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International Journal of Cartography
ISSN: 2372-9333 (Print) 2372-9341 (Online) Journal homepage: https://www.tandfonline.com/loi/tica20
Minimum distortion pointed-polar projections for
world maps by applying graticule transformation
János Györffy
To cite this article: János Györffy (2018) Minimum distortion pointed-polar projections for world
maps by applying graticule transformation, International Journal of Cartography, 4:2, 224-240, DOI:
10.1080/23729333.2018.1455263
To link to this article: https://doi.org/10.1080/23729333.2018.1455263
© 2018 The Author(s). Published by Informa
UK Limited, trading as Taylor & Francis
Group
Published online: 07 May 2018.
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Minimum distortion pointed-polar projections for world maps
by applying graticule transformation
János Györffy
Institute of Cartography and Geoinformatics, Eötvös Loránd University, Budapest, Hungary
ABSTRACT
Both the media and the geosciences often use small-scale world
maps for demonstrating global phenomena. The most important
demands on the projection of these maps are: (1) the map
distortions have to be reduced as much as possible; (2) the
outline shape of the mapped Earth must remind the viewer of the
Globe. If the map theme to be illustrated requires neither
equivalency (nor, which rarely happens, conformality) nor
prescriptions for the map graticule, an aphylactic non-conical
projection with simultaneously minimized angular and area
distortions is advisable. In this paper, a graticule transformation by
a parameterizable function helps to convert minimum distortion
pointed-polar pseudocylindrical projections for world maps into
general non-conical projections with further minimized
distortions. The maximum curvature of the outline shape will be
moderated at the same time in order to obtain a definitely
pointed-polar character.
RÉSUMÉ
Aussi bien les médias que les géosciences utilisent souvent les
cartes du monde à petite échelle pour illustrer des phénomènes
globaux. Les attentes les plus importantes à propos des
projections de ces cartes sont les suivantes : 1) la distorsion
géométrique doit être réduite autant que possible 2) la forme du
contour de la terre doit rappeler au lecteur l’aspect du globe. Si le
thème à cartographier ne requiert ni équivalence (ni conformité,
ce qui arrive rarement), ni des contraintes sur le quadrillage
cartographique, une projection aphylactique non conique
minimisant conjointement les distorsions des angles et des
surfaces est conseillée. Dans ce papier une transformation de
graticule par une fonction paramétrable aide à convertir des
projections polaire pointue pseudo-cylindriques pour planisphère
à distorsions minimales en projections non coniques standard
pour davantage minimiser les distorsions. La courbure maximale
du contour sera ainsi modérée afin d’obtenir également un
caractère résolument de type polaire pointu.
ZUSAMMENFASSUNG
Die Massenmedien sowie auch die Erdwissenschaften benutzen
meistens kleinmaßstäbige Karten zur Demonstration globaler
ARTICLE HISTORY
Received 15 June 2017
Accepted 19 March 2018
KEYWORDS
Minimum error projection;
pointed-polar projection;
non-conical projection;
pseudocylindrical projection;
graticule transformation
© 2018 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group
This is an Open Access article distributed under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives License
(http://creativecommons.org/licenses/by-nc-nd/4.0/), which permits non-commercial re-use, distribution, and reproduction in any
medium, provided the original work is properly cited, and is not altered, transformed, or built upon in any way.
CONTACT János Györffy terkep1@ludens.elte.hu
INTERNATIONAL JOURNAL OF CARTOGRAPHY
2018, VOL. 4, NO. 2, 224–240
https://doi.org/10.1080/23729333.2018.1455263
Phänomene. Die wichtigsten Erwartungen an die Netzentwürfe
dieser Karten sind: (1) die Verzerrungen der Abbildungen sollen
bestmöglich reduziert werden; (2) die Form der Konturlinie des
abgebildeten Globusses soll an die Erdkugel erinnern. Wenn die
darzustellende Thematik der Karte weder Flächentreue (oder sehr
selten Winkeltreue) noch Vorschrift für das Gradnetz erfordert,
kann eine vermittelnde unechte Abbildung mit gleichzeitig
minimierten Winkel- und Flächenverzerrungen empfohlen werden.
In diesem Aufsatz handelt es sich darum, unechte
Zylinderabbildungen mit Polpunkt und mit minimierten
Verzerrungen mit Hilfe einer Transformation des Gradnetzes
durch eine parametrierbare Funktion in allgemeine, unechte
Netzentwürfe mit Polpunkt und mit weiter verminderten
Verzerrungen umzuwandeln. Die maximale Krümmung der
Konturlinie wird gleichzeitig ermäßigt, um einen ausdrücklichen
Charakter mit Polpunkt zu erhalten.
1. Introduction
World maps often appear in the media and on the internet for global geographical
phenomena, and they are an integral part of atlases for experts or the general public,
too. The map user wants to perceive the size proportions concerning the distances,
areas and angles of represented map objects. This is unfavourably influenced by the
map distortions hindering the map reader from the right interpretation of the graticule
and thereby the map content. In contrast to the designed, sometimes bizarrely shaped
world maps spread in the media, when editing a small-scale map and selecting a
proper projection, these distortions should be kept at a low level (Snyder, 1993, p. 10).
Apart from map distortion claims of conformality or equivalency (necessitated mostly
by thematic maps), these maps, like geographic maps in atlases for the public or in
school atlases, have not any special requirements of distortions, but the diminishing of
scale, angular and area distortions, on the whole, are generally expected. On the other
hand, the requirement of the expressiveness comes to the front in small-scale world
maps; in other words, the map has to keep the shape of the mapped geographic
objects, for example, the continents, as far as possible (Baranyi, 1987, p. 13; Canters,
2002, p. 85). The raster maps used in web cartography make further demands on the
applied projections. So, the change or loss of information induced by the shape distortions
in the course of projection conversions (e.g. reprojection) of raster maps, Mulcahy (2000,
p. 8) have to be avoided or moderated. Both angular and area distortions can be elimi-
nated separately. Conformal world maps show large area distortions generating shape
deformations of the continents by disproportionate dimensional changes of different con-
tinent parts. Equal-area world maps have strong angular distortions, causing shape defor-
mations of the continents by twisting. The extremes are avoidable by application of
‘aphylactic’(‘compromise’: neither conformal nor equal-area) projections. This kind
keeps the balance between the two mentioned distortions, in this way they keep the
shape of the continents better and this explains why they are suggested for world
maps (Canters, 2002, p. 32).
Looking at a world map, the outline of the mapped Earth gives a first impression. The
map has to satisfy the principle of similarity for the whole Earth, too, which means that the
INTERNATIONAL JOURNAL OF CARTOGRAPHY 225
outline should possibly reflect the sphere (or the ellipsoid). This principle originates in Ptol-
emy’s Geography (Klinghammer, 2015, p. 16; Lelgemann, Kleineberg, & Marx, 2012; Mitch-
ell, 2007, p. 48; Snyder, 1993, p. 10). With regard to this, the circle- or oval (e.g. ellipse-
shaped) outlines may come into account. In the case of an elliptic shape, the outline con-
siderably curves near the equator, the curvature decreases at higher latitudes, and
becomes minimal at the poles.
It cannot be considered advantageous aesthetically if the curvature along the outline
changes very much. Breakpoints on the outline are even less favoured because the differ-
entiability demand of the mapping functions (‘projection equations’) fails in general at
them, and some of the distortions tend mostly to infinity in their environment, not to
speak of the aesthetic principle of geometric harmony (Mitchell, 2007, p. 48). This high-
lights the ‘pointed-polar’projections representing the pole as a point and overshadows
most of the ‘flat-polar’projections representing the pole as a (mainly straight) line, even
though the distortions around the poles are in general less disadvantageous. Demands
in connection with the similarity principle can set a limit to the minimization of the
distortions.
Battista Agnese constructed an aphylactic map projection in 1544 (known as Ortelius
oval projection), which represents the Earth in a roughly oval-shaped outline (Livieratos,
2016, p. 109). Since then, during the advance of cartography, numerous projections
were developed for World maps. Some of the often applied ones will be reviewed
briefly, according to graticule categories. The groups of projections are listed in order of
the increasing number of degrees of freedom taking into account that the pointed-
polar property means a force reducing the number of degrees of freedom (the
mapping function formulae of almost all of the 16 mappings are available in Snyder &
Voxland, 1989).
.Cylindrical projections (Equirectangular alias Plate Carrée, Web Mercator which is iden-
tical to Mercator conformal between +85.051129◦, conformal cylindrical with the stan-
dard parallel
w
s=+42◦)
.Pointed-polar pseudocylindrical projections (extended Apianus II, Mercator-Sanson equal-
area, Mollweide equal-area, Baranyi IV, formulae in Baranyi & Györffy, 1989, pp. 79–80,
pseudocylindrical with minimized distortion, labelled as ‘version (c)’later, formulae in
Györffy, 2016, p. 264)
.Mixed (partly pointed-polar, partly flat-polar) pseudocylindrical projection (Ortelius oval)
.Flat-polar pseudocylindrical projections (Eckert III, Eckert V, Kavrayskiy VII as well as
Robinson, formulae in Beineke, 1991, p. 93)
.Pointed-polar pseudoconical projection (van der Grinten)
.Pointed-polar non-conical projection (Aitoff)
.Flat-polar non-conical projection (Winkel Tripel with
w
s=arccos(2/
p
))
The cylindrical and pseudocylindrical projections illustrate expediently the zonal geo-
graphic phenomena (Baranyi, 1987, p. 15) which brings them to the forefront in the
case of thematic maps representing such topics, but non-conical projections with slightly
curved parallels (e.g. Winkel Tripel) are also suitable for this purpose. However, the distor-
tions of the non-conical projections can be further reduced because of more degrees of
freedom.
226 J. GYÖRFFY
Many investigations and user studies deal with the advantages and disadvantages of
flat-polar and pointed-polar projections from the aspect of world maps (Šavrič, Jenny,
White, & Strebe, 2015). Some of the projections, listed above were studied by Frančula
(1971, pp. 66–67) according to their distortions. One of his results was that the flat-polar
projections are generally less distorted than the pointed-polar ones, aside from the cylind-
rical projections. On the other hand, the interpretation of the pole line is not evident for
common people, mainly for schoolchildren (Szigeti & Kerkovits, 2018). Some investigations
show that map readers prefer to see the poles represented as points, not lines (Werner,
1993, p. 35). In a special case, the oblique aspect of the Winkel Tripel demonstrates that
understanding the intersection of the pseudopole line and the mapped graticule line is
difficult (Lapaine & Frančula, 2016, p. 49). The problems of representing the area near
the poles on global raster maps and their reprojection between the two above types of
projections were analysed in Steinwand, Hutchinson, and Snyder (1995).
This paper aims to construct new pointed-polar non-conical projections with minimized
distortions in the hope that significantly lower distortions can be reached than that of the
projections listed above. However, some observations show that in the case of minimum
distortion projections the curvature near the poles can be so little that the graticule is
barely distinguishable from a flat-polar projection (Györffy, 2016, p. 265; Snyder, 1985,
p. 130). Therefore, the curvature at the poles is aimed to rise at the expense of minimized
distortions by a further modification.
The substantive part of this study contains the following:
.an applied method of the measurement of overall map distortions for world maps
.a method to reduce distortions by transformation of pointed-polar pseudocylindrical
projection to a non-conical one
.a method to correct the outline shape of the obtained non-conical projection in order to
highlight the pointed-polar character of the mapped Earth
.numerical results and the finally achieved non-conical projection
2. Methodology
2.1. Measurement of overall map distortions for world maps
The aim is to rank projections of maps representing the same territory (in this case the
Earth) in respect to map distortions. In cartography, the distortions of scale, areas and
angles are taken into account. Let x(
w
,
l
)and y(
w
,
l
)denote the mapping functions,
which assign the map coordinates x,yto the geographic latitude
w
and the geographic
longitude λ. The map coordinates x,yare used in the normal mathematical way, in
addition the projection equations are considered with a unit spherical Earth and a unit
nominal map scale.
At a point on the map, the area distortion can be measured by the area scale p
p=a·b
and the angular distortion by the quotient
a
b,
INTERNATIONAL JOURNAL OF CARTOGRAPHY 227
where aand bare the maximal and minimal local linear scales (Canters, 2002), given by the
formulae
a=
∂x
∂
l
2
+∂y
∂
l
2
cos2
w
+∂y
∂
w
2
+∂x
∂
w
2
+2·
∂x
∂
l
·∂y
∂
w
−∂y
∂
l
·∂x
∂
w
cos
w
2
+
∂x
∂
l
2
+∂y
∂
l
2
cos2
w
+∂y
∂
w
2
+∂x
∂
w
2
−2·
∂x
∂
l
·∂y
∂
w
−∂y
∂
l
·∂x
∂
w
cos
w
2
and
b=
∂x
∂
l
2
+∂y
∂
l
2
cos2
w
+∂y
∂
w
2
+∂x
∂
w
2
+2·
∂x
∂
l
·∂y
∂
w
−∂y
∂
l
·∂x
∂
w
cos
w
2
−
∂x
∂
l
2
+∂y
∂
l
2
cos2
w
+∂y
∂
w
2
+∂x
∂
w
2
−2·
∂x
∂
l
·∂y
∂
w
−∂y
∂
l
·∂x
∂
w
cos
w
2
using the partial derivatives of the projection equations (expressions aand bprovide the
semi-major and semi-minor axes of the Tissot indicatrix ).
The third distortion type, the local linear scale, is considered as the consequence of the
above area and angular distortions (Györffy, 2016, p. 256). Namely, it can be expressed as a
function of them, therefore, it is sufficient to calculate only with the value of area and
angular distortions. So to obtain the local overall distortion at a point of the map, the
index number 12
Kbased on the principle of Kavrayskiy (Kavrayskiy, 1958; Bayeva, 1987)
will be used:
12
K=
ln2a·b()+ln2a
b
2
composed of the measure of the area and angular distortions ln2(a·b)and ln2(a/b).
Multiplying by 2 the quantity known as the Airy–Kavrayskiy index number of local distor-
tion 1
2·[ln2(a)+ln2(b)], the product gives the index number 12
K(Canters, 2002, p. 43).
Representing the same territory, in the case of conformal projections, the area scale
pcan grow large, and similarly, in the case of equivalent projections, the quotient a/bchar-
acterizing the angular distortion can grow large. In suitable aphylactic projections where
both area and angular distortions occur, the value of the index number 12
Kis generally
lower than in the equivalent and conformal projections. If the two distortions are
balanced, the index number 12
Kcan be minimal.
228 J. GYÖRFFY
Then the mean value E2
Kof the local overall distortions is calculated in the represen-
tation of the territory Ton Earth by the formula
E2
K=1
m
T()
·T12
KdT,
which is cited as the Airy–Kavrayskiy criterion further on. The surface integral represents
the aggregated local overall distortions 12
Kand is computable by the projection equations
x(
w
,
l
)and y(
w
,
l
). The division by the size
m
(T)of the territory Tresults in the averaging
(Györffy, 2016, p. 257).
This criterion gives a way to rank the projections of maps representing the whole Earth.
A projection with minimal criterion value E2
Kprovides the entity with minimum distortion
in a family of map projections. Since here the spherical Earth will be mapped, the surface
integral mentioned above turns into a double integral of the function 12
K·cos
w
. Further-
more, to avoid the distortion values tending to infinity at and near the poles, the values of
the function above assigned to the points of the 5◦−5◦environment of poles will be
omitted from the double integral (Frančula, 1971; Gede, 2011, p. 218; Grafarend & Nier-
mann, 1984, p. 104). So the final formula for E2
Kis:
E2
K=1
2·sin 85◦·85◦
−85◦180◦
−180◦
12
K·cos
w
d
l
d
w
.
The mean overall map distortions for the above listed 16 projections are compiled in
Table 1in descending order of the criterion values EK, arising from the two-dimensional
Simpson’s rule (segments of 1◦) (Davis & Rabinowitz, 1975, pp. 269–270; Kerkovits, 2017,
p. 123).
In Table 1the higher EKvalues (in the left column), apart from the three cylindrical pro-
jections, belong to pointed-polar ones, while the majority of projections with low EKvalues
(in the right column) represent the poles as lines (the EKvalue of the equirectangular
cylindrical projection above can diminish if
w
sis picking up +42.00◦for their standard par-
allels Grafarend & Niermann, 1984). Note that, for instance, there is a considerable differ-
ence between the EKvalues of nearly related (pointed-polar) Aitoff and (flat-polar) Winkel
projections in favour of the latter.
In addition, some projections were created representing the poles as concave curves
(bent towards the equator) by suitable renumbering of Aitoff and ordinary polyconic,
with even lower EKvalues 0.3336 and 0.3377, respectively (Frančula, 1971, p. 66). They
are ignored in the praxis of cartography because of their appearance.
Table 1. Mean overall distortion values EKof some often used projections (pp: pointed-polar; fp: flat-
polar; mixed: partly pointed-polar, partly flat-polar).
Projection EkPole Projection EkPole
Web Mercator 0.69104 fp Extended Apianus II 0.46485 pp
Mercator–Sanson 0.66474 pp Eckert V 0.42009 fp
van der Grinten 0.57682 pp Baranyi IV 0.40674 pp
conformal cylindrical (
w
s=+42◦)0.54896 fp Eckert III 0.40345 fp
Mollweide 0.53375 pp Robinson 0.39287 fp
Aitoff 0.52187 pp Kavrayskiy VII 0.36930 fp
Equirectangular 0.48864 fp Winkel Tripel (
w
s=50◦28′)0.36699 fp
Ortelius 0.47146 mixed Pseudocylindrical with minimized distortion 0.35184 pp
INTERNATIONAL JOURNAL OF CARTOGRAPHY 229
2.2. Minimization of map distortions for pointed-polar pseudocylindrical
projections
The final goal is to create an oval-shaped world map with minimal distortions, where the
double symmetry of the graticule is also preferred, and the poles are represented as a
point, accepting the slightly larger distortions in the environment of the poles. Namely,
the meridians in the pointed-polar projections are forced to converge at the poles,
which causes competition disadvantages because of distortion risings at higher geo-
graphic latitudes, but under +85◦. The effectiveness of the approximation of the
mapping functions depends substantially on the chosen approximating function.
Because of the double symmetry of the graticule, a simpler projection type, for
example, the pseudocylindrical offers itself as a start-up. In this way, the minimum distor-
tion projections for world maps will be achieved through pointed-polar pseudocylindrical
projections, where the lines of latitude are parallel straight lines on the map, that is the
map coordinate ydoes not depend on the longitude λ.
It can be proved that the minimum distortion pseudocylindrical projections have a true
scale central meridian (Györffy, 2016, pp. 258–259), that is
y=
w
,
while the mapping function x(
w
,
l
)can be approximated by a comparatively simple and
parameterizable product of a function of the latitude
w
and the function of the longitude λ
(both in radians) which is able to create a double symmetrical graticule with an oval-
shaped outline:
c1·1−2·
w
p
c2
1/c3
·
l
+c4·
l
3
,
where the coefficients c2and c3regulate the running down of the meridian arcs on the
map, furthermore c1and c4regulate the linear scale along the parallels. If there is a
special case when c1=1, c2=2, c3=2andc4=0, it would lead to the extension of
Apian’s second projection with true scale equator and central meridian, which represents
the Earth in the shape of an ellipse, and the mapped poles are points.
The coefficients ciwere calculated by the downhill simplex method,arobustmethod
of minimization working without derivatives but converging relatively slowly (Press,
Teukolsky, Vetterling, & Flannery, 1992, pp. 402–406). The author made efforts to
shorten the running time and to avoid the local minima in the course of computation.
For this, the initial functions were simpler projection equations (with fewer coeffi-
cients), and each new parameter was involved one by one, at first with zero as
its initial value. The resulting minimum function was inserted in the initial functions,
and the computer program was executed again; this phase was repeated several times.
2.3. Graticule transformation for conversion of the pseudocylindrical to a
minimumdistortion non-conical projection
Such pseudocylindrical projections will be converted into general non-conical projections
(without any restriction on the graticule). It is achieved with the help of a transformation
realigning the graticule lines inside of the mapped world outline. An auxiliary variable
230 J. GYÖRFFY
ψwill be substituted into the latitude
w
. Let ψbe a strictly increasing, odd function of
w
,
and an even function of λ, too, in favour of the double symmetry and the injectivity of the
projection. The transformation function
c
(
w
,
l
)which, in effect, is a type of generalized
renumbering of the cartographic grid (Canters, 2002, p. 119), needs to be parameteriz-
able, so a polynomial of variables
w
and λwith the above properties will be used.
The distortions along the central meridian are allowed to vary, changing the true scale
central meridian property of the minimum distortion pseudocylindrical projections. Thus
the function ψis not linear with respect to
w
if
l
=0. A possible form of the polynomial
ψwith the coefficients fij is:
c
=f11 ·
w
+f12 ·
w
3
+
l
2·f21 ·
w
+f22 ·
w
3
+
l
4·f31 ·
w
+f32 ·
w
3
,
where the expression within the first parentheses determines the distortions along the
central meridian.
In order to exclude the graticule lines running over the outline,
c
=+
p
/2 must be
assigned to the poles (
w
=+
p
/2)for each value of λby the function
c
(
w
,
l
), that is
c
+
p
2,
l
=+
p
2,
which implies the relation
f12 =1−f11
()·
2
p
2
.
To avoid the discontinuities and other anomalies around the poles, a second condition has
to be added to this equation: the multiplicator polynomials of
l
2and
l
4have to become
zero at the poles, that is
f21 ·
p
2+f22 ·
p
2
3
=0 and f31 ·
p
2+f32 ·
p
2
3
=0.
Consequently,
f22 =−f21 ·2
p
2
and f32 =−f31 ·2
p
2
.
In this way, the transformation function
c
(
w
,
l
)can be approximated as follows:
c
=c5·
w
+1−c5
()·
2
p
2
·
w
3
+
l
2·c6+
l
4·c7
·
w
−2
p
2
·
w
3
,
where the denotations c5=f11,c6=f21 and c7=f31 are used.
Then the function
c
(
w
,
l
)will be substituted into the projection equations for
w
, that is
the form of the transformed projection equations are:
x=c1·1−2·
cw
,
l
p
c2
1/c3
·
l
+c4·
l
3
,
y=
cw
,
l
.
As a result of this, the latitude line shapes are transformed from parallel straight lines to
INTERNATIONAL JOURNAL OF CARTOGRAPHY 231
curves. So, the pseudocylindrical projection changes to a general non-conical one, whilst
the outline of the map also varies slightly, adapting oneself to the minimization condition.
The pointed-polar character remains in the transformed projections, too.
Finally, the coefficients of the projection equations are selected by the minimization of
the Airy–Kavrayskiy criterion, with the help of the downhill simplex method.
A further similar transformation function could be established by substituting another
auxiliary variable
z
(
w
,
l
)into the geographic longitude λbut its effect on the reduction of
the E2
Kis an order of magnitude smaller and therefore it was ignored.
2.4. Reshaping the outline of the mapped Earth
As mentioned in the introduction, in the case of some of the known minimum distortion
pseudocylindrical and non-conical projections, the curvature of the outline is specifically
formed. Near the equator, the curvature is small or medium, while moving on towards
the poles the curvature increases. It reaches its maximum value, and decreases further
towards the poles to a small value, even to zero. In this case, the mapped Earth looks as
if it had been mapped in a flat-polar projection.
The outline can be corrected, if the maximal curvature can be reduced, while the
smaller curvatures rise, and the pointed-polar character becomes dominant. From now
on, the exact concept of the curvature in a point of a plane curve, known from the differ-
ential geometry, will be used. It can be defined by the reciprocal of the radius of the oscu-
lating circle at the point in question. The formula of the curvature (Stoker, 1989, p. 26) on
the outline of a world map given by the equation
l
=+
p
can be written:
k
l
=+
p
=
∂x
∂
w
·∂2y
∂
w
2−∂y
∂
w
·∂2x
∂
w
2
∂x
∂
w
2
+∂y
∂
w
2
3/2
l
=+
p
,
where the functions x(
w
,
l
=+
p
)and y(
w
,
l
=+
p
)give the parametric equation of the
outline. Since the graticule is mostly preferably double symmetrical, it was enough to cal-
culate the curvature only in one quarter of the whole outline, for example, on the meridian
of
l
=
p
from the Equator to the North Pole. The change of the curvature is relatively slow,
therefore it was sufficient to compute the curvature values taking the steps per 1◦, and
because of the singularity of the pole and the crosspoint of the outline and the
Equator, it was calculated from 1◦to 89◦, averaging them (
k
mean), and selecting the
maximal curvature
k
max assigned to the latitudes
w
k
max . Finally,
k
max was divided by the
averaged curvature
k
mean in order to eliminate the curvature changes which originate
in sizing ( enlarging–reducing) of the map.
So far the mean overall distortion EKwas the objective function value to be minimized.
In this examination phase the absoluteness of the aspect of criterion value decrease was
sacrified in order to correct the outline shape, and a new objective function was prepared
containing the maximal curvature, too, in the form of the product of the two
characteristics.
232 J. GYÖRFFY
Let gdenote the quotient
g=
k
max
k
mean
,
where the maximal curvature is normed by dividing it by the mean curvature, so the values
gbelonging to different projections can be compared. If in this way both the mean overall
distortion and the maximal curvature have to be reduced simultaneously, then for
example, the product
Err =EK·[
2i]g
(i=0,1,2,...)has to be minimized. The raising to the power of a factor influences its share
in the objective function. The less the value iis, the more dominant the effect of gis in the
product during the minimization. Consequently, the maximal curvature drops faster, and
the outline is closer to the oval shape while the distortion decreases to a lesser extent.
During these calculations, an increase of 1–2% of the mean overall distortion was
considered to be acceptable, which came true by choosing i=1, that means the square
root of g.
3. The attributes of the initial, transformed and outcome projections
Three versions were studied for initial projections according to the chosen coefficient
values of c2and c3, all of them are pointed-polar pseudocylindrical with minimized distor-
tions presented as follows. For two of them the distribution of angular and area distortions
were represented on the figures, where the angular distortions were characterized, as
usual, instead of a/b, by the maximum angular deformation 2
v
given by the formula
2
v
=2·arcsin a−b
a+b
=2·arcsin a/b−1
a/b+1
,
and the area distortions were given by p. The introduced versions with the mapping func-
tions (1) and (2) are Györffy (2016, p. 262):
ac2=2 and c3=2 in Expression (2). The coefficients in the first row of Table 2resulted
in E2
K=0.13510, EK=0.36756 (Figure 1).
The pointed-polar character of the graticule is manifested. On lower latitudes, the cur-
vature of the map outline is zero or almost zero and the graticule can favourably fit in
the rectangle of a map page.
bc2=2 and c3=2 in Expression (2). The second row of the Table 2lists the coefficients
resulting in E2
K=0.12406, EK=0.35222 (Figure 2).
The mean overall distortion is lower, the curvature of the map outline in the environ-
ment of the poles is zero or almost zero, therefore the pointed-polar character is not
so much visible, so it is more similar to a flat-polar projection.
cc2=2 and c3=2 in Expression (2). The coefficients in the third row of the Table 2
produced E2
K=0.12379, EK=0.35184.
The mean overall distortion is the lowest in this third case. The map outline is similar to
the previous one, but the curvature converges to infinity approaching the Equator
which causes an inconspicuous singularity on the outline.
INTERNATIONAL JOURNAL OF CARTOGRAPHY 233
The graticule transformation presented in Section 2.3. was adapted to the foregoing
minimum distortion pseudocylindrical projections (a) and (b). The criterion values arose
from the two dimensional Simpson’s rule (segments of 1◦), and the coefficients ciwere cal-
culated by the downhill simplex method. The numerical results of the coefficients ci, con-
cerning (d) and (e) (corresponding to (a) and (b)), for the mapping functions (4) and (5)as
well as the transforming function ψ(3) are available in the Table 2. The criterion values E2
K
and EKare given below, furthermore, the distribution of the area and angular distortions
are given by 2
v
and pon the figures:
(d) The shape of the formula (4) is in this case:
x=c1·
1−2·
c
(
w
,
l
)
p
c2
·
l
+c4·
l
3
and the coefficients in the fourth row of Table 2give E
K2
= 0.11943, E
K
= 0.34558
(Figure 3)
Figure 1. The isolines of the maximum angular deformation 2
v
(◦)and the area scale pfor the
minimum distortion pseudocylindrical projection (a).
Table 2. Coefficients ciand mean overall distortion values of the studied projections.
proj. c1c2c3c4c5c6c7Ek
(a) 0.73044 4.20041 2.00000 0.00471 –––0.36756
(b) 0.75762 2.00000 4.63375 0.00264 –––0.35222
(c) 0.76158 1.67084 5.17538 0.00272 –––0.35184
(d) 0.71416 3.79209 2.00000 0.00902 0.87550 0.01004 0.00273 0.34558
(e) 0.74532 2.00000 4.04753 0.00730 0.93884 0.00271 0.00450 0.31959
(f) 0.77172 2.00000 3.26655 0.00649 0.88525 0.00950 0.00305 0.32531
234 J. GYÖRFFY
(e) The shape of the formula (4) is now:
x=c1·1−2·
c
(
w
,
l
)
p
2
1
c3
·
l
+c4·
l
3
and the coefficients in the fifth row of Table 2results in E
K2
= 0.10214, E
K
= 0.31959
(Figure 4).
The transformation of version (c) was abandoned because there is not a notable differ-
ence between the criterion values of the version (b) and (c), and the outline of the
latter has the mentioned quasi-breakpoint. So the version (a) and (b) underlies the
developments.
Due to this graticule transformation, the mapped parallel curves become concave
upward on the northern hemisphere and concave downward on the southern hemi-
sphere, and their curvatures increase approaching the poles. Version (d) shows a definite
pointed-polar character, while version (e) shows it less. Apart from the nominal scale, the
entire length of the mapped central meridian does not change, but it is no longer divided
equally by parallels, resulting in larger linear scale along the central meridian towards the
poles compared to the equatorial regions. The entire length of the mapped equator is
about three-quarters of the real Earth length.
Both versions of the transformed projections have a substantially reduced (6%
respectively 9%) criterion value compared to the starting pseudocylindrical projections.
Figure 2. The isolines of the maximum angular deformation 2
v
(◦)and the area scale pfor the
minimum distortion pseudocylindrical projection (b).
INTERNATIONAL JOURNAL OF CARTOGRAPHY 235
Figure 3. The isolines of the maximum angular deformation 2
v
(◦)and the area scale pfor the
minimum distortion non-conical projection (d).
Figure 4. The isolines of the maximum angular deformation 2
v
(◦)and the area scale pfor the
minimum distortion non-conical projection (e).
236 J. GYÖRFFY
As it is foreseeable on the basis of the pointed-polar character, the stronger distortions
are concentrated at higher latitudes, while they barely increase towards the bounding
meridians. Both the low angular and area distortions occur in the temperate zones,
and they rise towards both the equator and the poles. The shape of the distortion isolines
on the transformed maps are similar to the ones on the starting maps, but the zones of
low distortions are wider.
It should be noted that the above introduced graticule transformation ψcan be applied
to other types of projections, for example, to the well-known Aitoff projection, too. It
shows a decline of 25% of the criterion value.
The appearance of the poles in version (e) gives a reason for reshaping the outline intro-
duced in 2.4. The projection coefficients were calculated in the same way as previously.
The results are:
(f) According to (3), (4) and (5) where the coefficients are taken from the sixth row of the
Table 2:E2
K=0.10583, EK=0.32531 (Figure 5).
The curvature values are:
k
max =0.71586,
w
k
max =44◦,
k
mean =0.61116, g=1.17131.
(The same values before the outline correction, for the version (e) are:
k
max =0.83436,
w
k
max =57◦,
k
mean =0.62635,g=1.33210.)
The distortion isolines hardly change on the whole, only the angular distortions grew
somewhat at lower latitudes near the bounding meridians.
To compare the results of the upper calculations, Table 2contains the coefficients ciof
the projection equations and the mean overall distortions EKfor all mentioned projections.
Figure 5. The isolines of the maximum angular deformation 2
v
(◦)and the area scale pfor the non-
conical projection (f) derived from the projection (e) with reduced maximal outline curvature.
INTERNATIONAL JOURNAL OF CARTOGRAPHY 237
4. Discussion
The 16 projections reviewed in the introduction together with the 6 studied projections
were compared with respect of the mean overall distortions EKand some characteristics of
the outline shape.
Table 1shows that, as expected, the reviewed conformal and equal-area projections
have the highest EKvalue, and the aphylactic van der Grinten fits in among them.
That’s why these projections are not recommended for geographic world maps
(Baranyi, 1987, pp. 11–12) except, at times, for cylindrical time-zone maps. The above con-
structed pointed-polar projections have the lowest EKvalue (see Table 2). If the flat-polar
property is highlighted, then the Winkel Tripel is by far the best (its EKvalue is roughly
equal to that of version (a)), and even more, the Kavrayskiy VII is also advisable, while
the Robinson and the two mentioned aphylactic projections of Eckert are just on the
line. If excluding the flat-polar projections, versions (d) and (f) are the main candidates
with their lowest EKvalue, and the Baranyi IV is barely acceptable. Note that a flat-polar
projection with an even slightly lower EKvalue can be generated by a method similar
to the one presented in Section 2.2.
An accepted method measures the shape deformations on the map by the distortion of
finite distances (Canters, 2002, p. 108). The distortion of distances is closely linked with the
scale distortions, whose mean can be measured, for example, by the Jordan–Kavrayskiy
criterion (Canters, 2002, p. 43). The correlation coefficient between the Jordan–Kavrayskiy
and Airy–Kavrayskiy criteria calculated for the 23 projections reviewed equals 0.8696,
which shows a close linear relationship between the two criteria. This confirms the low
shape deformations of the represented objects, for example, continents, in the case of
projections with minimal Airy–Kavrayskiy criterion.
The connections between the outline shapes and the mean overall distortions are worth
noting. The cylindrical projections with bounding meridians as straight lines (without any
curvature) have high EKvalue, therefore it does no good to take them into account. The pro-
jections with elliptic shape have a slightly higher EKvalue. The bounding meridians of Eckert
III, Ortelius and van der Grinten are semicircle arcs of constant curvature, with medium or
high mean overall distortions. The pointed-polar projections with minimized distortion
produce the peculiar outline shape with lower curvature near the equator and the poles
and higher curvature between them (see Figures 3and 4), as introduced in Section 2.4.
The shape of the mapped Earth in the renumbered Aitoff and ordinary polyconic as well
as the run of the parallels in the minimized distortion pointed-polar non-conical projection
together suggest that a minimized distortion projection, representing the pole as a line,
generates an outline with concave pole lines similar to that mentioned above. The unfami-
liar shape of the mapped Earth provokes the neglect of these projections.
In addition, the size of the angle at the intersection point of the pole line with the
tangent line of the bounding meridian at its endpoint was checked. The smaller the
angle is at the breakpoints, the more unfavourable they are considered. Apart from
the cylindrical projections with rectangular corners, this angle is under 150◦in the
case of the Eckert V. The greatest angle belongs to the Robinson (165.86◦), while the
same of the Winkel and Kavrayskiy VII are between 158◦and 163◦. In the case of the
concavely curved pole line mentioned above, the angles are between 100◦and 120◦,
and it is a further aesthetic disadvantage. The bounding meridians appearing as
238 J. GYÖRFFY
semicircular arcs in the case of the flat-polar Eckert III and Ortelius are linked to the pole
line by a smooth transition, without any breakpoint. The Mercator–Sanson is the only
listed pointed-polar projection with breakpoints at the poles constructing an angle of
144.69◦. The two bounding meridians of all of the others are linked at the poles
smoothly. Both the advantage of the flat-polar projections and the disadvantage of
the Mercator–Sanson are in this regard consistent with the mean overall distortion values.
5. Conclusions
A minimum distortion pointed-polar projection was sought for world maps, without any
additional restriction concerning the map graticule. The minimization of the distortions
was carried out by minimization of the Airy–Kavrayskiy criterion (giving the mean
overall distortion of the map), so the angular and area distortions were reduced at the
same time. Such a projection is inevitably aphylactic and non-conical. Some examples
of aphylactic pointed-polar projections often used for world maps are: van der Grinten
(EK=0.57687), Aitoff (EK=0.5218), Baranyi IV. pseudocylindrical (EK=0.4067), which
show that the demand of keeping distortions at a level as low as possible, is often ignored.
Starting from of a minimum distortion pointed-polar pseudocylindrical projection
(EK=0.3522)and changing it to a non-conical one by a polynomial graticule transform-
ation with suitably chosen coefficients, the mean overall distortion of a world map was
reduced to EK=0.31959 which shows a decrease by 9.3%. The fall is compared to some
flat-polar projections: 54%to the Web Mercator, 19%to the Robinson and 13%to the
Winkel Tripel one, see Table 1.
Zero or very small curvature of the map-outline in the environment of the poles was
raised by decreasing the maximal outline curvature, so the pointed-polar character of the
map became more visible (see Figure 5). The aesthetic outline correction caused a 1.8%
raise (to EK=0.32531)of the mean overall distortion on the other hand, but even so,
the shapes of the continents are kept favourably.
If the theme of the map requires first and foremost minimal overall distortions (e.g. world
political map), then the projection version (e) can be suggested. In the case of such maps
where the pointed-polar character is expected beyond minimal overall distortions (e.g.
any world maps in school atlases), version (f) is recommended. If there is relevant information
on higher latitudes (e.g. map of world wind currents) the use of version (d)) is advisable.
Disclosure statement
No potential conflict of interest was reported by the author.
ORCID
János Györffy http://orcid.org/0000-0001-5303-3090
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