Equidistant, Standard and Secant Parallels

The paper describes a study of equidistant, standard and secant paral lels in normal aspect cyl indrical and conical projections. First, the explanation of cyl indrical and conical projections as projections on cyl indrical or conical surfaces is not recommended because it leads to misunderstanding projection properties. Furthermore, equidistant, standard and secant paral lels are often assumed to be identical in references. After defining these three types of parallels, it is proved that it is necessary to differentiate them in the theory of map projections and teaching it.


Introduction
Let us agree that a map projection is a mapping of a curved surface, for example the Earth's sphere or ellipsoid, into a plane. Changes that occur in such mapping are called distortions. We can differentiate distortions of length, area, and angles. If there is no distortion at any point of a line/curve we say that it is a zero distortion line or a standard line. If the same is true of a parallel, we say that it is a standard parallel. At first glance, these are familiar, generally accepted definitions, but this paper will show that a major problem exists and provides a correct, mathematically based approach.
First of all, let us remember that developing a cylindrical or conical surface into a plane is isometric mapping. In other words, all distances should remain the same after the developing procedure.
Obviously, there is no reason to use a developable surface in a definition of a map projection if developing will cause additional distortions. According to Deetz and Adams (1969, pages 10-11): There are some surfaces, however, that can be spread out in a plane without any stretching or tearing. Such surfaces are called developable surfaces and those like the sphere are called nondevelopable. The cone and the cylinder are the two wellknown surfaces that are developable. Any curve drawn on the surface will have exactly the same length after development that it had before. In the literature on map projections, the common interpretation is that the Earth's sphere or ellipsoid is first mapped onto an auxiliary developable surface, which is then developed in a plane and thus becomes a map (see, for example, Deetz and Adams 1969, Richardus and Adler 1972, Slocum et al. 2009, Clarke 2015. This approach is generally unsatisfactory. This is because the use of developable surfaces in the definition of map projections applies to only a small number of projections. If a projection is called cylindrical, it does not mean that it is a mapping onto a cylinder. Such an understanding is very common but wrong. Correct thinking is: a projection is called cylindrical if a map produced in such a projection can be bend in a cylindrical surface. The same approach holds for conical projections (Close and Clarke 1911). In addition to cylindrical and conical projections, there are many other types, such as azimuthal, pseudocylindrical, pseudoconical, conditional, and so on, which cannot be interpreted by mapping onto a cylinder or conical surface. Some attempts have been made in that direction. For example, a pseudocylindrical projection is interpreted as mapping onto an oval surface (oval, ovoid) (Clarke 2015), without noticing that such a surface is undevelopable. It does not make sense to map a sphere, which is an undevelopable surface, onto another undevelopable surface. Any surface can be transformed by animating it into any other surface. This simply confuses the essence of map projections.
Lokalno mjerilo LS je produkt glavnog mjerila PS i lokalnog linearnog faktora mjerila c: Budući da lokalno mjerilo također ovisi o smjeru, umjesto (1) bilo bi bolje pisati gdje smo promatrani smjer označili s . Distorzija duljina je razlika između 1 i faktora lokalnog mjerila duljina. Dakle ona se izražava brojem, projection and any touching or secant cone. The name conical is given to the group embraced by the above definition, because, as is obvious, a projection so drawn can be round to form a cone. Lee (1944)  The projections are termed cylindric or conic because they can be regarded as developed on a cylinder or cone, as the case may be, but it is as well to dispense with picturing cylinders and cones, since they have given rise to much misunderstanding. Particularly is this so with regards to the conic projections with two standard parallels: they may be regarded as developed on cones, but they are cones which bear no simple relationship to the sphere. The most famous cartographers, for instance Mercator and Lambert, did not use any intermediate or developable surfaces in the projections known today as the Mercator projection and Lambert conformal conical projection. The introduction of developable surfaces in the theory of map projections is artificial and unnecessary, in general. One of the contributions of this paper is the author's effort to show that the theory of map projections should be approached mathematically, not dogmatically. We advocate thinking critically about broadly accepted, but sometimes wrong, ideas and statements. Figure 1, taken from Deetz and Adams (1969), shows the intersection of a cone and sphere along two parallels. For Deetz and Adams (and many others), standard parallels and secant parallels are identical. This paper disagrees with such an approach. Our main goals are to propose definitions of three different types of parallels: secant, standard and equidistant, and to show that some broadly accepted facts about secant and standard parallels found in many references are incorrect and should be revised. This requires a critical approach by the reader to established customs in map projection teaching and research. In several earlier published papers (Lapaine 2015(Lapaine , 2017a(Lapaine , 2017b(Lapaine , 2018(Lapaine , 2019 it was shown that there are conical and azimuthal projections with three and more standard parallels. Obviously, this could not be explained by using the secant projections approach, and led to differentiating between standard and secant parallels. This paper goes further and introduces equidistant parallels in order to facilitate understanding and indicate the need to abandon the secant projections approach.

Terminology
Let us agree that the principal (linear) scale PS is the ratio of the length in the plane of projection and its origin on the surface (sphere, ellipsoid) to be projected/mapped. PS is usually indicated on maps because it determines the general degree of reduction of the length on the map. On most maps, it is usually simply called 'scale' and is known as the map scale.
It is well known that scale changes from point to point, and at certain points usually depends on direction. This is the local scale. The local linear scale factor c is the ratio ofthe differential ofthe curve arc in the plane of projection and the differential of the corresponding curve arc on the ellipsoid or spherical surface (more details in section 5).
The local scale LS is the product of the principal scale PS and the local linear scale factor c: LS PS c (1) Since the local scale also depends on the direction, then instead of(1) it would be more correct to write where we denote the observed direction as . Length distortion is the difference between 1 and the local linear scale factor. Distortion is expressed by a number, so if c( ) = 1 for each , then instead of the expression 'no distortion' it would be better to say 'distortion is equal to zero'. This definition is found in the Multilingual Cartographic Dictionary (Borčić et al. 1977) and the Geodetic-Geoinformatic Dictionary (Frančula and Lapaine 2008): line without distortion Line on the map along which there is no distortion oflengths, area and angles or lines along the principal scale is preserved at all points in all directions. For perspective projections, these are the lines by which the auxiliary surface or the projection plane touches (touching parallel, touching meridian) or intersects (secant parallel, secant meridian) the sphere or ellipsoid. Remarks: (1) For the line without distortion, the term standard line is used, e.g. standard parallel (2) The point where the projection plane is tangent (touches) the surface of the ball or ellipsoid is called the touch point. See: projection, azimuthal. English: line, standard. German: Berührungslinie. This definition actually contains two: a line without distortion and a line along which the principal scale has tako da ako je c( ) 1 za svaki , tada je umjesto izraza "bez distorzije" bolje reći "distorzija jednaka nuli".

Točke, linije i površine s distorzijama jednakima nuli
Reći ćemo da je u nekoj točki distorzija jednaka nuli ako je faktor lokalnog mjerila duljina jednak 1, tj. ako je c( ) 1, za svaki gdje je c( ) definirano u (4) KiG No. 34, Vol. 1 9, 2020, https://doi.org/1 0.32909/kg.1 9.34.3 been preserved. Furthermore, introducing perspective projections and, in this respect, the tangent and secant parallel or meridian in the definition of a line without distortion is not good because the secant parallel does not need to be simultaneously parallel without distortion. Also, there is disagreement between languages about naming a line without distortion. For instance, in English it is called a standard line, and in German, the touch line (Berührungslinie).
In Enzyklopädischer Wörterbuch Kartographie in 25 Sprachen (Neumann 1997)  (2) tangent meridian; (3) secant parallel; (4) secant meridian. The terminology in the other 23 languages follows. In the above definition in German, Berührungslinie is literally translated as touching line, and the other terms are touching parallel, touching meridian, secant parallel and secant meridian. The auxiliary surface (Hilfsabbildugsfläche) has a decisive role, while distortion is not mentioned at all! In the English definition in same dictionary, a standard line is "a line in map projection along which the scale is equal to the principal scale". The auxiliary surface is not mentioned explicitly, but the terms tangent parallel, tangent meridian, secant parallel and secant meridian occur. In French, it is the "contact line or intersection with auxiliary projection surface", and in Russian the "zero distortion line, i.e. a line on the map where the principal scale at each point is preserved". The same understanding appears in the Portuguese definition.
In ESRI's Web Dictionary (ESRI 2017) we find this definition: "Standard line = [map projections] A line on a sphere or spheroid that has no length compression or expansion after being projected; usually a standard parallel or central meridian". Thus, only what happens along the line is important, not around it, although it is known that the linear scale factor or scale at each point generally depends on the direction (Tissot indicatrix). Snyder and Voxland (1989) say this ofperspective conical projection. "Scale. True along one or two chosen standard parallels, which may be on the same side of or both sides of the Equator... Distortion. Free of distortion only along the one or two standard parallels." These statements are incorrect because a simple perspective conical projection cannot have two standard parallels.
From the above, we can conclude that there is terminological confusion. The standard line and secant line are not uniformly defined terms. Confusion appears to arise from misunderstandings and results in the acceptance of unfounded assertions. We will attempt to resolve this confusion in this paper.
The same idea can be applied to the transverse and oblique aspects of both cylindrical and conical projections.
In order to be mathematically correct in all the sections that follow, we refer to published papers by Lapaine (2015Lapaine ( , 2017aLapaine ( , 2017bLapaine ( , 2018Lapaine ( , 2019.

Points, Lines and Areas with Zero Distortions
Let us say that at some point the distortion is zero if the local linear scale factor is equal to 1, i.e. if it is where c( ) is defined in (4) ( 4) where E, F and G are coefficients in the first differential form of the map projection. We use a sphere intentionally to facilitate understanding and to shorten derivations.
Note there is a difference between zero and none. Zero is a number, while none is not. Due to the fact that distortion is a number, it would be preferably to use "zero distortion" instead of "no distortion".
U posebnom slučaju kad je F 0 možemo se koristiti s h i k umjesto a i b.
(3,3) The expression a b 1 cannot be true at all points of a two-dimensional area on the map, as this would mean map projection without distortion. Leonhard Euler first proved that this was not possible (Euler 1777).
In a special case, when F 0, then we can use h and k instead of a and b.
If at all points of a meridian we have h 1, then it is not generally a zero-distortion meridian, but a meridian along which, or in the direction of which, the distortion is zero. We can say that the meridian is mapped equidistantly.
If at all points of a mapped area h 1, then we say that the map projection is equidistant along meridians. For example, normal aspect of the Postel projection is azimuthal projection equidistant along all meridians.
Ifat all points ofa parallel we have k 1, then it is not generally a zero-distortion parallel, but a parallel along which, or in the direction ofwhich, the distortion is zero. We can say that the parallel is mapped equidistantly.
If at all points of a mapped area k 1, then we say that the map projection is equidistant along parallels. For example, normal aspect of the orthographic projection is azimuthal projection equidistant along parallels.
If at all points of a meridian we have h k 1, then it is a zero-distortion meridian, or the distortion of that meridian is zero in all directions, i.e. this is a standard meridian.
Ifat all points ofa parallel we have h k 1, then it is a zero-distortion parallel, or the distortion ofthat parallel is zero in all directions, i.e. this is a standard parallel.
where the unit circle. This should be clear to everyone, although it can be inferred from the introductory chapter that it is not. In a special case, when F 0, the expressions (5) can be written like this h k 1, where h and k are local linear scale factors along meridian, and along parallel, respectively. Let us introduce for the sake of abbreviation: a c max , b c min .
Let us first note that a 0 and b 0 is always true. Table 1 shows all possible cases when a 1, b 1 or a b 1. This is very simple but new in the theory of map projections: (1,1) If at some point a 1, we can say that this point is locally equidistant in the direction of maximum local linear scale factor.
(1,2) Ifat some point b 1, we can say that this point is locally equidistant in the direction of minimal local linear scale factor.
(1,3) If at some point a b 1, we can say that this is a point with zero distortion, a zero-distortion point or a standard point.
(2,1) If at all points of a line a 1, then it is not generally a line with zero distortion, but we can say that this line is equidistant in the direction of maximum local linear scale factor.
(2,2) If at all points of a line b 1, then it is not generally a line with zero distortion, but we can say that this line is equidistant in the direction of minimal local linear scale factor.
(2,3) If at all points of a line a b 1, then it is a line with zero distortion, a zero-distortion line or a standard line.
(3,1) If at all points of an area a 1, then we can say that this area is equidistant in the direction of maximum local linear scale factor.
(9) and the first differential form is (10) The local linear scale factor squared for a mapping a sphere by using normal aspect cylindrical projection (8) is (11) Local linear scale factors along a meridian and a parallel, respectively are (12) For normal aspect cylindrical projections, along standard parallels with latitudes 1 it should be h( 1 ) k( 1 ) 1, or considering (12) and (9) (14) . and 1 ), if it is also and From the formula (15) and the properties oftrigonometric function cosine we can conclude that if the latitude of the standard parallel is 1 , then it should be 1 , too. The number of standard parallels in normal aspect cylindrical projections we can conclude based on the relations (14) and (15).
If 0 n 1 then two standard parallels exist ( 1 If n 1 then only one standard parallel ( 1 0) can exist, the Equator, providing that it is Ifn 1 then there is no 1 for which (15) holds, i.e. there is no standard parallel at all.
Let and be the latitudes of two parallels mapped by normal aspect cylindrical projection (8). The vertical distance between these two parallels on the sphere is 2R sin , and the distance between their images in the plane of projection (see Figure 2) is 2y( ). It is clear that 2y( )=2R sin , i.e. y( )=R sin is not true in general. It is true for the Lambert equal-area projection only. Thus, bending a map made in the normal aspect cylindrical projection onto a cylindrical surface and then putting it in a particular position with the sphere generally makes no sense. In other words, map projection and folding or unfolding of the cylindrical surface are two completely different actions. Since secant parallels do not have any particular property in terms of distortion distribution, we conclude that the use of cylindrical surfaces in the theory of map projections generally makes no sense and can lead to wrong conclusions about the identities of secant and standard parallels (Figure 3). The use of cylindrical surfaces Slika 3. Uobičajena pogrešna ilustracija: standardne paralele su istodobno presječne paralele u uspravnom aspektu cilindrične projekcije (Mercator 2009). Dokazali smo da je to nemoguće. Ta ilustracija je iluzija. Fig. 3 The usual erroneous illustration: standard parallels are at the same time secant parallel in normal aspect cylindrical projection (Mercator 2009). We have proved that this is impossible. The illustration is an illusion. .
Dvije paralele ( 1 i 1 ) su standardne paralele ako vrijedi (16)  Two parallels ( 1 and 1 ) are secant parallels if the distance between these two parallels on the sphere equals the distance between their images in the plane of projection: Two examples on cylindrical projections follow. are also mapped equidistantly . Since all the meridians in this projection are mapped equidistantly, the two equidistantly mapped parallels are also standard parallels The distance between two parallels that corresponds to latitudes and on a sphere is 2R sin (see Figure 2). If , then the distance between these two parallels on the sphere is . The distance of their images in the plane of projection is 2y( ) 2R( ), which for amounts (20) For that projection so we have It is immediately apparent that this projection is equal-area because it is hk 1 for any . Snyder and Voxland (1989) write that the Behrmann Projection is true along latitude 30 and that it is a projection on a cylinder secant at 30 (True along latitudes 30 , projection onto a cylinder secant at 30 ). The standard parallels of the Behrmann projection correspond to latitudes 30 . The distance between these parallels on the sphere with radius R is obviously R. In the plane of Behrmann's projection the distance between their images is which is makes sense in perspective projections on a cylinder only, but such projections are rarely applied.
To summarize about cylindrical projections given by (8)
[ , ], the constants are 0 n 1, R 0 and 0 [ , ], and the function Rf( ) is continuous, with positive values and monotone decreasing, or monotone increasing, and and are coordinates ofa point in the polar coordinate system in the plane. As can be seen, this is about mapping into the plane, not onto a conical surface. For such mapping we have (24) and the first differential form (25) The square of the local linear scale factor of normal aspect conic projection (23) is (26) From (26) we can read the local linear scale factor along the meridians and parallels, respectively (27) The minus sign in the formula for h was chosen because functions and f are monotone decreasing by hypothesis. If these functions are monotone increasing, then should be replaced by in the derivation that follows.

If for any
it is then all meridians are mapped equidistantly. If for some it is then parallels corresponding to these latitudes are mapped equidistantly. Along a standard parallel 1 it should be true that and (30) We cannot make any conclusions about the number of standard parallels in normal aspect conical projections as we could with cylindrical ones. This is because for the given n there are infinitely many continuous functions Rf( ), which are differentiable into a plane the Behrmann cylindrical surface that cuts the sphere along the parallels of 30 latitudes. Conversely, a map made in Behrman's projection cannot bend into a cylindrical surface that will intersect the sphere along the parallels of 30 latitudes. Identifying the equidistant or standard parallel and the parallel in which the cylindrical surface intersects the Earth's sphere is obviously not a valid procedure (Lapaine 2018).

Equidistant, standard and secant parallels in normal aspect conical projections
A conic projection is mapping defined by the formulae Rf( ), where (28) Slika 5. Ilustracija standardnih paralela kao paralela u kojima konusna ploha siječe Zemljin elipsoid. Izvor: Richardus i Adler 1 972, str. 94. To je općenito pogrešan pristup koji je popraćen netočnom tvrdnjom: "To je Lambertova konusna konformna projekcija s dvije standardne paralele. Konus siječe elipsoid u tim kružnim paralelama". Slika bez crvenih crta preuzeta je iz knjige Richardusa i Adlera, 1 972, str. 94, i to je jedna obmana. Precrtano crvenim je upotrijebljeno da bi se istaknulo da na toj slici nešto nije kako treba. Nakon gledanja slike i čitanja teksta, čitatelj bi trebao razlikovati činjenicu od iluzije. Fig. 5 Illustration of standard parallels as parallels in which the conical surface intersects the Earth's ellipsoid. Source: Richardus and Adler 1 972, p. 94. This is generally a misguided approach and is accompanied by the inaccurate statement, "This is the Lambert conical conformal projection with two standard parallels. The cone intersects the ellipsoid at these parallel circles". Figure without red cross was taken from Richardus and Adler 1 972, p. 94, and it is an illusion. Red cross was used to stress that something is wrong with the figure. After looking at the figure and reading the text, it is expected that the reader will be able to distinguish between fact and illusion.
However, the equality (33) is not true in general. Thus, it is not applicable to parallels mapped equidistantly and for standard parallels. This means that bending a map produced in the normal aspect conical projection into a conical surface and placing it in a particular position related to the sphere generally makes no sense. In other words, map projection and developing the conical surface into a plane are two quite different actions. Since secant parallels do not have any particular property in terms ofdistortion distribution, we conclude that the use of conical surfaces in the theory ofmap projections generally makes no sense and can lead to wrong conclusions about the identities ofsecant and standard parallels.
To summarize about conical projections given by (23) and two parallels Two parallels ( ' and ") are secant parallels if the distance between these two parallels on the (36) where [ , ], R 0 and 0 [ , ].
Since and we conclude that all meridians are mapped equidistantly. The condition (34) implies that two parallels corresponding to latitudes 0 and 60 are mapped equidistantly . Since all meridians in this projection are mapped equidistantly (h( )=1 for all ), the two equidistantly mapped parallels are standard parallels. The distance d between two parallels with latitudes ' and " on the sphere is given by (36). If ' 0 and " 60 , then the distance between them on the sphere is 2Rsin30 R. The distance between their images in the plane of projection is ( ') ( ") Rf( ') Rf( "), and for ' 0 and " 60 this is which is different from R. Thus, by bending the map into the conical surface and placing the cone so that its axis coincides with the straight line connecting the North and South Poles, it is not possible to obtain secant parallels identical with the standard parallels. Thus, we have shown that secant and standard parallels in conical projections generally cannot be identical.
Da bi konusna projekcija bila konformna, mora vrijediti h k, where C is a constant of integration.
Let 1 0 and 2 60 be two standard parallels. The conditions for standard parallels and (43) (44) and (45) and (46) If we now substitute values 1 0 and 2 60 we will get (47) The distance between the images of standard parallels with latitudes 1 0 and 2 60 in the plane of Lambert conformal conical projection equals (48) Let us bend the map into a conical surface and place the cone so that its axis coincides with the straight line connecting the North and South Poles on the sphere. It is not possible to obtain secant parallels identical with the standard parallels, because the shortest distance between standard parallels on the sphere measured along a straight line connecting the two parallels is equal to 1 (Figure 6). Bending (and developing) is isometry, that is, transformation that preserves distances.
If we bend a map made in the normal aspect conical projection into a conical surface and then try putting it in a particular position with the sphere, standard parallels in the projection will not coincide with standard parallels on the sphere. In other words, map projection and folding or unfolding of the conical surface are two completely different actions.
We have proved that secant parallels are not standard parallels, in general. depicting a cone that intersects the ellipsoid into two parallels ( Figure 5). Richardus and Adler wanted to show that these parallels were secant and standard parallels, but this cannot be true if we accept the definitions of secant and standard parallels recommended in this paper. Let us prove this using a simple example of a sphere of radius 1 and Lambert's conformal conical projection with two standard parallels corresponding to the latitudes 1 0 and 2 60 . First of all, there is no intermediate conical surface in the Lambert conformal conic projection. This is evident from Lambert's original work or the English translation (Lambert 1772).
Ifa map in the Lambert conic conformal projection has two standard parallels (right-hand part of Figure  5), then it is not possible to bend it into a conical surface which will cut the earth's sphere along these two parallels (left-hand part of Figure 5). Though this may surprise some people, it will be proved in this example.
In order for the conical projection to be conformal, it must be true that h k, where h and k are defined in (27). From there, because R 1, it follows and then from where, after integration, we get form a system of two equations with two unknowns, n and C, which can be transformed into from where it follows that . Slika 6. U Lambertovoj konusnoj konformnoj projekciji jedinične sfere udaljenost između dviju standardnih paralela kojima odgovaraju geografske širine 1 i 2 jednaka je 1 na sferi, a manja je od 1 u ravnini projekcije. Fig. 6 In conical conformal projection of a unit sphere the distance between two standard parallels latitudes 1 and 2 equals 1 on the sphere, and less then 1 in the plane of projection. gdje je C konstanta integracije.

Primjer 5
Snyder (1993) u svojoj čuvenoj knjizi Flattening the Earth, kaže na str. 123, "Kao i kod centralne ciloindrične projekcije opisane prije, konusna projekcija može se razviti geometrijskim projiciranjem globusa iz njegova središta na konus koji je KiG No. 34, Vol. 1 9, 2020, https://doi.org/1 0.32909/kg.1 9.34.3 (49) ... For the tangent form with one standard parallel, 1 may be equated to 2 . If the standard parallels favour the northern hemisphere, only part ofthe southern hemisphere can normally be shown". Snyder was apparently convinced that there was a gnomonic or central perspective projection on a cone with two standard parallels. He did not check the truth of this and accepted it as a fact, though it is erroneous. Here is the evidence! The local linear scale factor along the meridian in a normal aspect conical projection is calculated according to (27) (50) For a function ( ) defined by (49) which is different from 1, unless 1 2 . We have shown that a simple gnomonic perspective projection on a conical surface cannot have two standard parallels. In other words, if the conical surface intersects the sphere at two parallels, these two parallels cannot be standard parallels in the gnomonic perspective projection.

Conclusion
We argue that the explanation of cylindrical and conical projections as projections on cylindrical or conical surfaces is not a good approach, because it leads to misunderstanding some projection properties. Furthermore, standard and secant parallels are often considered identical, but this paper has shown that broadly accepted facts about secant and standard parallels found in many references are incorrect and should be revised. This requires a critical approach to established customs in teaching and researching map projection. In several previously published papers it was shown that conical and azimuthal projections existed with three and more standard parallels. Obviously, this could not be explained by using the secant projections approach and led to distinguishing between standard and secant parallels. The equidistant projections are known in the theory of map projections. The paper introduces equidistance in a broader sense. Equidistance is defined at a point, along a line and in an area. This allowed three types of parallels to be defined: secant, standard and equidistant. Explicit conditions for equidistant, standard and secant parallels are given for cylindrical and conical projections. Theoretical assertions are illustrated by appropriate examples.