Cyl i nd ri cal Projecti ons as a Li m i ti ng Case of Coni c Projecti ons

. Lambert (1 772) derived the equation of the Mercator projection as a limiting case of a conformal conic projection. In this paper, we give a derivation for equidistant, equal-area, conformal and perspective cylindrical projections as limiting cases of equidistant, equal-area, conformal and perspective conic projections. In this article the conic and cylindrical projections are not projections on a cone or a cylinder whose surfaces are cut and developed into a plane, but rather mappings of the sphere directly into the plane. Exceptions are projections that are defined as mappings on the surface of a cone or plane, as is the case with perspective projections. In the end, we prove that it is not always possible to obtain a corresponding cylindrical projection as a limiting case from a conic projection, as one might conclude at first glance. Therefore, the final conclusion is that it is not advisable to interpret cylindrical projections as limiting cases of conic projections.


Introduction
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 ones (Lee 1944, Kavrayskiy 1958, 1959, Jovanović 1983, Vakhrameyeva et al. 1986, Snyder 1987, Kuntz 1990, Canters 2022, Monmonier 2004, Serapinas 2005, but there are only a few attempts to prove it (Hinks 1912, Hoschek 1969, Daners 2012. Kimerling (2010) in his blog introduced a way of thinking about similarities among projections − that seemingly distinct projections may actually be parts ofa continuum ofprojections created by varying the parameters of a single pair of projection equations transforming the latitude and longitude coordinates on the sphere or ellipsoid into Cartesian coordinates on a flat projection surface. Such projection continuums are illustrated by animations that show the changes in the graticule and coastline as a certain projection parameter is varied systematically through a large range of values. Kimerling also provides animations showing the transition from conic to cylindrical or azimuthal projections.
In this paper, we will supplement the explanations and derivations of Hinks (1912), who applies Lambert's method of derivation (1772), although he does not mention Lambert.
The Lambert conformal conic projection is one of the most famous map projections. It is still used today in many countries. This projection was proposed by Lambert in his Notes and Supplements to the Establishment of Earth and Sky Maps published 250 Since the author is the journal's managing editor, the peer review process and independent editorial decision were performed by an external editor, Prof. Emer. Nedjeljko Frančula. We thank Prof. Emer. Nedjeljko Frančula for his help in addressing potential managing editor's conflict of interest.
Lambertova je konformna konusna projekcija jedna od najpoznatijih kartografskih projekcija. I danas se koristi u mnogim zemljama. Tu je projekciju predložio Lambert u svojim Bilješkama i dodatcima za uspostavljanje karata Zemlje i neba objavljenima prije 250 godina. Četvrti je pododjeljak Općenitija metoda predstavljanja sferne površine tako da svi kutovi zadrže svoje veličine. Taj je pododjeljak dalje podijeljen i počinje s §47 u kojem Lambert piše da S obzirom na to da je autor izvršni urednik ovoga časopisa, recenziranje je obavio i neovisnu uredničku odluku donio vanjski urednik prof. emer. Nedjeljko Frančula. Zahvaljujemo prof. emer. Nedjeljku Frančuli na pomoći vezanoj uz potencijalni sukob interesa izvršnog urednika. years ago. The fourth subsection is A more general method of representing a spherical surface so that all angles preserve their sizes. That subsection is further divided 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. In §48 Lambert derives the formula for the conformal projection of the unit sphere. After that he derived the Mercator projection as a limiting case of his conformal conic projection. Hinks (1912) uses the same method as Lambert for the construction ofa cylindrical projection as a limiting case of a conic projection. From the conic equidistant along the meridian (simple conic) he derives the cylindrical equidistant projection (in French projection plate carrée, in German quadratische Plattkarte). From the conformal conic (conic orthomorphic) he derives the conformal cylindrical projection (cylindrical orthomorphic -Mercator). For a simple equal-area projection with one standard parallel he does not give a derivation.
In this paper, we also give a derivation for the equal-area cylindrical projection as a limiting case of the equal-area conic projection. In addition, we give a derivation for the central perspective cylindrical projection as a limiting case of the central conic perspective projection.
It should be emphasized that Hinks defines a standard parallel as a parallel of true length: "One parallel, and sometimes a second, is made ofthe true length; that is to say, if the map is to be on the scale of one-millionth, the length of the complete parallel on the map will be one-millionth of the corresponding terrestrial parallel. This is called a Standard parallel." This definition does not correspond to today's understanding of distortions, according to which one should distinguish between standard parallels, equidistantly mapped parallels and parallels that have preserved their length in mapping.
Furthermore, Hinks implies in his derivations that a conic projection is a projection on a cone and that a cylindrical projection is a projection on a cylinder. In this paper, the conical and cylindrical projections are not projections on a cone or a cylinder whose surfaces are cut and developed into a plane, but rather mappings of the sphere directly into the plane. Exceptions are projections that are defined as mappings on the surface ofa cone, as is the case with perspective projections.
Finally, we prove that it is not always possible to obtain a corresponding cylindrical projection from a conic projection, as one might conclude at first glance. Although very simple, it is the most important contribution ofthis article.
The equations of normal aspect conic projections are usually given in the polar coordinate system in the plane ofthe projection: where and (-π,π) are the latitude and longitude, respectively, m the parameter, 0 m 1, and the polar coordinates in the plane ofprojection ( Figure 1).
Before that, let us recall that the factors of the local linear scale along the meridian and the parallel, respectively, for conic projections of the sphere of radius R are (Snyder 1987) Ifk( 0 ) 1 holds for some 0 , then according to (3) we have where we noted 0 ( 0 ). The equations ofany normal aspect cylindrical projection are (2), where and are the latitude and longitude, respectively, n the parameter, usually 0 n 1, and x and y the rectangular coordinates in the plane of projection ( Figure 2).
Ifwe write simply we obtained the equations of the cylindrical projection from the equations of the conic projection (1) without any problems and without any recalculation. However, it can be easily shown that in this way some properties of the conic projection (1), e.g. equal-area or conformality, will not be preserved. Therefore, the procedure described by relations (5) does not provide the desired solution.
If we substitute m 1 in (1), we will get the azimuthal projection equations: Ifwe substitute m 0 in (1), we will get 0, ( ), which would mean that the entire sphere was mapped to a straight line or part ofa straight line. So, to obtain a cylindrical projection from (1) with m 0, we must add another condition to prevent the image of the sphere being compressed into a straight line. We can achieve this, for example, by requiring that one parallel that is equidistantly mapped by a conic projection also be equidistantly mapped by a cylindrical projection. In the following sections, we will show how a cylindrical projection can be obtained as a limiting case of a conic one using examples of equidistant along the meridian, equal-area, conformal and central perspective projections.

Projections equidistant along meridians
For the normal aspect conic projection ofa sphere of radius R given by (1) to be equidistant along the meridians, the condition that the local linear scale factor along the meridians is equal to 1 must be met: Integrating equation (8) gives where C is a constant, to make 0 for each value of latitude. So, in the polar coordinate system, the equations of the conic projection that is equidistant along the meridians read: Let us suppose that 0 is the latitude of the equidistantly mapped parallel in that projection. Considering (4), we can write and from there we have and then Following Lambert (1772) and Hinks (1912), let us consider the difference 0 . Although both and 0 tend to infinity when m 0, their difference is finite regardless ofm: This allows us to write the equations of a cylindrical projection equidistant along the meridians . parametar, obično 0 n 1, a x i y pravokutne koordinate u ravnini projekcije (slika 2).
Faktori lokalnog linearnog mjerila uzduž meridijana odnosno paralele za cilindrične projekcije sfere radijusa R su (Snyder 1987) Izračunajmo Dakle, to je cilindrična projekcija ekvidistantna uzduž meridijana. Faktor lokalnog linearnog mjerila za tu projekciju uzduž paralele kojoj odgovara 0 je The factors of the local linear scale along the meridian and the parallel, respectively, for cylindrical projections ofthe sphere ofradius R are (Snyder 1987) Let us calculate So, it is a cylindrical projection equidistant along the meridians. The local linear scale factor for that projection along the parallel to which the latitude corresponds 0 is For this parallel to be equidistantly mapped, k( 0 ) 1 should be true, i.e. n R cos 0 .
Thus, the equations of the normal aspect cylindrical projection equidistant along the meridians, which has the same equidistantly mapped parallel 0 as the conic projection (13) which is equidistant along the meridian, are x R cos 0 • , y R( 0 ).
If we translate the image of the projection by the amount R 0 in the direction ofthe y axis, we will achieve that the image ofthe equator is on the coordinate axis x, as is usual in cartographic literature. So, the final equations of the normal aspect cylindrical projection equidistant along the meridian, which has the same equidistantly mapped parallel ( 0 ) as the conic projection (13) are x R cos 0 • , y R .
3 Equal-area projections For the normal aspect conic projection of a sphere given by (1) to be equal-area, the condition that the product of the factors of the local linear scales along the meridian and along the parallel is equal to 1 must be satisfied, i.e. that Integrating equation (21) gives where C is a constant. Let us assume that Let us suppose that 0 is the latitude of the equidistantly mapped parallel in that projection. Considering (4) x n , Dakle, jednadžbe uspravne cilindrične projekcije ekvidistantne uzduž meridijana koja ima istu ekvidistantno preslikanu paralelu ( 0 ) kao konusna projekcija (13) su x R cos 0 • , y R( 0 ).
So, in the polar coordinate system, the equations ofthe conformal conic projections read: Let us suppose that 0 is the latitude of the equidistantly mapped parallel in that projection. Considering (4), we can write From (36) it follows Now we can calculate When m 0 then the last fraction is of the form 0/0. Applying l'Hôpital's rule, we will get Now we can write the equations of the conformal cylindrical projection in the rectangular coordinate system in the projection plane If we want the parallel corresponding to the latitude 0 to be equidistantly mapped, then it should be n R cos 0 (formula (18)). The equations of the conformal cylindrical projection are:  Jednadžbe uspravne perspektivne projekcije na konus mogu se napisati u polarnom koordinatnom sustavu u obliku (Lapaine i Frančula 1992): The equations ofthe normal aspect central perspective projection on the cone can be written in the polar coordinate system in the form (Lapaine, Frančula 1992): where is a parameter that can be described geometrically as half the angle at the apex of the cone onto which the sphere is mapped and d the distance ofthe surface ofthe cone from the center (Figure 3). Let us write If 0, then and 0 tend to infinity, but their difference is finite lim( 0 ) d(tan tan ).
(48) 0 For the parallel corresponding to the latitude 0 to be equidistantly mapped onto the cone according to formulas (45), considering (4) it should be true From (46) and (49) it follows From (50) we can conclude that an equidistant mapped parallel in perspective conic projection exists ifd R and that d R cos 0 holds for 0.
Ifwe want the parallel corresponding to the latitude 0 to be equidistantly mapped in the cylindrical projection, then it must be n Rcos 0 , as shown before.
Thus, the equations of the normal aspect central perspective on the cylinder in the rectangular coordinate system in the plane are x R cos 0 • , y R cos 0 (tan tan 0 ).
If we translate the image of the projection by the amount Rsin 0 in the direction of the y axis, we will achieve that the image of the equator is on the coordinate axis x, as is usual in cartographic literature. So, the final equations of the normal aspect perspective cylindrical projection, which has the same equidistantly mapped parallel ( 0 ) as the perspecitve conic projection (45) reads x R cos 0 • , y R cos 0 tan .
6 Projections equidistant along parallels For the normal aspect conic projection of the sphere ofradius R given by (1) to be equidistant along the parallels, the condition that the local linear scale factor along the parallels is equal to 1 must be met: From equation (51) amount R cos 0 ln in the direction ofthe y axis, we will achieve that the image ofthe equator is on the coordinate axis x, as is usual in cartographic literature. So, the final equations of the normal aspect conformal cylindrical projection, which has the same equidistantly mapped parallel 0 as the conic projection (34) are Ako želimo da paralela kojoj odgovara geografska širina 0 bude ekvidistantno preslikana pri cilindričnoj projekciji, tada mora biti n R cos 0 , kao što je prije pokazano.
Dakle, jednadžbe uspravne centralne ili gnomonske cilindrične perspektivne projekcije u pravokutnom sustavu u ravnini projekcije glase x R cos 0 • , y R cos 0 (tan tan 0 ). Pomaknemo li sliku projekcije za iznos R 0 u smjeru osi y, postići ćemo da slika ekvatora bude na koordinatnoj osi x kao što je uobičajeno u kartografskoj literaturi. Dakle, konačne jednadžbe uspravne konformne cilindrične projekcije koja ima istu ekvidistantno preslikanu paralelu ( 0 ) kao konusna projekcija (45)  When m 0 then 0 . In the theory of map projections, it is usually assumed that the functions defining map projections are real, single-valued, continuous, and differentiable functions of and in some domain and that their Jacobian determinant does not vanish (Tobler 1962). Therefore, in the described way, a cylindrical projection equidistant along the parallels cannot be obtained as a limiting case ofa conic projection equidistant along parallels. Such a projection does not exist at all because in all normal aspect cylindrical projections all parallels are of equal length, and this is not the case on a sphere. This example proves that not all conic projections have a cylindrical limiting case as it seems at first glance.

Conclusion
Lambert (1772) derived the formula for the conformal conic projection. In the same publication, Lambert derived the equation ofthe Mercator projection as a limiting case of a conformal conic projection. Hinks (1912) used the same method as Lambert to construct a cylindrical projection as a limiting case of a conic projection. From the conic equidistant along the meridians (simple conic) he derives the cylindrical equidistant projection. For a simple equal-area projection with one standard parallel, Hinks does not give a derivation.
In this article, we give a derivation for the equalarea cylindrical projection as a limiting case of the equal-area conic projection. In addition, we give a derivation for the central perspective cylindrical projection as a limiting case ofthe central conic projection.
Furthermore, Hinks implies in his derivations that a conic projection is a mapping on a cone and that a cylindrical projection is a mapping on a cylinder. In this article, the conic and cylindrical projections are not projections on a cone or a cylinder whose surfaces are cut and developed into a plane, but mappings of the sphere directly into the plane. Exceptions are projections that are defined as mappings on the surface of a cone or plane, as is the case with perspective projections.
The main result ofthis paper is that it is not always possible to obtain a corresponding cylindrical projection as a limiting case from a conic projection, as one might conclude at first glance, and no one has noticed this so far. So, it is not advisable to interpret cylindrical projections as limiting cases ofconic projections.