research Centre) of the European Commission, resulted in the recommendation to the 
European Commission to adopt the Lambert conformal conic projection with two standard parallels 
(namely, 35°N and 65°N) for conformal pan-European mapping at the scales smaller or equal to 
1:500.000 (Annoni et al., 2003). It is not, however, clear the exact reason for such a choice. There 
is, however, an implicit indication within the final document of the workshop stating that an area 
with a large West-East extent is better represented in a normal conical projection (i.e. Lambert conformal 
conic - ETRS-LCC). 
The general rule of selecting a projection for conformal mapping is to achieve the least possible exag -
geration of the area within the region to be mapped. This paper aims to show that this requirement was 
not fulfilled by selecting the Lambert conformal conic projection for the pan-European area. T o support 
this claim, the CAMPREL (conformal adaptive mapping projection of rotation ellipsoid) projection 
(Nestorov, 1996) has been derived for the area of interest, and values of the projection quality criteria 
were calculated and compared with those of the Lambert conformal conic projection.
The CAMPREL projection (Nestorov, 1996 and 1997), tries to define the "best projection"
1
 for a subject 
area following Meshcheryakov (1968) criteria. It satisfies both Chebyshev's minimax criteria (Chebyshev, 
1856) and variational criteria based on numerical distortion measures. Such projection is not just a great 
candidate for the conformal projection of the area. Still, it can also answer what the smallest possible 
maximal linear distortion with which the area of interest can be conformally mapped is.
The next section presents the general idea of the CAMPREL projection, with theoretical background, 
derivation of equations for direct mapping, and numerical distortion criteria measures used to select the 
most optimal projection. In the following sections, a computer program and methodology finding the 
CAMPREL projection of the pan-European area is presented, and a comparison of the CAMPREL and 
1
 There are several ideas of "best map projection", among all possible projections, or among a certain class of projections. Some of them are:
1. Kavraysky's idea of best map projection: Map projection founded under unique conditions that maximal linear distortions on the whole area of 
mapping are minimal. (Kavrayskiy, 1958)
2. Milnor's idea of best map projection: Map projection where ratio between logarithms of maximal and minimal linear scales is minimal. Milnor's 
theorem proofs the existence of such projection for the mapping sphere on to the plane. (Milnor, 1969).
3. Chebyshev's idea of best map projection: Map projections having a minimal ratio between maximal and minimal linear scale on the whole area 
of mapping (Chebyshev, 1856). The last definition provides so-called minimax criteria and projection satisfying this criterion is called Chebyshev's 
projection or "the best projection in minimax sense" (Nestorov, 1996). The Chebyshev's - Grave’s theorem, also known as the theorem of conformal 
mapping, proofs the existence of such projection.
4. Meshcheryakov's idea of best map projection: Map projection satisfying both minimax and variational criteria. (Meshcheryakov, 1968). Map 
projection satisfying variational criteria is the one having minimal some of the numerical criteria measures like total distortion, range of the absolute 
distortion, relative linear scale and the like. 

| 216 |
| 64/2 | GEODETSKI VESTNIK  
RECENZIRANI ČLANKI | PEER-REVIEWED ARTICLES
SI | EN
Ivan Nestorov, Milan Kilibarda, Dragutin Protić | OPTIMALNA KONFORMNA PROJEKCIJA ZA VSEEVROPSKO KARTIRANJE | THE OPTIMAL CONFORMAL PROJECTION FOR PAN-EUROPEAN 
MAPPING | 214-226 |
ETRS-LCC projections is made based on numerical distortion measures. The concluding remarks and 
observation are given the final section.
2 THE IDEA OF CAMPREL PROJECTION
2.1 Theoretical background
The idea of the CAMPREL projection has been presented by Nestorov (1996) in the Serbian language. 
Nestorov (1997) summarized the idea in English. However, the full explanation of the projection for -
mulas has never been completely presented in the English language to the community. Therefore, this 
section presents the basics of the CAMPREL projection idea, following both references mentions above.
The concept of the CAMPREL projection is defined using the "reverse cartographic assignment method" 
(Urmayev, 1947; Meshcheryakov, 1968), which assumes that a new map projection can be developed if 
the conditions of the projection characteristics are known. Meshcheryakov's theorem (Meshcheryakov, 
1968) says that if one knows exactly four conditions of projection characteristics
2
, then the unique 
equations for direct and inverse mapping with positive Jacobian at the whole area of mapping exist. 
The projection equations and conditions can be defined analytically or by their numerical values (non-
analytically) at any locations of the mapping region using numerical methods with the desired accuracy.
A conformal map projection can be obtained by integrating Laplace's differential equation that describes 
a condition for conformal mapping of a rotational ellipsoid onto a plane:
 
( ) ( ) ∂∂
+=
∂∂
22
22
ln ln
0
vv
ql
 (1) 
where:
 
22
cos( )
;
1 sin ( )
p
vm
e
ϕ
ϕ
= ⋅
−⋅
 (2)
and:
 
( )
( )
1
2
0
1 sin
ln    and   .
4 2 1 sin
e
p
p
e
q tg l
e
ϕ
πϕ
λλ
ϕ
⋅ 
 −⋅
 
= +⋅ =− 



+⋅ 



 (3)
q and l - isometric latitude and isometric longitude,
ϕ and λ - geographic latitude and longitude,
λ
0
 - geographic longitude of the central meridian of the area,
m - linear scale distortion along the meridian,
e
p
 - eccentricity of meridian ellipsis.
2
 Characteristics of a projection are characterizing distortions created during the mapping. These functions can be linear scale in an arbitrary direction, 
extreme linear scales, principal directions, linear scales in the direction of the meridians and parallels, azimuths of the images of meridians and parallels, 
angle between the images of the meridians and parallels, the area scale, the distortion of azimuth, maximal angular distortion, etc.

| 217 |
GEODETSKI VESTNIK | 64/2 |
RECENZIRANI ČLANKI | PEER-REVIEWED ARTICLES SI | EN
Ivan Nestorov, Milan Kilibarda, Dragutin Protić | OPTIMALNA KONFORMNA PROJEKCIJA ZA VSEEVROPSKO KARTIRANJE | THE OPTIMAL CONFORMAL PROJECTION FOR PAN-EUROPEAN 
MAPPING | 214-226 |
The CAMPREL projection also utilizes Chebyshev-Grave's theorem
3
 (Chebyshev, 1856; Grave, 1911; 
Urmayev, 1947) stating that the best conformal map projection for a region is the one with the constant 
logarithm of linear scale along the region's boundary. Therefore, the following condition applies:
 
22
cos( )
ln( ) ln
1 sin ( )
p
vK
e
Γ
ϕ
ϕ


= ⋅

−⋅

 (4)
where:
Γ - the boundary of the mapping area;
K - the value of the constant linear scale assigned along the mapping boundary.
Since for conformal mapping is valid, that the linear scale c in any direction is constant and therefore 
m  c, the function of the linear scale in an arbitrary direction can be obtained by solving the Laplace's 
differential equation (1) under the boundary condition (4). The solutions of Laplace's differential equa -
tion, by definition, are harmonic functions, like a harmonic polynomial u
k
 of degree k in equation (5), 
where P
k
 and Q
k
 are the real and imaginary parts of harmonic polynomial u
k
.
 u
k
  (q + i  l )
k
  P
k
 + i  Q
k
;            i
2 
 1 (5)
One can obtain an expression for any degree of harmonic polynomials P
k
 and Q
k
 using Morozov's recur -
rent formulas (6) with given null terms P
0
  1 and Q
0
  0. Their equality can be proved by mathematical 
induction. (Mitrinović, 1989)
 P
j+1
  q  P
j
  l  Q
j
   P
0
  1; P
j+1
  q  P
j
  l  Q
j
 (6)
 Q
j+1
  q  Q
j
 + l  P
j
   Q
0
  0; Q
j+1
  q  Q
j
  l  P
j
 
The real P
k
 and imaginary Q
k
 parts are the solution of the of Laplace's differential equation (1) as well as 
their arbitrary linear combination presented in equation (7), where a
j
 and b
j
 are unknown Chebyshev's 
coefficients (Urmayev, 1947).
 
0
k
k jj j j
j
u aP bQ
=
= ⋅+⋅
∑
 (7)
Equation (7) can further be simplified if the boundary of the mapping area ( Γ ) and isocoles are symmetrical 
in relation to the central meridian. These symmetries exclude imaginary part Q
j
 of the polynomial and 
the solution takes a form of a symmetrical harmonic polynomial presented in equation (8).
 
0
k
k jj
j
u aP
=
= ⋅
∑
 (8)
Using the boundary condition (4) and the symmetrical harmonic polynomial above, one can express the 
more general solution of the Laplace's differential equation for the complete mapping area as:
3
 Chebyshev-Grave’s theorem reads precisely: In the class of all conformal map projections minimal ratio between maximal and minimal linear scale 
on the whole area of mapping, will be on the map projection having constant logarithm of linear scale along the contour of the mapping area .

| 218 |
| 64/2 | GEODETSKI VESTNIK  
RECENZIRANI ČLANKI | PEER-REVIEWED ARTICLES
SI | EN
Ivan Nestorov, Milan Kilibarda, Dragutin Protić | OPTIMALNA KONFORMNA PROJEKCIJA ZA VSEEVROPSKO KARTIRANJE | THE OPTIMAL CONFORMAL PROJECTION FOR PAN-EUROPEAN 
MAPPING | 214-226 |
 
22
0
cos( )
ln( ) ln
1 sin ( )
k
jj
j
p
c aP
e
φ
φ =


= ⋅−

−⋅

∑
 (9)
As explained by Nestorov (1997), equation (9) can be formulated for n  k  1 discrete points on the 
boundary of the mapping area as a system of equations presented with (10).
 
22
0
cos( )
ln( ) ln , 1,2,3,...,
1 sin ( )
k
i
jj
j
pi
K aP i n
e
ϕ
ϕ =


= ⋅− =

−⋅

∑
 (10)
With the constant linear scale, K assigned to the boundary of the mapping area, and with enough 
discrete points on the boundary, one can determine values of polynomial coefficients a
j
 using the least 
square method. 
Knowing the coefficients of the symmetrical harmonic polynomial makes it possible to calculate the 
values of the linear scale c for all the points of the mapping region using equation (11), and meridian 
convergence γ with equation (12) according to Nestorov (1996). In equation (11), e is the base of the 
natural logarithm.
 
22
0
cos( )
ln
1 sin ( )
k
jj
j p
aP
e
ce
φ
φ =


⋅−

−⋅
 
∑
=
 (11)
 
0
k
jj
j
aQ γ
=
= −
∑
 (12)
As already mentioned above, the four independent mapping characteristics in every point are needed 
for the complete map projection definition (Meshcheryakov, 1968). For the conformal mapping, it is 
enough to know two independent mapping characteristics, since the linear scale along the meridian 
m and the linear scale along the parallel n are both equal to the linear scale c, i.e. m  n  c and also 
the angle between a meridian and a parallel always remains right ( Θ  π/2). Therefore, two mapping 
characteristics c and γ, defined with equations (11) and (12), satisfy the Meshcheryakov's theorem, and 
one can determine unique equations of the direct and inverse mapping and also functions of all other 
mapping characteristics. 
2.2 Equations for direct mapping
When deriving equations of direct mapping based on isometric latitude ( q) and longitude (l ), x  x(q, l ), 
y  y(q, l ) the following two facts were taken into account:
1. The rectangular coordinates in the plane, x and y, are conjugated harmonic functions of isometric 
coordinates, and one can use an analytic function that determines the conformal mapping in the form 
of a harmonic polynomial:
 
1
1
n
jj j j
j
n
jj j j
j
y AP BQ
x AP BQ
=
=
= ⋅+⋅
= ⋅−⋅
∑
∑
 (13)

| 219 |
GEODETSKI VESTNIK | 64/2 |
RECENZIRANI ČLANKI | PEER-REVIEWED ARTICLES SI | EN
Ivan Nestorov, Milan Kilibarda, Dragutin Protić | OPTIMALNA KONFORMNA PROJEKCIJA ZA VSEEVROPSKO KARTIRANJE | THE OPTIMAL CONFORMAL PROJECTION FOR PAN-EUROPEAN 
MAPPING | 214-226 |
where A
i
 and B
i
 are real coefficients, and P
i
 and Q
i
 are real and imaginary parts of this harmonic poly -
nomial.
2. For conformal mapping, it is known (Bugayevskiy, 1974; Pedzich, 2005; T utić, 2009) to be valid:
 
( ) ( )
( ) ( )
cos , sin
sin , cos
ql
ql
xx
yy
νγ νγ
νγ νγ
=⋅ = −⋅
= ⋅= ⋅
 (14)
where x
q
, x
l 
, y
q
, y
l 
, are partial derivatives of the mapping function, γ is the meridian convergence and the 
function ν is previously given by expression (2).
The polynomial real coefficients, A
i
 and B
i
, can be determine using partial derivatives of the expression 
(13) by isometric latitude q.
 
( ) ( )
( ) ( )
1
1
n
q jj j j
qq
j
n
q jj j j
qq
j
y AP BQ
x AP BQ
=
=
= ⋅ +⋅
= ⋅ −⋅
∑
∑
 (15)
The partial derivative of a harmonic polynomial u
k
. by isometric latitude q is:
 (P
n
  i  Q
n
)
q
  n  (q  i  l )
n 1
  n  (P
n 1
  i  Q
n 1
). (16)
Therefore, the real and imaginary parts can be written as:
 (P
n
)
q
  n  P
n 1
 
(17)
 (Q
n
)
q
  n  Q
n 1
Hence, the expressions (15) becomes:
 
( )
( )
11
1
11
1
n
q jj j j
j
n
q jj j j
j
y jAP BQ
x jAP BQ
−−
=
−−
=
= ⋅⋅+⋅
= ⋅⋅−⋅
∑
∑
 (18)
On the other hand, the relations in equations (14) are valid at each point of the mapping area. The 
meridian convergence ( γ) and the function ν in these equations can be replaced using expressions (2), 
(11), and (12), and the numeric values of x
q
 and y
q
 can be calculated as follows:
 
0
0
0
0
sin
cos
k
ii
i
k
ii
i
k aP
q ii y
i
k aP
q ii x
i
y aQ T
x aQ T
e
e
=
=
⋅
=
⋅
=
∑

= ⋅ −⋅ =


∑

= ⋅ −⋅ =


∑
∑
 (19)
With known numeric values of partial derivatives y
q
 and x
q
 as T
y
, and T
x
, one can write for each point 
of the mapping area a system of two linear equations, where the coefficients A
j
 and B
j
 (i  1,2,3,...,n) 
are 2n unknown variables:

| 220 |
| 64/2 | GEODETSKI VESTNIK  
RECENZIRANI ČLANKI | PEER-REVIEWED ARTICLES
SI | EN
Ivan Nestorov, Milan Kilibarda, Dragutin Protić | OPTIMALNA KONFORMNA PROJEKCIJA ZA VSEEVROPSKO KARTIRANJE | THE OPTIMAL CONFORMAL PROJECTION FOR PAN-EUROPEAN 
MAPPING | 214-226 |
 
( )
( )
11
1
11
1
n
jj j j y
j
n
jj j j x
j
jAP BQ T
jAP BQ T
−−
=
−−
=
⋅⋅+⋅ =
⋅⋅−⋅ =
∑
∑
 (20)
To define the unknown coefficients A
j
 and B
j
 (j  1,2,3,...,n), it is necessary to use at least n 
different points over the mapping area and solve the system of 2 n linear equations with 2n un-
knowns. One can take a larger number of points than necessary and use the least-squares method 
to optimize the accuracy of the results obtained. Once the polynomial coefficients A
j
 and B
j
 are 
determined, the rectangular coordinates x and y can be calculated from the expression (13) for 
the whole mapping area.
For the calculation of geographic coordinates ϕ and λ based on known y and x coordinates, also known 
as inverse mapping, authors suggesting established numerical methods since the equations for the direct 
mapping are derived numerically.
2.3 Numerical distortion criteria for projection optimization
Since the idea of the CAMPREL is to find a projection satisfying both Chebyshev's (presented in the 
first subsection above) and variational criteria, this subsection presents numerical measures (distortion 
parameters) for projection optimization used to satisfy the lateral criterion. The "best projection" in a 
variational sense would, therefore, have minimal values of all these parameters.
All presented measures are based on linear distortion
4
. Besides maximal and minimal linear scales, c_max 
and c_min, the following measures were also utilized:
1. Jordan's total distortion criteria (Kavrayskiy, 1958):
 
2
22
0
1
( 1)
2
J
P
E c d dP
P
π
α
π
= −⋅⋅
⋅⋅
∫∫
 (21)
For the case of the finite number of points, the simplified form can be used (Nestorov, 1996):
 
2
1
( 1)
n
Ji
i
Ec
=
= −
∑
 (22)
2. Jordan-Kavraysky's total distortion criteria (Kavrayskiy, 1958):
 
( )
2
2
2
0
1
ln
2
JK
P
E c d dP
P
π
α
π
= ⋅⋅
⋅⋅
∫∫
 (23)
And in the simplified form of the above formula for the finite number of points (Nestorov, 1996):
 
2
1
(ln )
n
JK i
i
Ec
=
=
∑
 (24)
4
 When defining the projection quality criteria, it should be borne in mind that map projections generally cause inevitable distortions of the lengths, angles, 
and areas of the original surface, but also that mapping is possible in which eliminated either angular distortions (conformal projections) or distortions 
of area (equivalent projections). However, linear distortions are always present. Therefore, the overall measure of projection quality must contain linear 
distortions, namely linear distortions throughout the mapping area. In other words, the basic factor in defining the selection criteria and the quality of 
the projection should be the distribution of linear distortion over the entire mapping area.

| 221 |
GEODETSKI VESTNIK | 64/2 |
RECENZIRANI ČLANKI | PEER-REVIEWED ARTICLES SI | EN
Ivan Nestorov, Milan Kilibarda, Dragutin Protić | OPTIMALNA KONFORMNA PROJEKCIJA ZA VSEEVROPSKO KARTIRANJE | THE OPTIMAL CONFORMAL PROJECTION FOR PAN-EUROPEAN 
MAPPING | 214-226 |
3. Range of linear distortions:
 RLD  c_max  c_min (25)
4. Relative linear scale:
 
- c_max c_min
RLS
c_max
=
 (26)
5. The ratio of maximal and minimal linear scales:
 
max
min
c
RMMS
c
=
 (27)
6. Ratio of logarithms of maximal and minimal linear scales:
 
ln( )
ln( )
c_max
RLMMS
c_min
=
 (28)
7. Range of absolute linear distortions:
 RALD  d_absmax  d_absmin (29)
8. Average absolute linear distortion:
 
1
()
n
i
i
abs d
AALD
n
=
=
∑
 (30)
9. Root mean square of linear distortions:
 
2
1
n
i
i
d
RMSALD
n
=
=
∑
 (31)
where c, α, P, dP, d_max, d_min, d_absmax, and d_absmin are respectively: linear scale, azimuth, the 
area of mapping, a differential element of the area, maximal linear distortion, minimal linear distortion, 
the absolute value of maximal linear distortion and the absolute value of minimal linear distortion.
3 CAMPREL PROJECTION FOR THE AREA OF PAN-EUROPEAN MAPPING
3.1 Methodology for finding the CAMPREL projection 
To find the CAMPREL projection for pan-European mapping area - the area bounded by parallels of 
27°N and 71°N and meridians of 30°W and 45°E, (Annoni et al., 2001) - the task is to determine the 
degree (k) of the harmonic polynomial ( u
k
) and the constant value of the linear distortion along the area 
border (K) for which all numerical criteria presented in section 2.3 would be minimized. A computer 
program that generates and tests Chebyshev's projection variants were developed for this task.
Based on the boundary parallels and meridians, the program varies the degree of the harmonic polynomial 
(k), the constant value of the linear distortion along the area border ( K), and the number of points on the 
contour of the region ( n), all from expression (10), to create projection variants. It increases the degree of 

| 222 |
| 64/2 | GEODETSKI VESTNIK  
RECENZIRANI ČLANKI | PEER-REVIEWED ARTICLES
SI | EN
Ivan Nestorov, Milan Kilibarda, Dragutin Protić | OPTIMALNA KONFORMNA PROJEKCIJA ZA VSEEVROPSKO KARTIRANJE | THE OPTIMAL CONFORMAL PROJECTION FOR PAN-EUROPEAN 
MAPPING | 214-226 |
the harmonic polynomial ( k) until the difference of the calculated total distortion (based on formulas (22) 
and (24)) for the two successive degrees of the harmonic polynomials is less than 3x10
4
. For the initial 
value of the constant linear scale along the boundary meridians and parallels ( K), a value of 1.00700 was 
taken and varied to 1.00710 in steps of 0.00002. This range is based on the estimation that the maximum 
positive linear distortion is expected to be approximately equal to the maximum negative linear distortion. 
The number of boundary points ( n) varied from 36 to 60 until their number ceased to affect the accuracy 
of calculation of linear scales along the boundary meridians and parallels. The points along the border 
were evenly distributed. In total, the program generates 210 Chebyshev's projection variants.
For every variant, the program then calculates numerical criteria measures to test the projection quality (dis -
tortion parameters): Jordan's total distortion ( E
J
), Jordan-Kavraysky's total distortion ( E
JK
), maximal linear 
scale (c_max), minimal linear scale (c_min), range of linear distortions ( RLD), relative linear scale (RLS), the 
ratio of maximal and minimal linear scales ( RMMS), the ratio of logarithms of maximal and minimal linear 
scales (RLMMS), maximal linear distortion ( d_max), minimal linear distortion ( d_min), maximal absolute 
linear distortion ( d_absmax), minimal absolute linear distortion ( d_absmin), range of absolute linear distor -
tions (RALD), average absolute linear distortion ( AALD), and root mean square of absolute linear distortion 
(RMSALD). Each numerical measure is based on 3420 evenly distributed grid points (cross-sections of 
meridians and parallels for every degree of latitude and longitude) over the area of interest.
All variant cases are tested in less than one minute of CPU time.
3.2 The optimal CAMPREL projection and its comparison to ETRS-LCC projection 
From the 210 projection variants, Table 1 gives Chebyshev's coefficients, a
0
  a
10
, for the optimal CAM -
PREL projection designed for the pan-European mapping area. The total distortions of the projection have 
minimum values, and the absolute maximum positive linear distortion is approximately equal to the absolute 
maximum negative linear distortion value. The optimal variant has the degree of the harmonic polynomial 
10, the constant linear scale along the border is 1.00706, and 60 evenly distributed points on the boundary 
of the mapping area are used to define the coefficients. The precision of values presented in T able 1 is limited 
to the accuracy of the numerical methods used to generate the variant. Authors believe that the precision here 
is more than sufficient for the purpose of mapping in the scales smaller or equal to 1:500,000.
Table 1: Chebyshev's coefficients of the optimal CAMPREL projection for Pan-European area.
j a 
j
0 -0.567667536
1 -0.783326923
2 -0.083929018
3 0.011493356
4 0.055463877
5 -0.019925516
6 0.003009703
7 0.003545522
8 -0.006029762
9 0.000535393
10 -0.003798708

| 223 |
GEODETSKI VESTNIK | 64/2 |
RECENZIRANI ČLANKI | PEER-REVIEWED ARTICLES SI | EN
Ivan Nestorov, Milan Kilibarda, Dragutin Protić | OPTIMALNA KONFORMNA PROJEKCIJA ZA VSEEVROPSKO KARTIRANJE | THE OPTIMAL CONFORMAL PROJECTION FOR PAN-EUROPEAN 
MAPPING | 214-226 |
The numerical quality criteria measures (distortion parameters) for ETRS-LCC projection and the optimal 
CAMPREL projection were calculated from a grid of points for each degree of latitude and longitude 
within the pan-European mapping area. In total, 3420 points were used. The results are presented in 
Table 2.
All distortion parameters are convincingly better for the CAMPREL projection compared to the ETRS-
LCC. The Jordan's and Jordan-Kavraysky's distortion ratios of the two projections show that the CAM -
PREL projection is almost twice as good (the ratio is over 180%). Similarly, both mean values of linear 
distortion, average (AALD) and mean square (RMSALD), show the same outperformance. It should also 
be noted that the linear distortion range is about 39.17% better with the CAMPREL projection compared 
to the ETRS-LCC, as well as the absolute linear distortion range, which is improved for 55.70%. The 
linear distortion of the ETRS-LCC ranges from -0.034374 to 0.043704, while at CAMPREL projections 
symmetrically ranges from -0.028069 to 0.028069. With the CAMPREL projection, the maximum 
absolute linear distortion of conformally mapped pan-European area, therefore, drops to 0.028069.
Table 2: Numerical quality criteria measures (distortion parameters) for the ETRS-LCC and CAMPREL projections.
CRITERIA/PROJECTION ETRS-LCC CAMPREL ETRS-LCC/CAMPREL
Number of points: 3420 3420 -
Jordan’s total distortion - EJ 1.481778 0.821865 180.29%
Jordan-Kavrayskiy's total distortion - EJK 1.488330 0.826696 180.03%
Maximal linear scale - c_max 1.043704 1.028034 101.52%
Minimal linear scale - c_min 0.965626 0.971931 99.35%
Range of linear distortions - RLD 0.078078 0.056103 139.17%
Relative linear scale - RLS 7.48% 5.46% 137.08%
Ratio of maximal and minimal linear scales - RMMS 1.080857 1.057723 102.19%
Ratio of logarithms of maximal and minimal linear scales - 
RLMMS
-1.222914 -0.971120 125.93%
Maximal linear distortion - d_max 0.043704 0.028034 155.90%
Minimal linear distortion - d_min -0.034374 -0.028069 122.46%
Maximal absolute linear distortion - d_absmax 0.043704 0.028069 155.70%
Minimal absolute linear distortion - d_absmin 0.000000 0.000000 -
Range of absolute linear distortions - RALD 0.043704 0.028069 155.70%
Average absolute linear distortion - AALD 0.022567 0.011804 191.19%
Root mean square of absolute linear distortion - RMSALD 0.025338 0.014054 180.29%
Figure 1 and 2 shows linear scale isocole maps and absolute linear distortion isocole maps for the ETRS-
LCC and CAMPREL projections. These maps were generated from the values of linear scales, and absolute 
linear distortions in the grid of 3420 points used to calculate numerical criteria measures in Table 2.
When comparing projections in each figure, it is evident that the values of distortions are overall smaller 
in the CAMPREL projection compare to the ETRS-LCC, as it was indicated in T able 2. From the maps, 
one can also notice different distortion pattern of the CAMREL projection compares to the established 
ETRS-LCC. In the ETRS-LCC, the linear distortion changes only in one direction, south-to-north, 

| 224 |
| 64/2 | GEODETSKI VESTNIK  
RECENZIRANI ČLANKI | PEER-REVIEWED ARTICLES
SI | EN
Ivan Nestorov, Milan Kilibarda, Dragutin Protić | OPTIMALNA KONFORMNA PROJEKCIJA ZA VSEEVROPSKO KARTIRANJE | THE OPTIMAL CONFORMAL PROJECTION FOR PAN-EUROPEAN 
MAPPING | 214-226 |
and with fast decrease when approaching the standard parallels. In the CAMREL projection, the linear 
distortion changes radially and relatively evenly, in all directions, following the form of the border of the 
mapping area. This pattern is part of a very important feature of the CAMPREL projection that should 
be emphasized: the CAMPREL is an adaptive projection (Nestorov 1996 and 1997). What this means is 
that the distortion and its isocoles adapts and follow the boundary of the mapping region, which results 
in linear distortions optimally distributed and stretched within the subject area.
Figure 1: Isocole map of linear scale distribution (multiplied by 1000) for the ETRS-LCC projection (left), and for the optimal 
CAMPREL projection (right).
Figure 2: Isocole map of absolute linear distortion (multiplied by 1000) for the ETRS-LCC projection (left), and for the optimal 
CAMPREL projection (right).
4 CONCLUSION
The need for common European spatial datasets induces a strong demand for adopting standard map 
projections for pan-European mapping. The Lambert conformal conic projection (ETRS-LCC) was 
chosen for the scales equal or smaller to 1:500.000 that require the conformal mapping of an ellipsoid 
to a plane. However, if one adopts the general criteria when selecting a projection, which is "to select a 
projection in which the extreme distortions are smaller than would occur in any other projection used 
to map the same area" (Maling, 1989), then the general rule for selecting a projection for conformal 
mapping would be to achieve the least possible linear distortions. This paper proves that this rule has not 

| 225 |
GEODETSKI VESTNIK | 64/2 |
RECENZIRANI ČLANKI | PEER-REVIEWED ARTICLES SI | EN
Ivan Nestorov, Milan Kilibarda, Dragutin Protić | OPTIMALNA KONFORMNA PROJEKCIJA ZA VSEEVROPSKO KARTIRANJE | THE OPTIMAL CONFORMAL PROJECTION FOR PAN-EUROPEAN 
MAPPING | 214-226 |
been met with ETRS-LCC projection. It is shown that the CAMPREL projection defined for the area 
of interest has significantly better characteristics. The maximum absolute linear distortion is 0.028069 
compare to the maximum absolute linear distortion of 0.043704 for the ETRS-LCC. Also, linear distor -
tion distribution for the CAMPREL projection changes radially and relatively evenly, in all directions, 
following the form of the area boundary, while in ETRS-LCC, the linear distortion changes just in the 
direction of south-to-north and with faster decrease approaching the standard parallels.
Clearly, the CAMPREL projection designed for pan-European mapping better meets the projection 
selection criteria. Authors believe that the subject area cannot be better conformally mapped, that is, 
any other conformal projection will cause a maximum absolute linear distortion greater than 0.028069. 
They leave to other map projection experts to refute or confirm this claim.
Literature and reference: 
Annoni, A., Luzet, C., Gubler, E., Ihde, J. (Eds). (2001). Map Projections for Europe. 
Joint Research Centre, EUR 210120 EN.
Annoni, A., and Luzet, C. (Eds). (2000). Spatial Reference Systems for Europe. Joint 
Research Centre, EUR19575 EN. http://www.ec-gis.org/reports/spatial ref.pdf. 
Annoni, A., Smits, P . (2003). Main problems in building European environmental 
spatial data. International Journal of Remote Sensing, 24 (20), 3887–3902. 
DOI: https://doi.org/10.1080/0143116031000103763 
Bermejo, M., Otero, J. (2005). Minimum conformal mapping distortion according to 
Chebyshev's principle: a case study over peninsular Spain. Journal of Geodesy, 79 
(1–3), 124–134. DOI: https://doi.org/10.1007/s00190-005-0450-5 
Bugayevskiy, L. M., Bocharov, A.E. (1974). O primenenii asimetrichnyh ravnougolnnyh 
proekciy dlya sozdania kart na obširnie teritorii. Geodeziya i Kartografiya, 11, 
61–65.
Bugayevskiy, L. M., Snyder , J. P . (1995). Map projections. A reference manual. London: 
Taylor and Francis.
Chebyshev, P . L. (1856). Sur la construction des cartesg' eographiques. Reprinted in 
Ouvres de P .L. Tchebychef: NewYork: Chelsea Pub.Co., 1962.
Frankich, K. (1982). Optimization of geographic map projection for Canadian territory. 
Doctoral dissertation. Barnaby, B. C. Canada: Simon Fraser University. 
Grave, D. A. (1911). Demonstration d' un theoreme de T chebycheffgeneralise. Journal 
für die reine und angewandte Mathematik (Crelles Journal), 140, 247–251. DOI: 
https://doi.org/10.1515/crll.1911.140.247 
Habib, M. (2008). Proposal for developing the Syrian stereographic projection. Survey 
Review, 40 (307), 92–101. DOI: https://doi.org/10.1179/003962608x253547 
Hill, G. W . (1908). Application of T chebychef's principle in the projection of maps. The he 
Annals of Mathematics, 10 (2), 23–36. DOI: https://doi.org/10.2307/1967302 
ISO (2019). ISO 19111:2019, Geographic information - Referencing by coordinates, 
International Organization for Standardization. https://www.iso.org/
standard/74039.html.
Kavrayskiy, V . V .(1958). Isbrannyetrudy I, II, III Matematicheskaya kartografia, Moscow: 
Isdatelstvo Nedra.
Maling, D. H. (1989). Measurements from Maps. Oxford: Pergamon Press.
Nestorov, I. (1996). Nove optimalne kartografske projekcije. Belgrade: Zadužbina 
Andrejević.
Nestorov, I. (1997). CAMPREL: A New Adaptive Conformal Cartographic Projection. 
Cartography and geographic information systems, 24 (4), 221–227. DOI: https://
doi.org/10.1559/152304097782439295 
Maling, D. H. (1989). Measurements from Maps. Oxford: Pergamon Press.
Meshcheryakov I. A. (1968). Teoreticheskie osnovy matematicheskoi kartografii. 
Mossow: Isdatelstvo Nedra.
Mitrinović S. D. (1989). Kompleksna analiza. Belgrade: Građevinska knjiga.
Milnor, J. (1969). A Problem in Cartography. The American Mathematical Monthly, 
76 (10), 1101–1112. DOI: https://doi.org/10.1080/00029890.1969.12000424 
Orihuela, S. (2019). A proposal to include conformal harmonic polynomials in GIS 
software. Applied Geomatics, 11 (4), 331–338. DOI: https://doi.org/10.1007/
s12518-019-00262-x 
Pedzich, P . (2005). Conformal projection with minimal distortions. XXII International 
Cartographic Conference, La Coruña, ICA.
T obler , W . (2011). Notes and Comments on the Composition of T errestrial and Celestial 
Maps - translation of J.H.Lambert work. Preface, ESRI Press.
T utić, D.(2010). Konformne projekcije za Hrvatsku s najmanjim apsolutnim linearnim 
defirmacijama. Geodetski list, 3, 157–173.
T utić, D. (2009). Optimal Conformal Polynomial Projections for Croatia According to the 
Airy/Jordan Criterion. Kartografijai Geoinformacije, 8 (11), 48–67.
Urmayev, N.A.(1947). Metody iziskaniya novyh kartograficheskih proekciy. Moscow: 
Voyenno-topograficheskyoe upravlenie generalnogo shtaba vooruzhonnyh 
syl SSSR.
Vasilca, D., Badea, A. C., Badea, G. (2019). Comparison between the 1970 stereographic 
projection and a double-stereographic projection. 19th International 
Multidisciplinary Scientific GeoConference SGEM 2019, pp. 749–756.

| 226 |
| 64/2 | GEODETSKI VESTNIK  
RECENZIRANI ČLANKI | PEER-REVIEWED ARTICLES
SI | EN
Ivan Nestorov, Milan Kilibarda, Dragutin Protić | OPTIMALNA KONFORMNA PROJEKCIJA ZA VSEEVROPSKO KARTIRANJE | THE OPTIMAL CONFORMAL PROJECTION FOR PAN-EUROPEAN 
MAPPING | 214-226 |
Assoc. prof. dr. Ivan Nestorov, univ. grad. eng. of geod. 
University of Belgrade, Faculty of Civil Engineering–
Department of Geodesy and Geoinformatics 
Bulevar kralja Aleksandra 73, 11000 Belgrade, Serbia
e-mail: nestorov@grf.bg.ac.rs
Assist. prof. dr. Dragutin Protić, univ. grad. eng. of geod.
University of Belgrade, Faculty of Civil Engineering–
Department of Geodesy and Geoinformatics 
Bulevar kralja Aleksandra 73, 11000 Belgrade, Serbia
e-mail: protic@grf.bg.ac.rs
Assist. prof. dr. Milan Kilibarda, univ. grad. eng. of geod.
University of Belgrade, Faculty of Civil Engineering–
Department of Geodesy and Geoinformatics 
Bulevar kralja Aleksandra 73, 11000 Belgrade, Serbia
e-mail: kili@grf.bg.ac.rs
Nestorov I., Kilibarda M., Protić D. (2020). The optimal conformal projection for pan-European mapping. Geodetski vestnik, 64 (2), 214-226. 
DOI: https://doi.org/10.15292/geodetski-vestnik.2020.02.214-226