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A compromise aspect-adaptive cylindrical projection for world maps

Bernhard Jenny

a

*, Bojan Šavrič

a

and Tom Patterson

b

a

College of Earth, Ocean, and Atmospheric Sciences, Oregon State University, Corvallis,

OR, USA;

b

US National Park Service, Harpers Ferry, WV, USA

(Received 11 September 2014; final version received 7 November 2014)

There are two problems with current cylindrical projections for world maps. First,

existing cylindrical map projections have a static height-to-width aspect ratio and do

not automatically adjust their aspect ratio in order to optimally use available canvas

space. Second, many of the commonly used cylindrical compromise projections show

areas and shapes at higher latitudes with considerable distortion. This article introduces

a new compromise cylindrical map projection that adjusts the distribution of parallels

to the aspect ratio of a canvas. The goal of designing this projection was to show land

masses at central latitudes with a visually balanced appearance similar to how they

appear on a globe. The projection was constructed using a visual design procedure

where a series of graphically optimized projections was defined for a select number of

aspect ratios. The visually designed projections were approximated by polynomial

expressions that define a cylindrical projection for any height-to-width ratio between

0.3:1 and 1:1. The resulting equations for converting spherical to Cartesian coordinates

require a small number of coefficients and are fast to execute. The presented aspect-

adaptive cylindrical projection is well suited for digital maps embedded in web pages

with responsive web design, as well as GIS applications where the size of the map

canvas is unknown a priori. We highlight the projection with a height-to-width ratio of

0.6:1, which we call the Compact Miller projection because it is inspired by the Miller

Cylindrical projection. Unlike the Miller Cylindrical projection, the Compact Miller

projection has a smaller height-to-width ratio and shows the world with less areal

distortion at higher latitudes. A user study with 448 participants verified that the

Compact Miller –together with the Plate Carrée projection –is the most preferred

cylindrical compromise projection.

Keywords: aspect-adaptive cylindrical projection; Compact Miller projection; Miller

projection; adaptive composite map projections; Mercator; Flex Projector

1. Introduction

Cylindrical projections with equatorial aspect show meridians and parallels as parallel

straight lines. The projected parallels and meridians intersect at right angles, and the world

is mapped to a rectangle. Many cartographers do not recommend using cylindrical

projections for mapping the world because of the notion that rectangular world maps

mislead the map user’s interpretation of the world’s shape. Despite resounding opposition

to the use of cylindrical projections for world maps, the Mercator projection has become

the most frequently used projection for web maps in recent years, regardless of a ‘long

history of discussion about its inappropriateness for general-purpose mapping, particularly

at the global scale’(Battersby et al. 2014, p. 85).

*Corresponding author. Email: jennyb@geo.oregonstate.edu

International Journal of Geographical Information Science, 2015

http://dx.doi.org/10.1080/13658816.2014.997734

© 2015 Taylor & Francis

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We identify three arguments for the use of cylindrical projections: (1) certain phenom-

ena that change with longitude are presumably easiest to read on a map with straight

meridians, such as a map showing world time zones, (2) the appearance of land masses on

cylindrical map projections is familiar to all map users due to the widespread use of the

Web Mercator projection, and (3) the rectangular shape of cylindrical projections

considerably simplifies page layouts. Additionally, rectangular projections seem to be

preferred by some map-makers; however, it is unclear whether this is due to indifference,

ignorance or simply aesthetic preference. In 1993, before the widespread use of the

cylindrical Mercator projection for web maps, Werner studied projection preferences of

map users and found that projections with round shapes were preferred to maps with

rectangular outlines (Werner 1993). However, Šavričet al. (in press) found in a recent

user study that the cylindrical Plate Carrée is one of the most preferred projections among

nonexpert map users when comparing nine commonly used map projections for world

maps.

This article introduces a new compromise cylindrical map projection that adjusts the

distribution of parallels to the aspect ratio of the canvas. The following introductory

sections identify the rationale for developing another map projection, discuss existing

transformations that result in cylindrical projections with varying aspect ratios and

identify candidate projections for constructing an aspect-adaptive cylindrical projection.

The ‘Methods’section discusses the process for visually designing the members of the

aspect-adaptive cylindrical projection family and describes the method used for convert-

ing the visually designed projections to a polynomial expression. The first ‘Result’section

describes the aspect-adaptive map projection for aspect ratios between 0.3 and 1. The

second ‘Result’section describes the Compact Miller projection, a member of the aspect-

adaptive projection family with an aspect ratio of 0.6. The evaluation is split into two

sections. The first ‘Evaluation’section analyses distortion properties of the proposed

aspect-adaptive projections. The second ‘Evaluation’section discusses the Compact

Miller projection in further detail.

1.1. Rationale for the aspect-adaptive cylindrical map projection

We de v e l oped a new f a m i ly of cyl i n d rical ma p p r ojectio n s t hat adju s t t he distr i b u tion

of parallels to the aspect ratio of the map for two reasons. First, modern web pages

use responsive web design that adapts the layout to the viewing environment by using

flexible-sized text, vector graphics and raster images. Adapting a map to available

canvas space in such responsive layout systems is desirable in order to use available

space efficiently, particularly on mobile devices with small displays. Projections

adjusting their aspect ratios to available screen space could also be beneficial when

integrated with a desktop or Web-based GIS where screen size is unknown a priori or

where the size of windows changes frequently. We also expect aspect-adaptive cylind-

rical projections to simplify the workflow for print cartography, where a map’s

dimensions often need to be adjusted to the available size determined by the page

layout.

The second reason for developing a new projection stems from our dissatisfaction

with available cylindrical compromise projections. This may be surprising considering

the hundreds of map projections invented by cartographers in the past. For example,

John P. Snyder’s(1993) seminal inventory of the history map projections lists 265 major

projections, but it could be extended further with less commonly used projections.

However, the number of cylindrical projections in Snyder’s inventory is relatively

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small. Only 39 of the 265 projections are cylindrical projections. Of those 39

projections, 12 are for large-scale maps based on ellipsoids and are not useful for

world maps, 4 projections are specialized projections for mapping satellite tracks and

similar applications, 5 projections are transverse or oblique variants and therefore not

useful for world maps with equatorial aspect, and the central cylindrical projection is

only useful for didactical purposes due to its gross area distortion. Of the remaining 17

cylindrical projections, one is the conformal Mercator, one is the equal-area Lambert

cylindrical with three variations (Gall-Peters, Behrmann, and Trystan Edwards), one is

the equirectangular with a variation by Gall, and five are variations of perspective

cylindrical projections (Gall, Braun, Braun’s second, Kamenetskiy and BSAM [after

the Bolshoi Sovetskii Atlas Mira or the Great Soviet World Atlas]). Only six cylindrical

projections designed for world maps using other approaches remain in Snyder’s list. The

six projections are the Pavlov, Miller Cylindrical, Arden-Close, Kharchenko-Shabanova,

Kavrayskiy I and Urmayev Cylindrical III projections.

Five additional compromise cylindrical projections for world maps that are not

listed by Snyder (1993)andnotmembersoftheequal-area,equirectangularor

perspective projection families can be found in other cartographic literature. These

include the Miller Perspective Compromise, Miller II, Urmayev Cylindrical II,

Tob ler Cy lindr ical I a nd Tobl er Cylindrical II projections. Tabl e 1 orders the 11

cylindrical compromise projections by increasing aspect ratio; 5 of the 11 projections

in Tab le 1 have aspect ratios greater than 0.8 and were designed to compensate for

the huge polar distortion of the Mercator projection. However, projections with such

high aspect ratios should be avoided if possible because of their tendency to grossly

distort areas and shapes. Figure 1 shows the projections listed in Table 1 with aspect

ratios between 0.5 and 0.8, as well as the Plate Carrée and Braun Stereographic

projections.

We could not identify additional compromise cylindrical projections with aspect

ratios between 0.5 and 0.8 (excluding variations of equirectangular and perspective

projections). The reason for this shortage may be that ‘because of the very simplicity

of cylindrical projections in the normal aspect, they were generally ignored by

mathematicians and the more scientific map-makers especially attracted to the devel-

opment of new projections’(Snyder 1993, p. 104). With the discovery of the only

Table 1. Cylindrical compromise projections ordered by aspect ratio.

Equirectangular, stereographic and equal-area cylindrical projections are not

included.

Projection Aspect ratio Reference

Pavlov 0.421 Graur (1956), Snyder (1993)

Miller Perspective 0.543 Miller (1942)

Miller II 0.629 Miller (1942)

Urmayev Cylindrical II 0.698 Bugayevskiy and Snyder (1995)

Tobler Cylindrical I 0.706 Tobler (1997)

Miller Cylindrical 0.733 Miller (1942)

Arden-Close 0.803 Arden-Close (1947), Snyder (1993)

Tobler Cylindrical II 0.832 Tobler (1997)

Kharchenko-Shabanova 0.838 Maling (1960)

Kavrayskiy I 0.877 Graur (1956), Snyder (1993)

Urmayev Cylindrical III 0.922 Maling (1960)

International Journal of Geographical Information Science 3

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conformal cylindrical projection (Mercator) and the two families of equal-area and

equidistant cylindrical projections, only compromise cylindrical projections can be

devised. To customize these compromise cylindrical projections, parallels are distrib-

uted in different ways. Among the few who customized cylindrical projections are

various Soviet cartographers (see Maling 1960 for an overview), Osborn Maitland

Miller (1942; see Monmonier 2002 for the development of Miller’s projection) and

Wal d o R . To b ler (1997).

Plate Carrée, 0.5

Miller Perspective, 0.543

Miller II, 0.629

Braun Stereographic, 0.637

Miller, 0.733

Tobler I, 0.706

Urmayev II, 0.698

Figure 1. Compromise cylindrical projections with aspect ratios between 0.5 and 0.8.

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1.2. Transformations for compromise cylindrical projections

The main objective when developing the aspect-adaptive cylindrical projection was to

adjust the distribution of parallels to the aspect ratio of the map in a visually balanced

way. Other transformable compromise cylindrical projections that change their aspect

ratio and adjust the distribution of parallels are reviewed below. These transformable

projections served as inspirations for the aspect-adaptive cylindrical projection.

The equirectangular projection is an example of a transformable projection. The aspect

ratio can be adjusted by changing the standard parallel. However, at larger aspect ratios –

created with a standard parallel of 35° or higher –the equirectangular projection stretches

land masses in a visually disturbing way. The cylindrical stereographic projection, an

alternative projection where the aspect ratio can also be adjusted by varying the standard

parallel, has less drastic vertical stretching at large aspect ratios.

The Miller Cylindrical and the Miller II projections, developed by Miller (1942), are

two configurations of a continuous series of compromise cylindrical projections con-

structed by adding two terms (mand n) to the Mercator equation. By varying the values of

the two terms, a variety of cylindrical compromise projections can be constructed. The

limiting cases are the equirectangular projection on the one end and the (optionally

stretched or compressed) Mercator projection on the other. Miller’s projection series

shows areas close to the equator similar to the Mercator projection, which is a pleasing

characteristic to many cartographers and has made the Miller Cylindrical a popular choice

among map-makers.

Like Miller’s transformed Mercator, Canters (2002, p. 60) suggested modifying the

cylindrical equal-area projection. The equirectangular projection is also the limiting case

for this transformation. Miller’s and Canters’transformations can be combined by, for

example, choosing the Plate Carrée projection as the pivotal projection. Miller’s transfor-

mation is applied to maps with aspect ratios greater than 0.5, and Canters’transformation

is applied to maps with aspect ratios less than 0.5. The result is a smooth transition

between the Mercator projection for maps with an aspect ratio 1:1, and the Lambert equal-

area cylindrical projection for maps with an aspect ratio of 0.318. For both projections, the

mand nparameters can be computed for a given aspect ratio using the Newton–Raphson

method.

The drawback of the equirectangular, the stereographic, Miller’s transformed Mercator

and Canters’transformed Lambert projection families is that the polar areas are exces-

sively stretched or enlarged at larger aspect ratios. These projections are therefore not

viable options for an aspect-adaptive cylindrical projection.

1.3. Candidate projections for constructing the aspect-adaptive cylindrical projection

The cylindrical projections with variable aspect ratios discussed in the previous section all

have shortcomings. In order to identify candidate projections for inclusion in the new

aspect-adaptive cylindrical projection, this section examines existing projections and

evaluates their potential for inclusion.

The described combination of Miller’s transformed Mercator projection and Canters’

transformed Lambert cylindrical projection uses the Plate Carrée as a pivotal projection

(Figure 1, top left). The Plate Carrée is a reasonable choice because of its simplicity,

equidistance property along meridians and low linear scale distortion, and many map

users are likely familiar with this projection. The Plate Carrée is the standard projection

for disseminating raster data-sets of the Earth, and many maps employ the Plate Carrée ‘as

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is’with no additional projection transformations. Šavričet al. (in press) found that map

users prefer the Plate Carrée projection over eight other commonly used projections for

world maps. Prevalence, convenience and user preference aside, the Plate Carrée is not

ideal for general mapping because of the horizontal stretching it applies to higher

latitudes.

The Miller Perspective projection (which is not constructed with Miller’s Mercator

transformation described in Section 1.2) shows the world with an aspect ratio of 0.543

(Figure 1). This relatively compact aspect ratio compresses both middle and high lati-

tudes, giving land masses a shortened appearance. Similar alternative projections include

Braun Stereographic, a cylindrical stereographic projection that uses the equator as the

standard parallel (Figure 1, bottom left), and Miller II. Although both of these projections

improve upon the original Miller projection, polar areas are still too large compared to the

mid-latitudes and tropics. The Urmayev II and Tobler I projections –both with aspect

ratios close to 0.7 –are similar in appearance to the Miller Cylindrical. The Tobler I

projection was developed as a computationally more efficient alternative to the Miller

projection. Although both Urmayev II and Tobler I projections are more compact and

devote slightly less space to polar areas, they are fairly unknown and unavailable in most

mapping software (Figure 1, top right).

In our opinion, the projections in Figure 1 are the best available compromise cylind-

rical projections for a variety of aspect ratios up to now. However, these projections

generally dedicate too much canvas space to polar areas, considerably inflating the area of

higher latitudes, or apply too much compression to the mid-latitudes. For example,

southern South America looks unusually short in the Miller Perspective and Plate

Carrée projections.

2. Methods

To visually design the aspect-adaptive cylindrical map projection, eight different cylind-

rical projections with aspect ratios between 0.3:1 and 1:1 were produced using a custo-

mized version of Flex Projector (Jenny et al. 2010,2013). Flex Projector is a free, cross-

platform application for creating custom world map projections. The customized version

of Flex Projector differed from the standard version in that the user can lock the aspect

ratio for the projection that is designed. We adjusted, visually assessed and corrected the

vertical distances of parallels for the eight different projections with aspect ratios between

0.3:1 and 1:1 until no further improvement seemed possible. Each of the eight visually

designed cylindrical projections is defined by 18 vertical distances, one distance for every

5° of latitude between the equator and one pole. With eight projections and 18 values for

every projection, there are a total of 144 values. Coding equations with these many values

is impractical because of the increased likelihood of typographical errors. Additionally,

programmers would need to develop code to interpolate between vertical distances, which

would result in incompatible implementations if different interpolation methods were

applied. This is an issue for the Robinson projection, for example, which is defined by

two sets of numbers specifying the length and vertical distribution of parallels. As Šavrič

et al. (2011) point out, there are various incompatible interpolating and approximating

methods. Therefore, we used the least squares method to develop a polynomial expression

for the aspect-adaptive cylindrical projection (discussed in Section 3.1). The numerical

values defining the vertical distances of parallels for the eight projections served as the

basis for developing this polynomial expression. The polynomial expression provides the

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vertical distance of parallels from the equator for any given latitude and an aspect ratio

between 0.3:1 and 1:1.

Cylindrical projections, including projections by Pavlov, Urmayev, Kharchenko-

Shabanova and Tobler (see Table 1 for references), have also been defined using poly-

nomials. Šavričet al. (2011) provide details about creating polynomial expressions for

projections designed with Flex Projector using the Natural Earth projection as an example.

Similar to the aspect-adaptive cylindrical projection, the Natural Earth projection was first

visually designed with Flex Projector, and then, polynomial equations were developed

using the method of least squares.

The goal when developing polynomial equations for the aspect-adaptive cylindrical

projection was to simplify the mathematical model to reduce the number of required

parameters and to simplify the programming of projection equations. Various poly-

nomial forms with different degrees were compared. The weighted least square

adjustment method was used, and weights were adjusted in a trial-and-error procedure

until the resulting expression closely approximated the eight visually designed

projections.

To evaluate the distortion properties of the aspect-adaptive cylindrical projection, the

weighted mean error in areal distortion, the weighted mean error in the overall scale

distortion and the mean angular deformation were computed for a number of aspect ratios

using Equations (1), (2) and (3) (Canters and Decleir 1989, Canters 2002).

Dab ¼1

SX

k

i¼1

aq

iþbr

i

2#1

!"

cos ϕiΔϕΔλ(1)

Dan ¼1

SX

k

i¼1

2 arcsin ai#bi

aiþbi

!"

cos ϕiΔϕΔλ(2)

Dar ¼1

SX

k

i¼1

aibi

ðÞ

p#1ðÞcos ϕiΔϕΔλ(3)

where Dab is the weighted mean error in the overall scale distortion, Dan is the mean

angular deformation, Dar is the weighted mean error in areal distortion, aiand biare the

maximum and minimum scale distortions at the sample point, S¼P

k

i¼1

cos ϕiΔϕΔλis the

sum of the area weight factors, ϕiis the sample point latitude, Δϕand Δλare intervals in

the latitude and longitude (2:5&for all computations in this article), kis the number of

sample points, and p,qand rcoefficients are defined as

p¼1aibi'1

#1aibi<1

#;q¼1ai'1

#1ai<1

#;r¼1bi'1

#1bi<1

#

The three indices described above are the most commonly used measures in map projec-

tion literature and are applied, for example, by Snyder (1987,1993), Canters and Decleir

(1989), Canters (2002) and Šavričand Jenny (2014).

International Journal of Geographical Information Science 7

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3. Results

3.1. Result 1: the aspect-adaptive cylindrical map projection

Figure 2 shows the aspect-adaptive projection family for height-to-width aspect ratios

between 0.3:1 and 1:1. The highest aspect ratios (close to 1) resemble the Mercator

projection, but slightly reduce the extreme area distortion close to the poles. The projection

at the 0.6 aspect ratio is the Compact Miller, which is discussed in the following section.

The Plate Carrée projection is used for the aspect ratio of 0.5 for reasons outlined in Section

1.3. The lowest aspect ratio of 0.3 produces a map that is nearly equal-area with highly

flattened polar land masses.

Equation (4) defines the aspect-adaptive cylindrical map projection for aspect ratios

between 0.3:1 and 0.7:1. This is a polynomial surface with 12 polynomial terms deter-

mined with the method described in the previous section.

x¼λand y1φ;αðÞ¼

φ(k1þφ3(k2þφ5(k3

n(α(π(4)

where xand y1are the projected coordinates, φand λare the latitude and longitude, αis

the height-to-width aspect ratio of the map canvas, k1¼A1þA2(αþA3(α2þA4(α3,

k2¼A5þA6(αþA7(α2þA8(α3and k3¼A9þA10 (αþA11 (α2þA12 (α3, and nis a

normalization factor. The values of the 12 polynomial coefficients Aiare given in Table 2.

The coefficients k1,k2and k3are independent of φand λand they can be precomputed

during the initialization of the projection when the aspect ratio αis known. After

initialization, they can be treated as constants. The same is the case for the normalization

factor n, which is equal to φp(k1þφ3

p(k2þφ5

p(k3with φp¼π=2. The computational

cost per point consists of four multiplications and two additions when the polynomial for y

in Equation (4) is factorized as in Equation (5):

x¼λand y1φ;αðÞ¼φ(k1þφ2(k2þφ2(k3

$%$%

(5)

with ki¼kiαπ

n.

For the aspect ratio between 0.7:1 and 1:1, the vertical ycoordinate of the aspect-

adaptive cylindrical map projection is computed in two steps: (1) the y1coordinate is

computed with Equation (5) using the aspect ratio 0.7:1, and (2) y1for areas above 45° N

and below 45° S is modified by adding a four-term polynomial (Equation (6)).

y2φ;αðÞ¼y1φ;0:7ðÞþ

~

φ(k2;1þ~

φ2(k2;2

$% (6)

~

φ¼

φ#π

4;φ>π

4

0;φ

jj)π

4

φþπ

4;φ<#π

4

8

<

:

where y2φ;αðÞis the ycoordinate for the aspect ratio greater than 0.7, φis the latitude, αis

the aspect ratio of the map canvas, y1φ;0:7ðÞis the ycoordinate computed with Equation

(5) and an aspect ratio of 0.7, k2;i¼k2;iα#0:7ð Þ(π

n2, where k2;1¼B1þB2(α,

k2;2¼B3þB4(α, and n2is a normalization factor equal to π=4(k2;1þπ=4ðÞ

3(k2;2.

The values of the four polynomial coefficients Biare given in Table 3. The additional

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0.3

0.5 Plate Carrée

0.6 Compact Miller

0.7

0.8

0.4

1.0

0.9

Figure 2. The aspect-adaptive cylindrical projection for aspect ratios between 0.3:1 and 1:1.

International Journal of Geographical Information Science 9

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computational cost per point above or below the *45¶llel consists of three multi-

plications and two additions when the k2;iparameters are precomputed.

Figure 3 shows the polynomial surface defined by Equations (5) and (6). The vertical

axis shows the vertical distance of parallels to the equator. For a more accurate compar-

ison, the distances defined by Equations (5) and (6) have been divided by α(π, which

reduces distances to the range [0 …1]. Figure 4 shows the distance of parallels to the

Table 3. Polynomial coefficients Bifor

Equation (6).

Coefficient k2;1Coefficient k2;2

B10:0186 B3#1:179

B2#0:0215 B41:837

Table 2. Polynomial coefficients Aifor Equations (4)

and (5).

Coefficient k1Coefficient k2Coefficient k3

A19:684 A5#0:569 A9#0:509

A2#33:44 A6#0:875 A10 3:333

A343:13 A77:002 A11 #6:705

A4#19:77 A8#5:948 A12 4:148

–30°

–60°–45°

–15° 0°

30° 45° 60°

90°

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

–0.8

–0.6

–0.4

–0.2

0

0.2

0.4

0.6

0.8

–1

1

Parallel

Distance of parallel from equator

Aspect

–90° –75°

15°

75°

Figure 3. Polynomial surface defined by Equations (5) and (6) with distances normalized to [0...1].

The equator and aspect ratio of 0.5 (Plate Carrée) are highlighted.

10 B. Jenny et al.

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equator for selected aspect ratios between 0.3:1 and 1:1. For the aspect ratio of 0.5:1, the

distribution is linear and the resulting projection is the Plate Carrée. For aspect ratios less

than 0.5:1, the slopes of the curves indicate a gradual decrease of spacing between

parallels towards the poles. For aspect ratios greater than 0.5:1, the curves bend in upward

direction, indicating an increasing spacing between parallels. When designing the poly-

nomial expression, care was taken to obtain curves that gradually diverge towards higher

aspect ratios to avoid sudden discontinuities in the graticule.

A single polynomial equation cannot sufficiently approximate all reference projections

because the distributions of parallels between 55° N and S are almost identical for aspect

ratios between approximately 0.7 and 1. This phenomenon is visualized in Figure 4 by the

curves for aspect ratios 0.7, 0.8, 0.9 and 1. At lower latitudes, the four curves overlap; the

curves begin diverging at latitude 45°. When developing equations for the aspect-adaptive

cylindrical projection, it was therefore necessary to extend the polynomial in Equation (5)

with Equation (6) for higher aspect ratios despite extensive trials with different poly-

nomial degrees for Equation (5) and adjusting weights for the least square adjustment for

selected aspect ratios and latitude ranges.

3.2. Result 2: the Compact Miller projection

The Compact Miller projection is a particular case of the aspect-adaptive cylindrical

projection family with a 0.6 aspect ratio (Figure 5). The Compact Miller has been

carefully designed because it is likely to be used frequently for mapping due to its

favourable aspect ratio. We presume that many professional cartographers would select

this aspect ratio when asked to choose a cylindrical projection because it is relatively close

to the natural 1:2 ratio between the length of meridians and the equator, and it shows land

masses in a balanced manner.

The Compact Miller preserves the shape of equatorial and mid-latitude land masses

found on the Miller Cylindrical, which is a familiar projection to many users. Higher

latitudes are shown with a compromise between minimizing areal exaggeration and

retaining the characteristic shapes of land masses, such as Greenland. When designing

the Compact Miller projection, it was important that the distance between lines of latitude

3.25

2.75

2.5

2.25

2

1.75

1.5

1.25

0.75

0.5

0.25

3

1

0

0° 5° 10° 15° 20° 25° 30° 35° 40° 45° 50° 55° 60° 65° 70° 75° 80° 85° 90°

Latitude

y coordinate

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 4. Distance of parallels to the equator for aspect ratios between 0.3:1 and 1:1.

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did not decrease towards the poles. The Compact Miller projection duplicates the spacing

of parallels from the Miller Cylindrical projection between latitude 55° N and S. From

latitude 55° N and S to the poles, the Compact Miller projection maintains approximately

constant spacing. This construction principle is similar to the composite projection

proposed by Kavrayskiy for his first projection (Snyder 1993). Kavrayskiy used the

Mercator projection between latitudes 70° N and S and the equirectangular projection

beyond these latitudes (for equations, see Jenny and Šavričin press).

Equation (5) defines the Compact Miller projection with k1¼0:9902, k2¼0:1604

and k3¼#0:03054.

4. Evaluation

4.1. Evaluation of the aspect-adaptive cylindrical projection

The aspect-adaptive cylindrical projection smoothly adjusts the distribution of parallels

when the aspect ratio changes. The aspect-adaptive cylindrical projection shows the

central equatorial part with relatively small distortion, but accepts a compromise in higher

latitudes where shape and area distortions are larger. We consider aspect ratios between

0.55 and 0.70, a good compromise between acceptable areal distortion at high latitudes

and an overall pleasing appearance. Cylindrical maps with aspect ratios beyond this range

either introduce disproportionate areal distortion or distort the shape of map features

considerably. Deciding on a specific aspect ratio within the 0.55–0.70 range depends on

the map’s purpose, available space for the map in a graphical layout and personal taste. As

the aspect ratio increases from 0.70 to 1, areal distortion at high latitudes increases

dramatically while the distribution of parallels closer to the equator remains constant

and mid-latitude land masses, up to latitude 55° N and S, maintain a familiar appearance

(Figure 2). Latitudinal compression becomes stronger with aspect ratios less than 0.5. The

Areal Distortion Maximum angular

distortion

3

4

3

4

1.5

5

3.5

2.5

2

1

0

2

5

2.5

1.5

1

3.5

20º

120º

80º

60º

40º

100º

10º

5º

0º

20º

120º

80º

60º

40º

10º

5º

100º

Figure 5. Compact Miller projection with areal (left) and maximum angular distortion isolines

(right) and Tissot indicatrices (centre).

12 B. Jenny et al.

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lowest aspect ratio (0.3) produces a map that is nearly equal-area with highly flattened

polar land masses.

Figures 6–8show the weighted mean error in areal distortion, the mean angular

deformation and the weighted mean error in the overall scale distortion for aspect ratios

between 0.3 and 1 for four transformable families of cylindrical projections –equal-area,

stereographic, equirectangular, Canters’transformed Lambert and Miller’s transformed

Mercator (described in Section 1.2, with m¼n)–as well as for selected cylindrical

projections with fixed aspect ratios.

Figure 6 shows the weighted mean error in areal distortion. For the aspect-adaptive

projection, the weighted mean error in areal distortion increases almost linearly with the

aspect ratio. It is interesting to note that almost all compromise cylindrical projections

Aspect ratio

2.5

2.25

1.75

1.5

0.5

1.25

0.25

0.75

2

1

0

0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

Pavlov

Miller Perspective

Miller II

Miller

Urmayev II Tobler I

Tobler II

Kharchenko-Shabanova

Kavrayskiy I

Urmayev III

Plate Carrée

Compact Mill er

Equal-area

Equirectangular

Canters’ transformed Lambert

Aspect-adaptive

Stereographic

Miller’s

transformed

Mercator

Figure 6. Weighted mean error in areal distortion Dar for the aspect-adaptive cylindrical map

projection (solid line) and other selected cylindrical map projections.

50

45

40

35

30

25

20

15

10

5

0

0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

Aspect ratio

Plate Carrée

Pavlov

Miller II Kharchenko-

Shabanova

Compact Miller

Kavrayskiy I

Urmayev II Urmayev III

Tobler II

Tobler I

Miller

Miller Perspective

Aspect-adaptive

Equal-area

Equirectangular

Stereographic

Canters’

transformed

Lambert

Miller’s

transformed

Mercator

Figure 7. Mean angular deformation Dan for the aspect-adaptive cylindrical map projection and

other selected cylindrical map projections.

International Journal of Geographical Information Science 13

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align in Figure 6. The areal distortion of the aspect-adaptive projection for aspect ratios

above 0.65 is high when compared to the equirectangular and the stereographic. This is

expected because the aspect-adaptive projection increases the spacing of parallels from the

equator towards the poles, which is a general characteristic of conformal projections and

projections with small angular distortion. The mean angular deformation of the aspect-

adaptive projection is low in comparison with other projections. Figure 7 illustrates that

the aspect-adaptive projection has less angular distortion than most of the other projec-

tions for aspect ratios greater than 0.5. For aspect ratios less than or equal to 0.5, angular

distortion is similar to that of other cylindrical projections.

Figure 8 shows that scale distortion of the aspect-adaptive cylindrical projection is

similar to all projections with a fixed aspect ratio. The stereographic and equirectangular

projection families have lower scale distortion values.

4.2. Evaluation of the Compact Miller projection

When designing the Compact Miller with Flex Projector, the aspect ratio was not initially

predefined. Through a process of continuous graphical improvements aiming to optimize

the balance between aspect ratio and distribution of parallels, the 0.6 aspect ratio resulted

‘naturally’. Larger aspect ratios either disproportionally stretched higher latitudes and

wasted canvas space or stretched mid-latitudes, creating an elongated appearance of

equatorial land masses, such as Africa, that would appear ‘unnatural’to most map

users. The 0.6 aspect ratio is close to the golden ratio (0.618) frequently found in classical

architecture and art. However, research suggests that world maps with a golden aspect

ratio are not innately preferred by map users (Gilmartin 1983).

Figure 5 shows distortion characteristics of the Compact Miller projection with areal

and maximum angular distortion isolines and Tissot indicatrices. The equator is the

standard parallel without areal or angular distortion. Weighted mean error in areal

0.65

0.6

0.5

0.4

0.3

0.2

0.1

0.55

0.45

0.35

0.25

0.15

0.05

0

0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

Aspect ratio

Pavlov

Plate Carrée

Tobler II Kavrayskiy I

Urmayev III

Kharchenko-

Shabanova

Miller

Tobler I

Miller

Perspective

Urmayev II

Compact

Miller

Miller II

Canters’

transformed

Lambert

Aspect-adaptive

Equal-area

Equirectangular

Stereographic

Miller’s transformed

Mercator

Figure 8. Weighted mean error in the overall scale distortion Dab for the aspect-adaptive cylind-

rical map projection and other selected cylindrical map projections.

14 B. Jenny et al.

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distortion, mean angular deformation and weighted mean error in the overall scale

distortion indices of the Compact Miller are marked on Figures 6,7and 8.

To evaluate the opinions of the general map users and map-makers regarding the

Compact Miller projection, we asked 355 map users and 93 map projection experts,

cartographers and experienced GIS users to do a pairwise comparison of the Plate Carrée,

Braun Stereographic, Mercator, Miller and Compact Miller projections. This online

survey was part of the larger user study about user preferences for world map projections

(Šavričet al. in press). Participants were recruited through Amazon’s Mechanical Turk,

online forums and social networks. Participants were shown all possible 10 pairs created

from the set of 5 projections and asked to select the projection they preferred in each pair.

Details about the user study survey process, recruiting and statistical analysis can be found

in Šavričet al. (in press).

Tab le 4 shows the results for general map users. Of the 355 participants, 52%

preferred the Plate Carrée to the Compact Miller. The Compact Miller was preferred

to the Braun Stereographic by 59% of the participants; 70% preferred the Compact

Miller to the Miller and 86% preferred the Compact Miller to the Mercator. Tabl e 5

shows the results for the map projection experts, cartographers and experienced GIS

users. Of the 93 participants, 58% preferred the Compact Miller to the Plate Carrée, 70%

preferred it to the Braun Stereographic, 83% preferred it to the Miller and 94% preferred

it to the Mercator. The overall test of equality (David 1988) was used to determine

whether any of the map projections had a significantly different preference compared to

Table 4. Pairwise preference of five cylindrical compromise pro-

jections by general map users. The names of the projections are

arranged in both rows and columns according to the total scores.

Each row shows the percentage of participants that have a pre-

ference for the projection in the row to other projections listed in

the column. Compact Miller projection is marked in bold.

12345

1-Plate Carrée 52% 57% 71% 85%

2-Compact Miller 48% 59% 70% 86%

3-Braun Stereographic 43% 41% 75% 87%

4-Miller Cylindrical 29% 30% 25% 87%

5-Mercator 15% 14% 13% 13%

Table 5. Pairwise preference of five cylindrical compromise pro-

jections by projection experts, cartographers and GIS users. The

table has the same ordering and units of measure as Table 4. The

Compact Miller projection is marked in bold.

12345

1-Compact Miller 70% 58% 83% 94%

2-Braun Stereographic 30% 49% 77% 92%

3-Plate Carrée 42% 51% 62% 88%

4-Miller Cylindrical 17% 23% 38% 94%

5-Mercator 6% 8% 12% 6%

International Journal of Geographical Information Science 15

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all other projections. The test showed that differences in preference existed for both

study participants groups: χ2

4;0:01 ¼13:28 and Dn¼928:86 for general map users and

χ2

4;0:01 ¼13:28 and Dn¼328:52 for map projection experts, cartographers and experi-

enced GIS users. To determine which graticules were significantly different in prefer-

ences, a post hoc analysis was performed with the multiple comparison range test

(David 1988). The results, displayed in Figure 9, are arranged with the most preferred

projections on the left. Projections circled individually and not grouped with other

projections were significantly different in preferences.

General map users preferred the Plate Carrée, Compact Miller and Braun

Stereographic projections to the Miller and Mercator projections. There was an almost

even split in preference for the Plate Carrée (52%) and Compact Miller (48%) projections.

Because the total scores of the Plate Carrée and Compact Miller projections were very

close for general map users, we cannot assume that either one of these two projections

was preferred more to the other. Experts most frequently preferred the Compact Miller

projection to the other four projections. Based on the post hoc analysis with a multiple

comparison range test, this preference for the Compact Miller projection is significant

(Figure 9).

5. Conclusions

Our advocacy of cylindrical projections with moderate aspect ratios and less exaggerated

polar areas parallels the famous Peters controversy. In 1973, Peters reintroduced the Gall

projection, an equal-area cylindrical projection with an aspect ratio of 0.636 but flawed by

grossly stretched land masses, as a reactionary response to the perceived ‘Eurocentric’

Mercator projection (Sriskandarajah 2003, Vujakovic 2003). The Gall-Peters projection

was lauded by popular media and found advocates among international organizations

despite objections from respected professional cartographers (e.g. Robinson 1985,1990).

Some cartographic associations, in an attempt to thwart the popularity of the Gall-Peters,

took the extreme position of denouncing all rectangular (i.e. cylindrical) world maps

(American Cartographic Association et al.1989). Neither position won the argument: the

Gall-Peters projection has largely become a historical curiosity, and cylindrical projections

dominate today’s online mapping services. The Mercator projection is ubiquitous once

again. Our call for using cylindrical projections with moderate aspect ratios, less extreme

polar area distortion and recognizable continental shapes is a new attempt at finding

Miller

Cylindrical Mercator

Compact

Miller

Plate

Carrée

Braun

Stereographic

General map users

Compact

Miller

Plate

Carrée

Braun

Stereographic

Miller

Cylindrical Mercator

Projection experts, cartographers and GIS users

Figure 9. Significant differences in the preferences between the projections in paired comparison

test for each participants group. Projections are arranged with the most preferred on the left.

Projections circled individually and not grouped with other projections were significantly different

in preferences.

16 B. Jenny et al.

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acceptable alternatives to the Mercator projection for world maps. Sociopolitical concerns

about the exaggerated size of industrialized countries in the ‘north’compared to devel-

oping tropical countries are a valid argument against using cylindrical world maps with

high aspect ratios. Because many other projections show the entire world with consider-

ably less distortion, cylindrical projections should only be used with compelling reason-

ing. If using compromise cylindrical projections, we recommend avoiding those with

aspect ratios less than 0.55 or more than 0.7. If possible, we recommend using the

Compact Miller projection with an aspect ratio of 0.6 or one of its close neighbours of

the aspect-adaptive cylindrical family with a similar aspect ratio. We believe these

projections achieve an acceptable compromise between distortion properties and visual

appearance, and the Compact Miller was well received by both general map users and

experts in map projections, cartography and GIS.

The aspect-adaptive cylindrical projection family extends the adaptive composite map

projection framework by Jenny (2012), which automatically selects projections based on

the geographic extent of a map. Adjusting projections to the aspect ratio of maps could be

extended in future research to include aspect-adaptive pseudocylindrical projections or

world map projections with curved parallels. We hope to see aspect-adaptive projections

of various types being used on websites with responsive design and within GIS software.

Acknowledgements

The support of Esri is greatly acknowledged, including valuable discussions with David Burrows,

Scott Morehouse, Dawn Wright and others. The authors also thank Brooke E. Marston, Oregon

State University, for editing the text of this article, and Christine M. Escher and Eugene Zhang, both

from Oregon State University, for their help in finding polynomial equations. The authors also thank

the anonymous reviewers for their valuable comments.

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