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The Natural Earth II world map projection
Bojan Šavrič
a,b
, Tom Patterson
c
and Bernhard Jenny
d
a
College of Earth, Ocean, and Atmospheric Sciences, Oregon State University, 104 CEOAS Administration
Building, Corvallis, OR, USA;
b
Esri Inc., Redlands, CA, USA;
c
U.S. National Park Service, Harpers Ferry Center,
Harpers Ferry, WV, USA;
d
School of Mathematical and Geospatial Sciences, RMIT University, Melbourne,
Australia
ABSTRACT
The Natural Earth II projection is a new compromise
pseudocylindrical projection for world maps. The Natural Earth II
projection has a unique shape compared to most other
pseudocylindrical projections. At high latitudes, meridians bend
steeply toward short pole lines resulting in a map with highly
rounded corners that resembles an elongated globe. Its
distortion properties are similar to most other established world
map projections. Equations consist of simple polynomials. A user
study evaluated whether map-readers prefer Natural Earth II to
similar compromise projections. The 355 participating general
map-readers rated the Natural Earth II projection lower than the
Robinson and Natural Earth projections, but higher than the
Wagner VI, Kavrayskiy VII and Wagner II projections.
ARTICLE HISTORY
Received 21 January 2015
Accepted 9 September 2015
KEYWORDS
Natural Earth II projection;
map-reader preference;
world projection; Natural
Earth projection; Flex
Projector
1. Introduction
This article introduces the Natural Earth II projection (Figure 1), a new compromise pseudo-
cylindrical projection for world maps. Pseudocylindrical projections, such as the popular
Robinson projection, are characterized by straight parallels and curved meridians. The
Natural Earth II extends the work done on the Natural Earth projection (Jenny, Patterson,
& Hurni, 2008;Šavrič, Jenny, Patterson, Petrovič, & Hurni, 2011), which applies moderate
rounding to the four corners where bounding meridians and pole lines meet. This
concept was taken one step further by eliminating all indication of corners on a world
map for the Natural Earth II projection. At high latitudes, the bounding meridians bend
steeply inward toward the central meridian and join seamlessly with relatively short pole
lines. The result is a pseudocylindrical projection with a more rounded form resembling
an elongated globe (Figure 1).
The design process, polynomial equation and characteristics of the Natural Earth II
projection are described in the following section. A user study was conducted to evalu-
ate whether map-readers prefer the Natural Earth II projection to similar compromise
projections with straight parallels. The design of the user study, statistical significance
tests and results are presented. The article concludes with a short discussion of our
findings.
© 2016 International Cartographic Association
CONTACT Bojan Šavričsavricb@geo.oregonstate.edu
INTERNATIONAL JOURNAL OF CARTOGRAPHY, 2015
VOL. 1, NO. 2, 123–133
http://dx.doi.org/10.1080/23729333.2015.1093312
2. Natural Earth II
2.1. Design and polynomial equations
The Natural Earth II projection (Figure 1) is a compromise pseudocylindrical projection
designed by Tom Patterson. Flex Projector, a freeware application for the interactive
design and evaluation of map projections (Jenny & Patterson, 2014), was used to design
Natural Earth II. For developing projections, Flex Projector takes a graphic approach that
was first introduced by Arthur H. Robinson during the design of his well-known, epon-
ymous projection (Jenny & Patterson, 2013; Jenny et al., 2008; Jenny, Patterson, & Hurni,
2010; Robinson, 1974). In Flex Projector, the user adjusts the length, shape, and spacing
of parallels and meridians for every 5° of latitude and longitude. The Natural Earth II pro-
jection was designed by adjusting the relative length of parallels and the relative distance
of parallels from the equator, while spacing of meridians remained constant and no
bending was applied to parallels. The design process required trial-and-error experimen-
tation and visual evaluation of the resulting map projection. By shortening the relative
length of the pole lines, smoothly rounded corners of the bounding meridians were
created, which resulted in a rounded graticule resembling an elongated globe.
When the Natural Earth II projection design was completed, the least squares adjust-
ment method was used to develop a polynomial expression of the projection. Šavrič
et al. (2011) detail the derivation of the polynomial expression for the Natural Earth pro-
jection, a similar projection designed by Tom Patterson (Jenny & Patterson, 2014). The
graphical design in Flex Projector defined the Natural Earth II projection with 37 control
points distributed over the complete range of the bounding meridian for every 5°. Each
point defined the relative length of parallels and the relative distance of parallels from
the equator. Control points were used to approximate the graphical design with two
polynomials. To ensure the symmetry about the xand yaxes, the xcoordinate contains
only even powers of latitude
w
, and the ycoordinate consists of only odd powers of
Figure 1. The Natural Earth II projection.
124 B. ŠAVRIČET AL.
latitude
w
. By multiplying the even powers of the latitude with longitude
l
, the meridians
remained equally spaced curves. Details on polynomial derivations are presented by
Canters (2002, p. 133).
Three additional constraints were used in the least square adjustment. The first two
constraints enforced the distance of the pole line from the equator and the length of
the equator to the values selected in Flex Projector. The third constraint fixed the slope
of the ycoordinate equation to 0° at poles, which ensured smoothly rounded corners
where bounding meridians meet the pole lines. Šavričet al. (2011) detail how constraints
are applied and included in the least square adjustment method. Equation (1) is the result-
ing polynomial equation for the Natural Earth II projection.
x=R·
l
·(A1+A2
w
2+A3
w
12 +A4
w
14 +A5
w
16 +A6
w
18)
y=R·(B1
w
+B2
w
9+B3
w
11 +B4
w
13)
(1)
where xand yare the projected coordinates,
l
and
w
are the latitude and longitude in
radians, Ris the radius of the generating globe, and A1to A6and B1to B4are polynomial
coefficients given in Table 1. The polynomial expression for the Natural Earth II projection
consists of 10 coefficients, 6 for the xcoordinate and 4 for the ycoordinate. With an appro-
priate factorization, the polynomial equation can be simplified to 15 multiplications and 8
additions per point. To convert Cartesian coordinates to spherical coordinates, the
Newton–Raphson method is used to find the latitude
w
from the yequation; longitude
l
is computed by inverting the xequation.
2.2. Projection characteristics
The Natural Earth II projection has short pole lines that are 0.226 times as long as the
equator. The smoothly rounded corners of the bounding meridians give the graticule a
rounded appearance. The graticule is symmetric about the central meridian and the
equator. The length of the equator is 0.847 times the circumference of a sphere. The
central meridian is a straight line 0.535 times as long as the equator. Other meridians
are equally spaced polynomial curves and the smoothness of their ends at the pole line
decreases toward the central meridian. The parallels are straight and unequally spaced.
The ratio between the lengths of pole lines and the equator as well as the ratio
between the length of the central meridian and the equator are a result of the projection
design and the derivation of the polynomial equations.
As a compromise projection, Natural Earth II is neither conformal nor equal area, but its
distortion characteristics are comparable to other established projections. The distortion
values of Natural Earth II fall between those of the Kavraiskiy VII, Robinson, Winkel
Table 1. Coefficients for the polynomial expression Equation (1) of the Natural
Earth II projection.
xEquation yEquation
A10.84719 B11.01183
A2–0.13063 B2–0.02625
A3–0.04515 B30.01926
A40.05494 B4–0.00396
A50.02326
A60.00331
INTERNATIONAL JOURNAL OF CARTOGRAPHY 125
Tripel, Wagner VI and Natural Earth projections. Table 2 compares the weighted mean
error in the overall scale distortion index Dab , the weighted mean error in areal distortion
index Dar, and the mean angular deformation index Dan of the Natural Earth II projection to
other compromise and equal-area projections commonly used for small-scale mapping
(for details on how indices are defined, see Canters & Decleir, 1989, pp. 42–43 and
Canters, 2002, p. 48). The local linear scale along meridians and parallels, areal scale and
distortion of angles can be computed from the Gaussian fundamental quantities (for
details on equations, see Canters, 2002, pp. 9–16).
As with all compromise projections, the Natural Earth II projection exaggerates the size
of high-latitude areas. Figure 2 illustrates the distortion characteristics of Natural Earth II. In
the top image, Tissot’s indicatrices are placed at the intersection of the 30° meridians and
parallels that make up the graticule. The area of indicatrices increases toward the poles,
indicating an exaggeration of the size of high-latitude areas.
The middle and bottom images of Figure 2 show isocols of area distortion and
maximum angular distortion. Areal distortion increases with latitude, but does not
change with longitude. Therefore, all isocols of areal distortion are parallel to the
equator. Isocols of maximum angular distortion increase with latitude, which follows the
general pattern common to most pseudocylindrical projections.
Compared to the Natural Earth projection, Natural Earth II has a larger height-to-width
ratio, shorter pole lines and a rounder appearance (Figure 3). It has a similar overall scale
distortion index Dab, a slightly better weighted mean error in areal distortion index Dar and
a larger mean angular deformation index Dan (Table 2). Unlike the Natural Earth projection
which has standard parallels at 33°18′N/S, the Natural Earth II projection has standard par-
allels at 37°4′N/S.
Besides the Natural Earth projections, there are several other compromise pseudocy-
lindrical projections with rounded corners. Winkel II is one example that is similar in
appearance, commonly used and available in many GIS and mapping applications. The
Winkel II projection with standard parallels at approximately 29°41′N/S (Figure 3) has
the same height-to-width ratio as the Natural Earth II projection. Winkel II also has a
Table 2. The weighted mean error in the overall scale distortion index D
ab
the
weighted mean error in areal distortion index D
ar
and the mean angular
deformation index D
an
for the Natural Earth II projection and other world map
projections.
Projection Dab Dar Dan
Kavrayskiy VII 0.23 0.279 19.15
Natural Earth 0.251 0.194 20.54
Natural Earth II 0.254 0.175 21.43
Winkel Triple 0.256 0.179 23.28
Wagner VI 0.263 0.342 20.41
Robinson 0.265 0.275 21.26
Winkel II (
w
S29W41′) 0.268 0.194 21.49
Plate Carrée 0.285 0.571 16.84
Wagner II 0.315 0.116 26.88
Eckert IV 0.363 0 28.73
Miller Cylindrical 0.393 1.303 32.28
Mollweide 0.394 0 7.63
Note: Lower values indicate better distortion characteristics. The Natural Earth II projection is
marked in bold.
126 B. ŠAVRIČET AL.
similar mean angular deformation index Dan, but Natural Earth II has a better overall scale
distortion index Dab and weighted mean error in areal distortion index Dar (Table 2).
Because the Natural Earth II projection has shorter pole lines and a rounder appearance,
areas near the poles are stretched less in the east-west direction than with the Winkel II
projection.
Figure 2. Tissot’s indicatrices (top), isocols of area distortion (middle) and isocols of maximum angular
distortion (bottom) for the Natural Earth II projection.
INTERNATIONAL JOURNAL OF CARTOGRAPHY 127
3. User study
In a previous study of world map esthetics, student test subjects strongly preferred ellip-
tical map graticules to rectangular ones (Gilmartin, 1983). We conducted a user study to
test whether the highly rounded shape of the Natural Earth II projection (which is not
Figure 3. Comparing the Natural Earth II projection with the Natural Earth projection and the Winkel II
projection with standard parallels at 29°41′N/S.
128 B. ŠAVRIČET AL.
elliptical) appeals to map-readers. Participants compared the Natural Earth II projection to
five commonly used compromise projections with straight parallels. The study was part of
a larger user study about map-readers’preferences for world map projections Šavrič,
Jenny, White, & Strebe, 2015; Jenny, Šavrič, & Patterson, 2015.
3.1. Design, process and statistics
The user study used a paired comparison test to evaluate map-readers’preference for the
Natural Earth II projection. The paired comparison test compared each map projection
with every other map projection in the set. The set contained the Kavrayskiy VII, Natural
Earth, Robinson, Wagner II, Wagner VI and Natural Earth II projections (Figure 4). All pro-
jections in the set have compromise distortion and straight parallels. The difference
Figure 4. The six small-scale map projections used in the user study. They are arranged in descending
order, from top left to bottom right, based on general map-reader preference.
INTERNATIONAL JOURNAL OF CARTOGRAPHY 129
between the projections lies mostly in the shape of meridians, height-to-width ratio,
length of pole lines and the representation of pole line corners.
The study contained all 15 possible projection pairs. Participants of the online study
were recruited through Amazon’s Mechanical Turk, online forums and social networks. Par-
ticipants were categorized into two groups of subjects: (1) general map-readers with
limited or no map-making experience, and (2) experts in map projections, cartographers
or experienced GIS users. Each participant evaluated all pairs of map projections by select-
ing the map in a presented pair that he or she personally preferred. Study participants
were also asked demographic questions regarding their gender, age, education level, car-
tographic experience and how often they use maps. Details about the user study survey
process, recruiting, selection of valid responses and participants’demographics can be
found in Šavričet al., 2015.
For the statistical analysis of the paired comparison study, two non-parametric tests of
significance, proposed by David (1988), were used. The overall test of equality (David,
1988) determined which map projections were significantly different from each other
based on how many times users selected a projection from the set. In the post hoc analy-
sis, the multiple comparison range test (David, 1988)identified which graticules were
significantly different from each other based on which graticules participants preferred.
The
x
2test for the 2 ×Ctable, where Crepresents the number of categories for each
participant’s demographic characteristics, evaluated the influence that demographic
characteristics and frequency of map use had on participants’preferences. The
x
2test
was used on all 15 map projection pairs separately. The overall test of equality, multiple
comparison range test and
x
2test were performed separately for both groups of
subjects.
3.2. Results
A total of 448 participants submitted valid responses. Participants were categorized into
two groups: (1) general map-readers with 355 participants, and (2) projection experts,carto-
graphers or experienced GIS users with 93 participants.
General map-readers selected the Robinson and Natural Earth projections more
often than any other projection. Following the Robinson and Natural Earth projections
were the Natural Earth II, Wagner VI and Kavrayskiy VII projections. Wagner II was
the least preferred projection. The six projections are arranged according to general
map-readers’preferences in Figure 4.Table 3 shows results of the pairwise preference
Table 3. Pairwise preference by general map-readers for six pseudocylindrical compromise projections.
123456
(1) Robinson 47% 58% 57% 62% 85%
(2) Natural Earth 53% 52% 57% 61% 86%
(3) Natural Earth II 42% 48% 52% 55% 80%
(4) Wagner VI 43% 43% 48% 48% 86%
(5) Kavrayskiy VII 38% 39% 45% 52% 84%
(6) Wagner II 15% 14% 20% 14% 16%
Note: Names of projections are arranged in both rows and columns according to the total scores. Each row shows the per-
centages of participants that prefer the projection in the row to other projections listed in the column. TheNatural Earth II
projection is marked in bold.
130 B. ŠAVRIČET AL.
as a percentage for general map-readers. Of the 355 general map-reader participants,
58% preferred Robinson and 52% preferred Natural Earth to the Natural Earth II
projection. Natural Earth II was preferred to Wagner VI by 52%, to Kavrayskiy VII
by 58%, and to Wagner II by 80% of participants. An overall test of equality
(
x
2
5,0.01 =15.09, Dn=877.8) showed statistically significant differences in general
map-readers’preferences. Results from a post hoc multi comparison range test are dis-
played in Figure 5. Projections are arranged left to right according to user preference
and any projection that is not outlined by the same line is significantly different based
on user preferences.
Of the 93 participants that made up the group of projection experts, cartographers or
experienced GIS users, 74% preferred the Natural Earth II projection to the Wagner II pro-
jection. Sixty-seven percent preferred Robinson, 62% preferred Kavrayskiy VII and Natural
Earth, and 61% preferred Wagner VI to Natural Earth II. The Robinson and Natural Earth
projections were selected most frequently, followed by Kavrayskiy VII and Wagner VI.
Natural Earth II ranked second to last and Wagner II was the least preferred projection.
The overall test of equality (
x
2
5,0.01 =15.09, Dn=284.04) showed statistically signifi-
cant differences in map-readers’preferences for projections. Figure 5 shows results from
the post hoc multi comparison range test (Table 4).
The
x
2tests on the 15 map projection pairs did not show any differences in user pre-
ference due to gender, age, education level, background in cartography or cartographic
experience for either group of participants. There were no differences attributable to par-
ticipants’frequency of using web or virtual globe maps, or related to the type of map most
often utilized by participants.
Table 4. Pairwise preference by projection experts, cartographers or experienced GIS users for six
pseudocylindrical compromise projections.
123456
(1) Robinson 51% 57% 63% 67% 88%
(2) Natural Earth 49% 62% 52% 62% 86%
(3) Kavrayskiy VII 43% 38% 52% 62% 94%
(4) Wagner VI 37% 48% 48% 61% 89%
(5) Natural Earth II 33% 38% 38% 39% 74%
(6) Wagner II 12% 14% 6% 11% 26%
Note: The table has the same ordering and units of measure as Table 3. The Natural Earth II projection is marked in bold.
Figure 5. Projection preference summary. Projections are arranged with the most preferred on the left.
Preference differences are significantly greater between outlined groups than within.
INTERNATIONAL JOURNAL OF CARTOGRAPHY 131
4. Conclusion
The method for creating Natural Earth II involved three steps: (1) using the graphical user
interface of Flex Projector to design a rough draft projection with a desired look; (2) devel-
oping the polynomial equation that approximates the Flex Projector draft; and (3) fine
tuning the polynomial equation for projection appearance, distortion characteristics and
computational efficiency. The resulting Natural Earth II projection is a rounder pseudocy-
lindrical projection with compromise distortion characteristics similar to other projections
in its class.
Creating the Natural Earth II projection was partially motivated by the idea that map-
readers would prefer a pseudocylindrical projection with rounded corners that more
closely resembles the spherical shape of Earth. However, the user study found that partici-
pants did not have a strong preference for the Natural Earth II projection. Preferences for
the Natural Earth projection, which has slightly rounded corners, and the Robinson projec-
tion, which has more defined corners, were too close statistically to assume that either pro-
jection was preferred to the other. Both the Natural Earth and Robinson projections were
preferred to the highly rounded Natural Earth II projection.
Wagner II was the least preferred projection. Wagner II differs from the other projec-
tions included in the set because meridian lines have a more sinusoidal shape, which
results in the bulging of areas along the equator. Šavričet al. (2015) found that map-
readers generally dislike meridians that strongly bulge outwards at the equator. Our
user study confirms these findings.
Projection experts, cartographers and experienced GIS users strongly preferred four
map projections –Robinson, Natural Earth, Karayskiy VII and Wagner VI –all of which
look more similar to the Robinson projection than to the Natural Earth II and Wagner II pro-
jections. In contrast, general map-readers liked the Natural Earth II projection, preferring it
immediately behind the Robinson and Natural Earth projections. The discrepancy between
the two groups is possibly due to different exposures to the Robinson projection. We
speculate that mapping professionals who routinely use the Robinson projection may
regard it as the standard against which all other pseudocylindrical projections are
measured, thereby favoring projections most similar to it. General map-readers who are
less familiar with the Robinson projection may presumably be more open to projections
with alternative shapes.
The popularity of world map projections changes over time and varies according to the
taste of publishers. For those looking for a new pseudocylindrical projection with accep-
table distortion characteristics, Natural Earth II is available as an option.
Acknowledgments
The authors thank all participants for taking the user study, Brooke E. Marston, Oregon State Univer-
sity, and Jillian A. Edstrom, Esri Inc., for editing the text of this article, as well as the anonymous
reviewers for their valuable comments.
Disclosure statement
No potential conflict of interest was reported by the authors.
132 B. ŠAVRIČET AL.
Notes on contributors
Bojan Šavričis a Software Development Engineer at Esri, Inc. He holds a Ph.D. in geography and a
minor in computer science from Oregon State University. He received his Diploma degree in geode-
tic engineering from the University of Ljubljana and his graduate certificate in geographic infor-
mation science from Oregon State University. His main research interests are map projections,
mathematical techniques in cartography, and the development of tools for cartographers. He is
the author of Projection Wizard, an online map projection selection tool, and a member of the Inter-
national Cartographic Association Commission on Map Projections.
Tom Patterson is Senior Cartographer at the U.S. National Park Service, Harpers Ferry Center. He has
an M.A. in Geography from the University of Hawai‘iatMānoa. He maintains the www.ShadedRelief.
com website and is the co-developer of the Natural Earth cartographic dataset. Tom is a former pre-
sident of the North American Cartographic Information Society and is now Vice Chair of the Inter-
national Cartographic Association, Commission on Mountain Cartography.
Bernhard Jenny is a Senior Lecturer at the School of Mathematical and Geospatial Sciences of RMIT
University, Melbourne, Australia. He obtained a Ph.D. degree and a post-graduate certificate in com-
puter graphics from ETH Zurich, and a M.S. degree in Rural Engineering, Surveying and Environ-
mental Sciences from EPFL Lausanne. His research combines computer graphics, geographic
information science, and cartographic design principles to develop new methods for the visual rep-
resentation and analysis of geospatial information.
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