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RESEARCH ARTICLE

The Equal Earth map projection

Bojan Šavrič

a

, Tom Patterson

b

and Bernhard Jenny

c

a

Esri Inc., Redlands, CA, USA;

b

U.S. National Park Service, Harpers Ferry, VW, USA;

c

Faculty of Information

Technology, Monash University, Melbourne, Australia

ABSTRACT

The Equal Earth map projection is a new equal-area pseudocylind-

rical projection for world maps. It is inspired by the widely used

Robinson projection, but unlike the Robinson projection, retains

the relative size of areas. The projection equations are simple to

implement and fast to evaluate. Continental outlines are shown in

a visually pleasing and balanced way.

ARTICLE HISTORY

Received 6 April 2018

Accepted 23 July 2018

KEYWORDS

Equal Earth projection;

world map projection;

pseudocylindrical projection;

Robinson projection; Gall-

Peters projection

Introduction

A wave of news stories that ran in late March 2017 motivated the creation of the Equal

Earth map projection. Boston Public Schools announced the switch to the Gall-Peters

projection for all classroom maps showing the entire world (Boston Public Schools 2017).

The media reporting by major national and international news outlets, such as The

Guardian (Walters 2017), The Huﬃngton Post (Workneh 2017), National Public Radio

(Dwyer 2017)orNewsweek (Williams 2017), largely focused on these all-too-familiar

themes: the Mercator projection is bad for world maps because it grossly enlarges the

high-latitude regions at the expense of the tropics (true); nowadays, the Mercator

projection is still the standard for making world maps (false

1

); and only maps using

the equal-area Gall-Peters projection can right this wrong (false) (Sriskandarajah 2003,

Vujakovic 2003, Monmonier 2004).

The reaction among cartographers to this announcement, and to others like it in

years past, was predictable: frustration (Vujakovic 2003, Monmonier 2004, Crowe 2017,

Giaimo 2017, Mahnken 2017). It is noteworthy that most of the news stories did not

publish comments from professional cartographers. Our message –that Gall-Peters is

not the only equal-area projection –was not getting through.

We searched for alternative equal-area map projections for world maps, but could not

ﬁnd any that met all our aesthetic criteria. Hence the idea was born to create a new

projection that would have more ‘eye appeal’compared to existing equal-area projec-

tions and to give it the catchy name Equal Earth.

The next sections detail the rationale of our aesthetic criteria, present the Equal Earth

projection, and the approach and mathematical details of the development of the Equal

Earth projection. The ﬁnal section compares the Equal Earth projection to similar

projections. An Appendix mathematically proves its equal-area property.

CONTACT Bernhard Jenny bernie.jenny@monash.edu

INTERNATIONAL JOURNAL OF GEOGRAPHICAL INFORMATION SCIENCE

https://doi.org/10.1080/13658816.2018.1504949

© 2018 Informa UK Limited, trading as Taylor & Francis Group

Aesthetic criteria

The ﬁrst step in developing the Equal Earth projection for world maps was deciding on

its basic characteristics. To create a world map with an appearance familiar to as many

people as possible, it must have an equatorial aspect and north-up orientation. We

rejected developing another equal-area cylindrical projection, such as the Gall-Peters.

Transforming the spherical Earth to ﬁt in a rectangle introduces excessive shape distor-

tions. In the case of Gall-Peters, the continents in mid-latitude and tropical areas are

highly elongated on the north–south axis. Conversely, the pole lines that stretch across

the entire width of the map severely elongate polar regions in the east–west direction

(Figure 1).

We also rejected the concept of an equal-area projection that depicts the poles as

points, such as the Mollweide and sinusoidal projections. On these projections, the

meridians that steeply converge towards the poles present a practical problem for

cartographers. Land areas are pinched on the horizontal axis at high latitudes, which

limits the number of map labels that can be placed there (Figure 1). Aesthetic preference

was another factor. Recent research by Šavričet al.(2015) indicates that map-readers

prefer projections with straight pole lines over those with protruding pole points. The

same study also found a clear preference for pole lines among professionals in carto-

graphy and GIS, while for general map-readers no clear preference for the representa-

tion of poles was found.

Map-readers also prefer projections with meridians that do not excessively bulge

outwards as do the meridians on the sinusoidal or many projections with straight pole

lines, such as the Eckert VI, McBryde and McBryde-Thomas ﬂat-polar series, Putniņš P4ʹ

and P6ʹ, and Wagner IV (same as Putniņš P2ʹ) projections (Šavričet al.2015).

Figure 1. The Equal Earth projection (lower left) and other projections with short pole lines strike a

balance between pole-point projections, such as the sinusoidal and Mollweide, and cylindrical

projections, such as the Gall-Peters. (All projections are equal-area.)

2B. ŠAVRIČET AL.

We decided that the overall shape of the graticule should resemble the one created

by the Robinson projection, because it is preferred by many map-readers (Šavričet al.

2015) and widely used by cartographers (Monmonier 2004, p. 128, Kessler, pers. comm.,

October 2014 and April 2018

2

). The Robinson (1974) projection meets all of our criteria:

it is a pseudocylindrical projection with pole lines and meridians that do not excessively

bulge outwards, has regularly spaced meridians, and a height-to-width ratio close to 1:2.

However, it is not an equal-area projection.

Equal Earth

Equal Earth (Figure 2) has an overall shape resembling that of the Robinson projection

(Figure 6). The meridians of Equal Earth are equally spaced and do not excessively curve

outwards; they instead loosely approximate elliptical arcs that mimic the overall appear-

ance of a globe. The height-to-width aspect ratio of 1:2.05458 is very close to the natural

ratio of a sphere, and pole lines are 0.59247 times the length of the equator. Its graticule

results in a subjectively pleasing appearance of continental land masses.

The equations for the Equal Earth projection for a unit sphere are

sin θ¼ﬃﬃﬃ

3

p

2sin ϕ(1)

x¼2ﬃﬃﬃ

3

pλcos θ

3A1þ3A2θ2þθ67A3þ9A4θ2

y¼θA1þA2θ2þθ6A3þA4θ2

In Equation (1), xand yare the projected coordinates, λand ϕare the longitude and the

latitude, respectively, θis a parametric latitude, and A1to A4are polynomial coeﬃcients

deﬁned as A1¼1:340264, A2¼0:081106, A3¼0:000893, and A4¼0:003796.

For projecting geographic coordinates on an ellipsoid, such as WGS 1984, we suggest

transforming geographic latitudes to authalic latitudes (Snyder 1987, p. 16) before

projecting with the spherical Equal Earth projection.

To convert Cartesian coordinates back to geographic coordinates, the Newton–

Raphson method is used to ﬁnd the parametric latitude θfrom the yequation; the

Figure 2. Equal Earth map projection.

INTERNATIONAL JOURNAL OF GEOGRAPHICAL INFORMATION SCIENCE 3

latitude ϕis computed by inverting the equation for the parametric latitude θ; the

longitude λis computed by inverting the xequation.

Development of the Equal Earth projection

The development of the Equal Earth projection consisted of two steps. We ﬁrst graphi-

cally designed the projection with a custom-made tool for combining two existing

projections. This visual approach allowed us to ﬁnd a subjectively pleasing appearance

of the graticule and continental land masses. The second step consisted in developing

equations. We identiﬁed a set of conditions that determined the structure of the

resulting equations, and then developed a polynomial model for the vertical distribution

of parallels using parametric latitudes θ, an approach similar to Wagner’sUmbeziﬀern

transformation (Wagner 1949).

Graphical design

The development of the Equal Earth projection started by blending various pairs of

existing equal-area projections with Flex Projector (Jenny et al.2008,2010). The projec-

tions resulting from Flex Projector’s Cartesian coordinate blending are not equal-area

(Jenny and Patterson 2013), but the convenient user interface allowed us to identify

potential source projections. We then further explored source projections with a tech-

nique that Strebe (2017,2018)) introduced recently. This technique consists of a combi-

nation of (1) a forward projection from a sphere to Cartesian space, (2) a scaling of the

resulting map in Cartesian space, (3) an inverse projection back to spherical coordinates

(usually using the inverse of the ﬁrst projection), (4) a second forward projection to

Cartesian space, and (5) a second scaling to compensate for shrinking introduced by the

ﬁrst scaling. The major novelty of this method is that if the two source projections are

equal-area, the resulting combined projection is also equal-area. The relative weight of

the two projections in the ﬁnal projection is adjusted by the value (between 0 and 1) of

the ﬁrst scale factor.

Figure 3 shows the tool that we developed to combine pairs of existing projections

with Strebe’s technique. The graphical user interface allows the user to select the

projections involved (for our purpose the ﬁrst forward projection and the inverse

Figure 3. Graphical design of Equal Earth with Strebe’s technique.

4B. ŠAVRIČET AL.

projection are identical) and the ﬁrst scale factor for Strebe’s technique (the second scale

factor is determined programmatically). The height-to-width ratio can also be adjusted

by an area-preserving aﬃne transformation that scales xcoordinates by a user-deﬁned

stretch factor, and divides ycoordinates by the same factor. We added the pseudocy-

lindrical projections that we found promising when blending projections with Flex

Projector, and experimented extensively with diﬀerent combinations and values for the

two scale factors, taking the visual criteria into account that we describe in the preced-

ing section.

We decided to use the Putniņš P4ʹprojection (Putniņš 1934) for the ﬁrst forward

projection and the inverse projection, and the Eckert IV (Eckert 1906) projection for the

second forward projection (Figure 4). The value of the scale factor applied to the

Cartesian coordinates resulting from the ﬁrst forward projection is 0.87. Selecting an

appropriate amount of ‘equal-area stretch factor’for the ﬁnal aﬃne transformation was a

critical consideration (Figure 5). A value of 0.97 results in a compact Earth shape but with

excessively elongated continents (north to south). With a value of 1.07, the continents

look better but the entire Earth is too wide. An acceptable compromise was found

at 1.02.

Figure 4. The Equal Earth projection blends characteristics of the Eckert IV and Putniņš P4ʹ

projections.

Figure 5. North–south elongated (left) and excessive width (right) resulting from small and large

candidate stretch factor; acceptable compromise (center) with a stretch factor of 1.02.

Figure 6. The popular Robinson projection and Equal Earth projection have similar overall shapes.

INTERNATIONAL JOURNAL OF GEOGRAPHICAL INFORMATION SCIENCE 5

While the transformation technique by Strebe is a rigorous mathematical model, the

two forward projections and the inverse projection result in relatively complex mathe-

matical equations. In addition, the Eckert IV projection, which we use as the second

forward projection, requires iterative computations (Snyder 1987). It was therefore

deemed necessary to develop simpler and compact equations for the Equal Earth

projection.

Conditions for equations

When developing equations for the Equal Earth projection, we took the following four

considerations into account.

Condition 1: Equal-area. The equal-area condition for any projection on the sphere

is (after Snyder 1987, p. 28):

@x

@λ@y

@ϕ@x

@ϕ@y

@λ¼R2cos ϕ(2)

In Equation (2) and the following equations, xand yare the projected coordinates, λand

ϕare longitude and latitude, respectively, and Ris the radius of the sphere.

Condition 2: Straight parallels. The pseudocylindrical graticule has straight parallels

that are unequally spaced. This means that ycoordinates solely depend on the latitude

ϕand are independent of the longitude λ(Snyder 1985, p. 37; Canters 2002, p. 141,

Werenskiold 1945). In Equation (3), fyϕðÞis a function depending on the latitude ϕ.

y¼RfyϕðÞ!

@y

@λ¼0 (3)

Condition 3: Regularly distributed meridians. Meridians are equally spaced along

parallels, which means that xcoordinates are a linear function of longitude λ(Snyder

1985, p. 37; Canters 2002, p. 139). In Equation (4), fxϕðÞis a function depending on the

latitude ϕ:

x¼RλfxϕðÞ (4)

Condition 4: Bilateral symmetry. The graticule is symmetric relative to the horizontal

x-axis and the vertical y-axis.

Development of equations for Equal Earth

From conditions (1), (2), and (3), one arrives at the following three general equations for

equal-area pseudocylindrical projections, where fy0ϕðÞis the derivative of fyϕðÞ(see also

Snyder 1985, p. 121; Canters 2002, p. 137; and Werenskiold 1945, p. 4 for similar

formulations):

fxϕðÞfy0ϕðÞ¼cos ϕ(5)

x¼RλfxϕðÞ

y¼RfyϕðÞ

6B. ŠAVRIČET AL.

From conditions (2), (3), and (4), one can approximate the spacing between parallels

with a polynomial using only odd powers of latitude ϕ(Snyder 1985, p. 121; Canters

2002, p. 138–141). The least squares adjustment is used to develop a polynomial

expression of fyϕðÞ.Šavričet al.(2011) provide details for a similar derivation using

least squares adjustment. The relative distances of parallels from the equator are

calculated for every degree using the projection equations of Strebe’s technique with

the parameters discussed above. Those values are then used as a basis in the least

squares adjustment. Equation (6) is the resulting polynomial equation for ycoordinates,

where A1to A4are polynomial coeﬃcients.

fyϕðÞ¼ϕA1þA2ϕ2þA3ϕ6þA4ϕ8

(6)

To avoid representing poles as points (for details see Snyder 1985, p. 124), latitudes in

Equation (6) are renumbered to parametric latitudes θ. We use the approach applied by

Wagner for his Umbeziﬀern transformation (Wagner 1931,1932,1941,1949,1962,1982,

Canters 2002) and by Putniņš for his P4ʹprojection (Putniņš 1934, Snyder

1993): sin θ¼msin ϕ, where mis a factor between 0 and 1 (see Šavričand Jenny

(2014) for a recent detailed description of Wagner’s transformation method).

The spacing between parallels is approximated with a new function dependent on

the parametric latitude θ:fyϕðÞ¼fyθðÞ¼θA1þA2θ2þA3θ6þA4θ8

. Polynomial

coeﬃcients are recalculated using the same relative distances and the least squares

adjustment approach as above. This changes the equal-area condition in Equation (5) to

fy0ϕðÞ¼fy0θðÞfθ0ϕðÞ¼fy0θðÞmcos ϕ

cos θ(7)

fxϕðÞfy0θðÞ¼

cos θ

m

In Equation (7), fy0θðÞis the derivative of fyθðÞand fθ0ϕðÞ¼

mcos ϕ

cos θis the deriva-

tive of sin θ¼msin ϕ.

With fyθ

ðÞand the equal-area condition, the equation for fxϕ

ðÞ

is

fxϕðÞ¼ cos θ

mA1þ3A2θ2þ7A3θ6þ9A4θ8

(8)

The ﬁnal equations for the unit sphere are given in Equation (1).

Evaluation

The Robinson and Equal Earth projections share a similar outer shape (Figure 6). Upon close

inspection, however, the way that they diﬀer becomes apparent. The Equal Earth with a

height-to-width ratio of 1:2.05 is slightly wider than the Robinson at 1:1.97. On the Equal

Earth, the continents in tropical and mid-latitude areas are more elongated (north to south)

and polar areas are more ﬂattened. This is a consequence of Equal Earth being equal-area

in contrast to the Robinson that moderately exaggerates high-latitude areas.

INTERNATIONAL JOURNAL OF GEOGRAPHICAL INFORMATION SCIENCE 7

Equal Earth shares similarities with the Eckert IV and Wagner IV, both of which are

also equal-area pseudocylindrical projections (Figure 7). Equal Earth is an intermediate

form. Unlike the Eckert IV that has a rounded shape, Equal Earth has pointed corners

where the pole lines and lateral meridians meet, as do the Wagner IV and Robinson

projections. On the other hand, Equal Earth has lateral meridians closer to Eckert IV and

Robinson than those on Wagner IV that bulge outwards.

Other equal-area pseudocylindrical projections with even greater lateral bulging

include the Putniņš P4ʹand Eckert VI (Figure 7). On these projections, the orientation

of land areas close to their steeply inclined lateral meridians is highly skewed, and users

look with disfavour on the aesthetics of highly bulging projections (Šavričet al.2015).

The Putniņš P4ʹand Eckert VI nevertheless oﬀer one potential beneﬁt. Their shorter pole

lines give Antarctica a more compact form that better matches its actual shape.

Table 1 shows distortion indices of the Equal Earth projection, the Gall-Peters projec-

tion, and commonly used pseudocylindrical equal-area projections with pole lines. Scale

and angular distortion indices for the Equal Earth projection compare favourably with

distortion indices of similar pseudocylindrical projections, and they outperform the

indices for the Gall-Peters projection.

Figure 7. The Equal Earth compared to similar equal-area pseudocylindrical projections.

Table 1. Distortion indices for pseudocylindrical equal-area projections with pole lines and the Gall-

Peters projection: the weighted mean error in the overall scale distortion index D

ab

and the mean

angular deformation index D

an

(Canters and Decleir, 1989).

Projection Scale distortion D

ab

Angular deformation D

an

McBryde-Thomas Flat-Pole Sine (No. 2) 0.32 26.38

Eckert IV 0.36 28.73

Equal Earth 0.37 29.08

Wagner IV 0.38 30.39

Putniņš P4ʹ0.39 31.54

Wagner I 0.39 31.92

Eckert VI 0.4 32.45

Gall-Peters 0.46 33.06

Indices were computed with Flex Projector (Jenny et al.2008,2010). Lower values indicate less distortion.

8B. ŠAVRIČET AL.

Conclusion

Equal Earth is a possible solution for Boston Public Schools and other organizations

wanting world maps that show all countries at their true relative sizes. As professional

cartographers, though, we know that equal-area projections are not the panacea that

these organizations might think. For example, continental shapes suﬀer. And there are

the many compromise projections (such as the Robinson projection) that are not quite

equal-area but still highly suitable for making world maps. Nevertheless, when an equal-

area map must be used, we oﬀer the Equal Earth projection as an alternative to the Gall-

Peters and other cylindrical and pseudocylindrical equal-area projections. Its key features

are its resemblance to the popular Robinson projection and continents with a visually

pleasing appearance similar to those found on a globe.

Notes

1. For example, Monmonier (2004, p. 127) found that none of the 12 atlases at his local

bookstores used the Mercator projection (with the exception of a single time-zone map). As

for web maps, the Mercator projection is almost exclusively used (Battersby et al.2014), but

these maps are not designed to visualize the entire globe.

2. Fritz C. Kessler, Pennsylvania State University, and Daniel R. Strebe, Mapthematics Inc.,

counted the number of projections in 11 English-language atlases and 1 Russian atlas

published between 2000 and 2011. The Robinson was the most frequent projection.

Acknowledgements

The authors wish to thank Fritz Kessler (Pennsylvania State University) for sharing his atlas research

results, Daniel “daan”Strebe (Mapthematics LLC) for the help implementing his new technique,

and the anonymous reviewers for their valuable comments.

Disclosure statement

No potential conﬂict of interest was reported by the authors.

ORCID

Bojan Šavričhttp://orcid.org/0000-0003-4080-6378

Bernhard Jenny http://orcid.org/0000-0001-6101-6100

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Appendix 1. Proof of equivalence

This appendix proves that the Equal Earth projection preserves the relative size of areas. To prove

it, we need to show the equivalence of Equation (2) (after Snyder 1987, p. 28). First, we compute

partial derivatives of xand ycoordinate functions starting with the partial derivatives of the y

coordinate with respect to latitude ϕ. Because the ycoordinate is a function of the parametric

latitude θand the parametric latitude is a function of latitude ϕ, the chain rule is used to compute

the derivative of this composition: @y

@ϕ¼@y

@θ@θ

@ϕ. A partial derivative of the ycoordinate with respect

to the parametric latitude θis derived from Equation (1):

@y

@θ¼RA1þ3A2θ2þθ67A3þ9A4θ2

Similarly, the partial derivative of the parametric latitude θwith respect to latitude ϕin

Equation (1) is

@θ

@ϕ¼ﬃﬃﬃ

3

pcos ϕ

2cos θ

The partial derivative of the ycoordinate function with respect to the latitude ϕis

@y

@ϕ¼@y

@θ@θ

@ϕ¼RA1þ3A2θ2þθ67A3þ9A4θ2

ﬃﬃﬃ

3

pcos ϕ

2cos θ

Next, we ﬁnd the partial derivative of the ycoordinate function with respect to the longitude λ.

Because the ycoordinate function is independent of longitude λ, the derivative equals 0, that is,

@y

@λ¼0, and we do not need to compute the partial derivative of the xcoordinate function with

respect to latitude ϕ. Regardless of the @x

@ϕderivative, the following product of two partial

derivatives will always be 0:

@y

@λ¼0!@x

@ϕ@y

@λ¼0

Finally, we require the partial derivative of the xcoordinate function with respect to longitude

λ. From Equation (1), the derivative is

INTERNATIONAL JOURNAL OF GEOGRAPHICAL INFORMATION SCIENCE 11

@x

@λ¼R2ﬃﬃﬃ

3

pcos θ

3A1þ3A2θ2þθ67A3þ9A4θ2

Next, we separately compute the left side of the equal-area condition:

@x

@λ@y

@ϕ@x

@ϕ@y

@λ¼@x

@λ@y

@ϕ¼

¼R2ﬃﬃﬃ

3

pcos θRA1þ3A2θ2þθ67A3þ9A4θ2

ﬃﬃﬃ

3

pcos ϕ

3A1þ3A2θ2þθ67A3þ9A4θ2

2cos θ

The above formulation simpliﬁes to the equal-area equivalence in Equation (2):

@x

@λ@y

@ϕ¼R2ﬃﬃﬃ

3

pcos θRA1þ3A2θ2þθ67A3þ9A4θ2

ﬃﬃﬃ

3

pcos ϕ

3A1þ3A2θ2þθ67A3þ9A4θ2

2cos θ¼R2cos ϕ

12 B. ŠAVRIČET AL.