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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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