| 363 |
| 67/3 |
RECENZIRANI ČLANKI | PEER-REVIEWED ARTICLES
V
G 2023
GEODETSKI VESTNIK | letn. / Vol. 67 | št. / No. 3 |
SI  |  EN
ABSTRACT IZVLEČEK 
KLJUČNE BESEDE KEY WORDS
map projection, conic projection, cylindrical projection kartografska projekcija, konusna projekcija, cilindrična projekcija 
Miljenko Lapaine | OD KONUSNIH DO CILINDRIČNIH KARTOGRAFSKIH PROJEKCIJ | FROM CONIC TO CYLINDRICAL MAP PROJECTIONS  | 363-373 |
UDK : 528.235:528.9  
Klasifikacija prispevka po COBISS.SI: 1.03
Prispelo: 10. 2. 2023
Sprejeto: 12. 6. 2023
DOI: 10.15292/geodetski-vestnik.2023.03.363-373
OTHER SCIENTIFIC ARTICLE
Received: 10. 2. 2023
Accepted: 12. 6. 2023
Miljenko Lapaine
OD KONUSNIH 
DO CILINDRIČNIH 
KARTOGRAFSKIH PROJEKCIJ
FROM CONIC TO 
CYLINDRICAL MAP 
PROJECTIONS
In books and textbooks on map projections, cylindrical, 
conic and azimuthal projections are usually considered 
separately. It is sometimes mentioned that cylindrical and 
azimuthal projections can be interpreted as limiting cases 
of conic, but this is rarely proven. The goal of this article is 
to show in a rigorous and systematic way how to generally 
approach solving the problem of transition from a conic to 
a corresponding cylindrical projection. This article points to 
the fact that J. H. Lambert showed as early as 1772 that a 
conformal cylindrical projection is created from a conformal 
conic projection. Following his idea, this paper shows that 
not only conformal, but also equal-area and equidistant 
cylindrical projections can be derived from corresponding 
conic map projections. Although it seems that the paper 
deals with quite well known and intuitive property of 
conic projections, it will also show that the transition from 
the conic to the corresponding cylindrical projection is not 
always possible. 
V knjigah in učbenikih o kartografskih projekcijah se 
cilindrične, konusne in azimutne projekcije običajno 
obravnavajo ločeno. Včasih se omenja, da je mogoče 
cilindrične in azimutne projekcije interpretirati kot robne 
primere konusnih, vendar je to redko dokazano. Cilj tega 
članka je natančno in sistematično pokazati, kako na 
splošno pristopiti k reševanju problema prehoda iz konusne 
v ustrezno cilindrično projekcijo. Članek opozarja, kako 
je J. H. Lambert že leta 1772 dokazal, da je konformna 
cilindrična projekcija ustvarjena iz konformne konusne 
projekcije. Na podlagi te zamisli je prikazano, kako je 
mogoče iz pokončnih konusnih izpeljati ne le konformne, 
temveč tudi ekvivalentne (enakoploščinske) in ekvidistantne 
(enakodolžinske) cilindrične kartografske projekcije. Čeprav 
se zdi, da prispevek obravnava precej dobro znano in 
intuitivno lastnost konusnih projekcij, bo tudi pokazal, 
da prehod iz konusne v ustrezno cilindrično projekcijo ni 
vedno mogoč. 

| 364 |
| 67/3 | GEODETSKI VESTNIK  
RECENZIRANI ČLANKI | PEER-REVIEWED ARTICLES
SI | EN
Miljenko Lapaine | OD KONUSNIH DO CILINDRIČNIH KARTOGRAFSKIH PROJEKCIJ | FROM CONIC TO CYLINDRICAL MAP PROJECTIONS  | 363-373 |
1 MOTIVATION AND INTRODUCTION
In books and textbooks on map projections, cylindrical, conic and azimuthal projections are usually 
considered separately, and sometimes it is mentioned that cylindrical and azimuthal projections can be 
interpreted as limiting cases of conic projections (Lee, 1944; Kavrayskiy, 1958, 1959; Jovanović, 1983; 
Vakhrameyeva et al., 1986; Snyder, 1987; Kuntz, 1990; Canters, 2002; Monmonier, 2004; Serapinas, 
2005), but it is rarely attempted to prove (Lambert, 1772; Hinks, 1912; Hoschek, 1969; Daners, 2012). 
This fact was the motivation for the research that preceded the writing of this article.
Lambert (1772) finished his derivation of the transformation from the conformal conic projection to the 
Mercator projection without any explanation of the connecton of his final expression and the Mercator 
projection.
Hinks (1912) uses Lambert's approach for the construction of cylindrical projections as the limiting 
cases of a conic projections. For a simple equal-area projection, with one standard parallel he missed to 
give a derivation. 
Hoschek (1969) derives the equations of cylindrical equidistant, equal-area and conformal projections as 
limiting cases of corresponding conic projections, whereby he interprets conic projections as projections 
of a sphere onto a cone, not into a plane. In doing so, he uses expansions in the Taylor series, but when 
neglecting terms of a higher order, he makes the mistake of assuming that the indefinite expression 0∙∞ 
is equal to 0, which in general does not have to be the case.
Daners (2012) derives the Mercator projection equations as the limiting case of a conformal conic 
projection of a sphere onto a cone that touches the sphere along a parallel. Daners does not deal with 
equidistant, equal-area or other projections.
The goal of this article is to show in a rigorous and systematic way how to generally approach solving 
the problem of transition from a conic to a corresponding cylindrical projection. We will also show that 
the transition from the conic to the corresponding cylindrical projection is not always possible. The 
transition from the conical to the corresponding azimuth projection is direct, there are no problems or 
open questions with it, so we will not deal with it.
The following section demonstrates Lambert’s derivation of the Mercator projection as a special case of 
the conformal conic projection which is further explained in the section 3. Section 4 presents generic 
derivations of cylindrical as special case of conic projections having conformal, equal-area and equidistant 
properties. Final section presents concluding remarks.
2 LAMBERT'S DERIVATION OF THE MERCATOR PROJECTION AS A SPECIAL CASE OF THE 
CONFORMAL CONIC PROJECTION
The beginnings of the theory of mapping one surface to another belong to Johann Heinrich Lambert 
(1728–1777), who dealt with the general problem of mapping a sphere and an ellipsoid into a plane in 
the chapter Anmerkungen und Zusätze zur Entwerfung der Land und Himmelscharten (Notes and addi-
tions to the establishment of maps of the Earth and the Sky) printed in to the third part of his Beyträge 
zum Gebrauche der Mathematik und deren Anwendung (Contributions to the use of mathematics and its 
application) (Lambert, 1772).

| 365 |
GEODETSKI VESTNIK | 67/3 |
RECENZIRANI ČLANKI | PEER-REVIEWED ARTICLES SI | EN
Miljenko Lapaine | OD KONUSNIH DO CILINDRIČNIH KARTOGRAFSKIH PROJEKCIJ | FROM CONIC TO CYLINDRICAL MAP PROJECTIONS  | 363-373 |
Lambert was the first mathematician who dealt with the general properties of map projections. He 
first considered the properties of conformality and equivalence and pointed to the fact that these two 
properties are mutually exclusive. In the aforementioned book, he published seven new map projections 
that he did not name, but today they are known as:
1. The Lambert conformal conic
2. The transverse Mercator
3. The Lambert azimuthal equal-area
4. The Lagrange projection
5. The Lambert cylindrical equal-area
6. The transverse cylindrical equal-area
7. The Lambert conic equal-area
The Lambert conformal conic projection is one of the most famous map projections. It is still used 
today in many countries. E.g. in Croatia, it is the official projection for topographic maps of smaller 
scales.
This projection was proposed by Lambert in his Notes and Supplements to the Establishment of Earth and 
Sky Maps published 250 years ago. The fourth subsection is entitled A more general method of representing 
a spherical surface so that all angles preserve their sizes. That subsection is further divided into paragraphs 
and begins with §47 in which Lambert writes that the stereographic representation of the spherical 
surface, as well as Mercator nautical charts, have the property that all angles retain the size they had on 
the surface of the globe. This gives the greatest resemblance that an arbitrary small figure can have in the 
plane to that drawn on the surface of the sphere.
Until then, the question of whether this property appears only in the two mentioned projections or 
whether it can be reached from one to the other, whose representations are so different, can be reached 
with the help of intermediate steps. In §48 Lambert derives the formula for the conformal projection 
of the unit sphere and arrives at the equation 
1
(tan )
2
m
x ε = , i.e. with today's usual notation 
 tan (45 ),
2
m
ϕ
ρ = − ° (1)
where ϕ is latitude, m parameter, 0 < m < 1, and ρ polar radius in the polar coordinate system in the 
plane of projection. In addition to (1), the equation (2) should also be written 
 θ = mλ, (2)
where λ is longitude, and θ polar angle in the polar coordinate system in the plane of projection. Equations 
(1) and (2) define the Lambert conformal conical projection of the unit sphere for 0 < m < 1. For m = 1 it 
is a polar aspect stereographic projection (conformal azimuthal projection) (Snyder, 1987; Peterca, 2001; 
Frančula, 2004). However, for m = 0 it is not clear at first glance that it will be the Mercator projection 
(normal aspect conformal cylindrical projection). In fact, for m = 0, (1) and (2) give ρ = 1 and θ = 0, 
and these are obviously not the equations of the Mercator projection. Here's how Lambert approaches 
it. In §50 it is written (with today's notation):
For the Mercator projection, m = 0. This only gives ρ = 1. We can write

| 366 |
| 67/3 | GEODETSKI VESTNIK  
RECENZIRANI ČLANKI | PEER-REVIEWED ARTICLES
SI | EN
Miljenko Lapaine | OD KONUSNIH DO CILINDRIČNIH KARTOGRAFSKIH PROJEKCIJ | FROM CONIC TO CYLINDRICAL MAP PROJECTIONS  | 363-373 |
 
1 tan
2
tan 45
2
1 tan
2
m
m
ϕ
ϕ
ρ
ϕ

−


= −=
 


+

° 
which gives
 
22
11
tan tan tan . tan tan .
222
11
222
m
mm
m m etc m m etc
ϕϕϕϕ
ρ
−+
  
=−+ −−+ −
  
  
so that for m = 0 it is obtained
 
357
1 111 1
tan tan tan tan . lncot 45 .
2 23 25 27 2 2 2
etc
m
ρϕϕϕϕ ϕ −

= + + + + = °−


Here 
1
m
ρ −
 shows the degrees calculated from the equator and they increase proportionally with 
lncot 45 lntan 45
22
ϕϕ

°− = °+


. This concludes Lambert's derivation for the Mercator projection. 
3 THE MERCATOR PROJECTION AS A SPECIAL CASE OF THE LAMBERT CONFORMAL 
CONIC PROJECTION
The first thing we can notice is that every map projection is determined by two equations, usually in 
the form
 x = x(ϕ, λ), y = y(ϕ, λ), (3)
where ϕ, λ are latitude and longitude, respectively. For normal aspect cylindrical projections, such as 
the Mercator, the equations are simpler
 x = x(λ), y = y(ϕ), (4)
if we have chosen a mathematical coordinate system. However, in deriving the formula for the Mercator 
projection, Lambert does not mention the expression for the function x = x(λ). He probably thought 
that x = λ was too simple or obvious to not mention it. This equation can be derived from the Cauchy
-Riemann conditions for conformal mappings, which he probably knew about because they were first 
used by d'Alembert (1717−1783). Regardless of the aforementioned, in §71 from the same Lambert 
book, we can read that the Mercator projection is described by the equations
 
, lntan 45 .
2
xy
ϕ
λ

= = °+


 (5)
Lambert finished his derivation of the transformation from the conformal conic projection to the Mer-
cator projection without comment or explanation about the connection between the limiting value of 
the fraction 
1
m
ρ −
 and the Mercator projection. So, we will do that in this section.
Let us first notice that when Lambert says that he will calculate 
1
m
ρ −
 for m = 0, he actually means the 
limiting value of that fraction when m → 0. In doing so, he uses series development, which was common 

| 367 |
GEODETSKI VESTNIK | 67/3 |
RECENZIRANI ČLANKI | PEER-REVIEWED ARTICLES SI | EN
Miljenko Lapaine | OD KONUSNIH DO CILINDRIČNIH KARTOGRAFSKIH PROJEKCIJ | FROM CONIC TO CYLINDRICAL MAP PROJECTIONS  | 363-373 |
for him and at that time. Also, he does not use limes label for calculation the limit value, as is common 
nowadays. Instead of using series development, it would be simpler to see that when m → 0 then ρ → 1. 
This means that 
0
1
lim
m
m
ρ
→
−
= is of the form 
0
0
, so it can be solved by applying the l'Hôpital rule:
 
00 0
1
lim lim limtan 45 lntan 45 lncot 45
1 22 2
lntan 45 .
2
m
mm m
d
dm
m
ρ
ρ ϕϕϕ
ϕ
→→→
−
−

= = − °− °− = °− =



= °+


 (6)
We can reach the same result, but without applying l'Hôpital's rule, if we observe that lnρ can be de-
veloped in the series
 
23
( 1) ( 1)
ln 1
22
ρρ
ρρ
−−
= − − + − ⋅⋅⋅
 (7)
and that lnρ is closer to ρ − 1 the closer ρ is to 1. So,
 
0 00
lntan 45
1 ln 2
lim lim lim lntan 45 lntan 45 .
22
m
m mm
mm m
ϕ
ρρ ϕ ϕ
→→→

°−

−
 
= − = − = − °− = °+


 (8)
In order to better understand why Lambert considered the fraction 
1
m
ρ −
, we will use the explanation of 
Hinks (1912). On page 70 of his book, Hinks (1912) states that a normal aspect cylindrical projection is 
a special case of a normal aspect conic projection when the pole of the projection goes to infinity. Parallels 
then become straight lines, ρ is infinite for all parallels, m is zero. Formulas for conic projections should 
be adapted to give not ρ
0
 and ρ, but ρ
0
 − ρ, the distance between any chosen parallel and the standard 
parallel. On page 103 of the same book, he examines the Mercator projection in more detail. It begins 
with the general case of a conformal conic projection with one standard parallel (in our notation and 
with a sphere radius equal to 1):
 
tan 45
2
m
K
ϕ
ρ

= °−


 (9)
with
 
0
cos
.
tan 45
2
m
K
m
ϕ
ϕ
=

°−


 (10)
If our cone becomes a cylinder tangent to the sphere along the equator, it is clear that K and ρ become 
infinitely large, and that the cone constant m is equal to zero. But in that case, it is
 
0
cos0
1.
tan (45 )
Km = =
°
 (11)
Furthermore,
 
00
() . K ρ ρϕ = = (12)

| 368 |
| 67/3 | GEODETSKI VESTNIK  
RECENZIRANI ČLANKI | PEER-REVIEWED ARTICLES
SI | EN
Miljenko Lapaine | OD KONUSNIH DO CILINDRIČNIH KARTOGRAFSKIH PROJEKCIJ | FROM CONIC TO CYLINDRICAL MAP PROJECTIONS  | 363-373 |
Now we can write
 
0
1 tan 45
2
tan 45 1 tan 45
22
1 tan 45
2
m
mm
m
K K K Km
m
m
ϕ
ϕϕ
ρρ
ϕ

− °−

   
− = − °− = − °− = =
  
  

− °−


=
 (13)
and due to (8) and (5) we have
 
0
00
1 tan 45
2
lim( ) lim lntan 45 .
2
m
mm
m
ϕ
ϕ
ρρ
→→

− °−

 
− = = °+


 (14)
Since the equations of the normal aspect Mercator projection are (5) then we clarified the result of Lam-
bert's approach to the Mercator projection (conformal cylindrical) as a special case of the conformal conic.
4 CYLINDRICAL PROJECTION AS A SPECIAL CASE OF CONIC PROJECTION − GENERAL 
CASE
A more general question now arises: how we from some normal aspect conic projection given by the 
equations in the polar coordinate system in the plane of the projection
 θ = mλ, ρ = ρ(ϕ) (15)
could derive the equations of a normal aspect cylindrical projection in a rectangular system in the pro-
jection plane
 x = nλ, y = y(ϕ) (16)
which will have the same properties as the conic given by equations (15)? For example, if the equal-area 
conic projection is given by equations (15), what are the equations (16) of the corresponding equal-area 
cylindrical projection. Or vice versa, if the equal-area cylindrical projection is given by equations (16), 
what are the equations of corresponding equal-area conic projections? We will give the answer in this 
section.
Let the normal aspect conic projection be given by the equations in the polar coordinate system in the 
plane of projection (15) where ϕ and λ are latitude and longitude, respectively, on the unit sphere, m 
is a parameter, 0 < m < 1. It is known from the literature that the local linear scale factors along the 
meridian and parallel are (Peterca, 2001; Frančula, 2004)
 
,.
cos
dm
hk
d
ρρ
ϕϕ
= −=
 (17)
Furthermore, the equations of the normal aspect cylindrical projection in the rectangular coordinate 
system in the projection plane are (16) where ϕ and λ are latitude and longitude, respectively, on the 
unit sphere, n is a parameter, usually n > 0. It is known from the literature that the local linear scale 

| 369 |
GEODETSKI VESTNIK | 67/3 |
RECENZIRANI ČLANKI | PEER-REVIEWED ARTICLES SI | EN
Miljenko Lapaine | OD KONUSNIH DO CILINDRIČNIH KARTOGRAFSKIH PROJEKCIJ | FROM CONIC TO CYLINDRICAL MAP PROJECTIONS  | 363-373 |
factors along the meridian and parallel are (Peterca, 2001; Frančula, 2004)
 ,.
cos cos
dy dx n
hk
dd ϕ ϕλ ϕ
= = = (18)
4.1 Conformal projections
Let h = k, i.e. we are working with conformal mappings. From (17) it follows
 
,
cos
dm
d
ρρ
ϕϕ
−=
 (19)
or
 
.
cos
d md ρϕ
ρϕ
−=
 (20)
Similarly, from (18) it follows
 
,
cos
dy n
dϕϕ
=
 (21)
or
 
.
cos
nd
dy
ϕ
ϕ
=
 (22)
By comparing relations (20) and (22) we have
 
nd
dy
m
ρ
ρ
= −
 (23)
and from there after integration
 
ln .
n
y const
m
ρ = −+
 (24)
Vice versa,
 
dm
dy
n
ρ
ρ
= −
 (25)
so it is after integration
 
ln .
m
y const
n
ρ= −+
 (26)
which is equivalent to (24).
If we substitute Lambert's ρ from (1) in (24), we get
 ln lntan 45 lntan 45 ,
22
n
yn n
m
ϕϕ
ρ

= − = − °− = °+


 (27)
where we chose y = 0 for ϕ = 0, i.e. const. = 0. For all normal aspect cylindrical projections, it is (Peterca, 
2001; Frančula, 2004)

| 370 |
| 67/3 | GEODETSKI VESTNIK  
RECENZIRANI ČLANKI | PEER-REVIEWED ARTICLES
SI | EN
Miljenko Lapaine | OD KONUSNIH DO CILINDRIČNIH KARTOGRAFSKIH PROJEKCIJ | FROM CONIC TO CYLINDRICAL MAP PROJECTIONS  | 363-373 |
 x = n(λ − λ
0
). (28)
Thus, by applying formula (24), we obtained the equation for y = y(ϕ) of the normal aspect Mercator 
projection directly from the equation for ρ = ρ(ϕ) of the normal aspect Lambert conformal projection. 
Equations (27) and (28) are the equations of the normal aspect Mercator projection of the sphere (Snyder, 
1987; Peterca, 2001; Frančula, 2004).
4.2 Equal-area projections
Let hk = 1, i.e. we are working with equal-area mappings. From (17) it follows
 
1,
cos
dm
d
ρρ
ϕϕ
−=
 (29)
or
 
cos
.
d
d
m
ϕϕ
ρρ −=
 (30)
Similarly, from (18) follows
 
1,
cos
dy n
dϕϕ
=
 (31)
or
 
cos
.
d
dy
n
ϕϕ
=
 (32)
By comparing relations (30) and (32) we have
 ,
m
dy d
n
ρρ = − (33)
and from there after integration
 
2
.
2
m
y const
n
ρ
= −+ (34)
Vice versa,
 
n
d dy
m
ρρ = −
 (35)
so it is after integration
 
2
.
2
n
y const
m
ρ
= −+
 (36)
which is equivalent to (34). 
By solving the differential equation (30), we can obtain the equation of the normal aspect conic equal-area 
projection (Lapaine and T riplat Horvat, 2014)

| 371 |
GEODETSKI VESTNIK | 67/3 |
RECENZIRANI ČLANKI | PEER-REVIEWED ARTICLES SI | EN
Miljenko Lapaine | OD KONUSNIH DO CILINDRIČNIH KARTOGRAFSKIH PROJEKCIJ | FROM CONIC TO CYLINDRICAL MAP PROJECTIONS  | 363-373 |
 
2
sin , C
m
ρϕ = − (37)
where C is a constant. If we substitute that ρ in (26), we get
 
2
sin
.,
2
m
y const
nn
ρϕ
= −+=
 (38)
where we chose y = 0 for ϕ = 0, i.e. .
2
Cm
const
n
= . The equation (28) remains to be valid also for the 
normal aspect equal-area cylindrical projection.
Thus, by applying formula (34), we obtained the equation y = y(ϕ) of the normal aspect equal-area cylin-
drical projection directly from the equation ρ = ρ(ϕ) of the normal aspect equal-area conic projection. 
Equations (38) and (28) are the equations of the normal aspect equal-area cylindrical projection of the 
sphere (Snyder, 1987; Frančula, 2004).
4.3 Equidistant projections along meridians
Let h = 1, i.e. we are working with equidistant mappings along meridians. It follows from (17)
 
1,
d
d
ρ
ϕ
−=
 (39)
or
 −dρ = dϕ. (40)
Similarly, from (18) it follows
 
1,
dy
d ϕ
=
 (41)
or
 dy = dϕ. (42)
By comparing relations (40) and (42) it is not difficult to find a connection
 dy = −dρ (43)
and from there after integration
 y = −ρ + const. (44)
Vice versa,
 dρ = −dy (45)
so it is after integration
 ρ = − y + const. (46)
which is equivalent to (44).

| 372 |
| 67/3 | GEODETSKI VESTNIK  
RECENZIRANI ČLANKI | PEER-REVIEWED ARTICLES
SI | EN
Miljenko Lapaine | OD KONUSNIH DO CILINDRIČNIH KARTOGRAFSKIH PROJEKCIJ | FROM CONIC TO CYLINDRICAL MAP PROJECTIONS  | 363-373 |
By solving the differential equation (40), we can obtain the equation of the normal aspect conic pro-
jection equidistant along meridians
 ρ = C − ϕ, (47)
where C is a constant. If we substitute that ρ in (44) we can get
 y = −ρ + const. = ϕ (48)
where we chose y = 0 for ϕ = 0, i.e. const. = C. Thus, by applying formula (44), we obtained the equation 
of the normal aspect cylindrical projection equidistant along meridians directly from the equation of the 
normal aspect conic projection equidistant along meridians. Equations (48) and (28) are the equations 
of a normal aspect cylindrical projection equidistant along meridians (Snyder, 1987; Frančula, 2004).
4.4 Equidistant projections along parallels
Let k=1, i.e. we are working with equidistant mappings along parallels. It follows from (17)
 1,
cos
m ρ
ϕ
= (49)
or
 
cos
,
m
ϕ
ρ =
 (50)
Such a projection is not found in cartographic literature, except for m=1 when it comes as the ortho-
graphic polar projection. Thus, we can consider it a generalization of the orthographic projection. From 
(18) it follows
 
1,
cos
n
ϕ
=
 (51)
or
 n = cosϕ. (52)
Since n is a constant quantity, and (52) should be valid for all values of ϕ, it is immediately clear that 
from a normal aspect conic projection equidistant along parallels it is not possible to obtain a normal 
aspect cylindrical projection equidistant along parallels, because such does not exist.
5 CONCLUSION
Johann Heinrich Lambert (1772) derived the equations of a normal aspect conformal conic projection 
and showed how these equations can be used to get the equations of a normal aspect conformal cylindrical 
(Mercator) projection. In this paper Lambert's derivation is discussed and further explained. This gave 
the motivation to investigate the possibility of transition from some conic to corresponding cylindrical 
projection with the same properties (conformality, equal-area, equidistance). In this sense, the appropriate 
formulas for conformal, equal-area and equidistant projections along meridians are derived. In addition, 
it was shown that an analogous procedure is not possible for projections equidistant along parallels. 

| 373 |
GEODETSKI VESTNIK | 67/3 |
RECENZIRANI ČLANKI | PEER-REVIEWED ARTICLES SI | EN
Miljenko Lapaine | OD KONUSNIH DO CILINDRIČNIH KARTOGRAFSKIH PROJEKCIJ | FROM CONIC TO CYLINDRICAL MAP PROJECTIONS  | 363-373 |
Acknowledgement
The author thanks the anonymous reviewers for their detailed reading and very useful comments on the 
first and second versions of the article.
Literature and references:
Canters, F . (2002), Small-Scale Map Projection Design, CRC Press, London.
Daners, D. (2012). The Mercator and Stereographic Projections, and Many in Between, 
The American Mathematical Monthly, 119:3, 199-210, doi: 10.4169/amer.
math.monthly.119.03.199 
Frančula, N. (2004). Kartografske projekcije. Skripta. Geodetski fakultet Sveučilišta 
u Zagrebu, Zagreb.
Hinks, A. R. (1912). Map projections: Cambridge, Eng., Cambridge Univ. Press, 126 
p. [2nd ed., 1921, 158 p.]
Hoschek, J. (1969). Mathematische Grundlagen der Kartographie: Mannheim, Zürich, 
Bibliographisches Institut, 167 p. [2nd ed., 1984, 210 p.]
Jovanović, V . (1983). Matematička kartografija, Vojnogeografski institut, Beograd.
Kavrayskiy, V . V . (1958). Izbrannye trudy, T om II, Matematicheskaya kartografiya, Vyp. 
1, Obshchaya teoriya kartograficheskih proekciy. Izdanie Upravleniya nachal'nika 
Gidrograficheskoy Sluzhby VMF .
Kavrayskiy, V . V . (1959). Izbrannye trudy, T om II, Matematicheskaya kartografiya, Vyp. 
2, Konicheskie i cilindricheskie proekcii, ih primenenie. Izdanie Upravleniya 
nachal'nika Gidrograficheskoj Sluzhby VMF .
Kuntz, E. (1990). Kartennetzentwurfslehre, Wichmann, Karlsruhe.
Lambert, J. H. (1772). Beyträge zum Gebrauche der Mathematik und deren 
Anwendung, Dritter Theil, VI. Chapter: Anmerkungen und Zusätze zur Entwerfung 
der Land- und Himmelscharten. Berlin. Reprint in German, 1894, Ostwald’s 
Klassiker der Exakten Wissenschaften, no. 54: Leipzig, Wilhelm Engelmann, 
uredio Albert Wangerin. Translated into English and wrote the introduction W 
R Tobler: Notes and Comments on the Composition of Terrestrial and Celestial 
Maps: Ann Arbor, Univ. Michigan, 1972, Mich. Geographical Publication no. 8, 
125 p. Second edition Esri Press, Redlands, 2011.
Lapaine, M., Triplat Horvat, M. (2014). Determination of Standard Parallels Based on 
Parameters of Normal Aspect Conical Projections. 5th International Conference 
on Cartography and GIS, Riviera, Bulgaria, 15.–21. 6. 2014. Proceedings (eds. 
T emenoujka Bandrova, Milan Konecny), ISSN 1314-0604, Bulgarian Cartographic 
Association, 143–153.
Lee, L. P . (1944). The nomenclature and classification of map projections: Empire 
Survey Review, v. 7, no. 51, p. 190-200.
Monmonier, M. (2004). Rhumb Lines and Map Wars: A Social History of the Mercator 
Projection, University of Chicago Press, Chicago, 2004.
Peterca, M. (2001). Matematična kartografija / Kartografske projekcije, Univerza v 
Ljubljani, Fakulteta za gradbeništvo in geodezijo, Ljubljana.
Serapinas, B. B. (2005). Matematicheskaya kartografiya, Academa, Moscow.
Snyder , J. P . (1987). Map Projections – A Working Manual, U.S. Geological Survey Prof. 
Paper 1395, 383 p. Reprinted 1989 and 1994 with corrections.
Vakhrameyeva, L. A., Bugayevskiy, L. M., Kazakova, Z. L. (1986). Matematicheskaya 
kartografiya, Moscow, Nedra.
Prof. emer. Miljenko Lapaine 
University of Zagreb, Faculty of Geodesy 
Kačićeva 26, 10000 Zagreb 
e-mail: mlapaine@geof.hr
Lapaine M. (2023). From Conic to Cylindrical Map Projections. Geodestski vestnik, 67 (3), 363-373. 
DOI: https://doi.org/ 10.15292/geodetski-vestnik.2023.03.363-373