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12 Number 59, Winter 2008
cartographic perspectives
Flex Projector–Interactive Software for
Designing World Map Projections
Bernhard Jenny
Institute of Cartography
ETH Zurich, Switzerland
jenny@karto.baug.ethz.ch
Tom Patterson
US National Park Service
Harpers Ferry, WV, USA
tom_patterson@nps.gov
Lorenz Hurni
Institute of Cartography
ETH Zurich, Switzerland
hurni@karto.baug.ethz.ch
Initial submission, December 4, 2007; revised submission, February 29, 2008;
final acceptance, March 4, 2008
INTRODUCTION
Flex Projector is a free, open-source, and cross-platform software
application that allows cartographers to interactively design custom
projections for small-scale world maps. It specializes in cylindrical, and
pseudocylindrical projections, as well as polyconical projections with
curved parallels. Giving meridians non-uniform spacing is an option for
all classes of projections. The interface of Flex Projector enables car-
tographers to shape the projection graticule, and provides visual and
numerical feedback to judge its distortion properties. The intended
users of Flex Projector are those without specialized mathematical
expertise, including practicing mapmakers and cartography students.
The pages that follow discuss why the authors developed Flex Projector,
give an overview of its features, and introduce two new map projections
created by the authors with this new software: the A4 and the Natural
Earth projection.
Flex Projector is available at www.flexprojector.com.
espite the central importance of projections to mapmaking, prior
to the release of Flex Projector few cartographers have ever created
a map projection. Explanations for this lack of involvement include the
ready availability of existing map projections; the time and tedium as-
sociated with designing projections, with no guarantee of success; and,
the general lack of mathematical expertise needed to devise projections.
It is an opaque undertaking to all but a few. Not that these barriers have
prevented cartographers from informally experimenting with new projec-
tion designs. In the pre-digital era, pencils, graph paper, French curves,
and optical devices were the tools of choice for such tinkering. Today,
programs such as Adobe Photoshop and Illustrator offer innumerable
graphical tools for changing the appearance of a projection with just the
click of a mouse. What cartographer in an uninhibited moment has not
thought about adjusting the width-to-height proportions of a map so that
it would fit better in a graphical layout, or perhaps applying a transforma-
tion filter to portray the world with a unique new shape? It is completely
natural that mapmakers should want control over the look of world map
projections beyond what is possible by adjusting the parameters of exist-
ing map projections. In an era when nearly all aspects of mapmaking are
customizable by the user, map projection design has been a bastion of
specialization.
Taking a cue from the way cartographers work, Flex Projector offers a
suite of graphical tools and interactive feedback for the design of custom
world map projections. Guiding the software design was the idea that
shape and form are the primary determinants for selecting a projection—
“In an era when nearly all
aspects of mapmaking are
customizable by the user, map
projection design has been a
bastion of specialization.”
cartographic perspectives 13
Number 59, Winter 2008 cartographic perspectives
an acknowledgement that maps are inherently graphical. Flex Projector,
however, is more than just a glorified graphical application for reshaping
how the world looks. It alters the internal geometry of existing projections
to create new projections, provides the user with detailed information
about the angular, areal, and scale distortion properties, imports and ex-
ports data in a variety of graphics and GIS formats, and saves new projec-
tions as text files that others can reproduce. It is a mapping application.
Lessons from the Robinson Projection
Flex Projector uses a graphical approach to map projection design similar
to that used by Arthur H. Robinson for devising the famous projection
that shares his name. In 1961, Robinson was commissioned by Rand Mc-
Nally to design a world map projection that, among other criteria, was un-
interrupted, had limited distortion, and was pleasing to the eye of general
viewers (Robinson, 1974). He came up with a very simple idea: instead
of devising a mathematical formula that relates longitude and latitude
intersections on the sphere to X/Y coordinates on the map, he developed
two sets of tabular parameters by trial and error. The first table described
the length of parallels for every five degrees of increasing latitude (the
horizontal arrows in Figure 1). The second table of parameters defined the
distance of each parallel from the equator, also in steps of five degrees of
increasing latitude (the vertical arrows in Figure 1). Interpolation deter-
mined the coordinates of points for intervals finer than five degrees.
Figure 1. Two sets of tabular parameters define the Robinson projection shown here with horizontal
arrows (length of parallels) and vertical arrows (distance of parallels from equator). Note: not all
parameters appear in the illustration.
Robinson used an iterative process to create his pseudocylindrical pro-
jection, evaluating the appearance and relative relationships of landmass-
es in a succession of drafts. He started by estimating values for the length
and spacing of parallels and then plotted the positions of continents on the
resulting graticule. When the look of the projection was less than satisfac-
tory, as was typically the case early on, Robinson made compensating
adjustments and drafted a new projection. He repeated this process, a sort
of graphic successive approximation, until it became obvious that further
adjustments would produce no improvement, at least to the eyes of the
author (Robinson, 1974, p. 151–152). The Robinson projection was well
received by cartographers and widely used, including by the National
Geographic Society (Garver, 1988).
“. . . instead of devising a
mathematical formula that
relates longitude and latitude
intersections on the sphere to
X/Y coordinates on the map,
Robinson developed two sets of
tabular parameters by trial and
error.”
14 Number 59, Winter 2008
cartographic perspectives
Arthur Robinson’s task would have been greatly simplified had he
employed Flex Projector. The graphical user interface of Flex Projector
allows the user to alter the length of parallels and their distance from the
equator, just as Robinson did. The results immediately appear on screen
with a graticule and sample coastline dataset. Flex Projector extends
Robinson’s methodology in two major ways. (1) Robinson’s projection
uses straight parallels. With Flex Projector, bending parallels to concave or
convex curves is possible, in a manner similar to the arced parallels seen
on the popular Winkel Tripel projection. (2) Robinson’s projection distrib-
uted meridians with even spacing along the equator. This resulted in a true
pseudocylindrical projection, where meridians are equidistant on all paral-
lels (Snyder, 1993, p. 189). Flex Projector provides the option to distribute
meridians with uneven spacing.
Popular map projections, which give shape to our mental image of the
world, fall in and out of vogue over time. Take for example National
Geographic Society (NGS), which used the Van der Grinten I as its world
map projection from 1922 to 1988, a notably long run. In 1988, the NGS
switched to the Robinson projection, originally called the orthophanic
projection, meaning “pleasing to the eye”. The Robinson projection is still
popular today mostly because of the balanced appearance of major land-
masses. It has a classic shape that looks the way a world map should look
to the eyes of many readers. The Robinson era at National Geographic,
however, came to an end in 1998 when the staff chose the Winkel Tripel
projection as its replacement, primarily because its compact form fit better
on a two-page atlas spread. Other map publishers have followed suit and
the Winkel Tripel has risen from relative obscurity to become common
today. Readers of National Geographic will no doubt see a switch to another
world map projection in the future.
Considering that hundreds of map projections already exist, is there
really a need for an application like Flex Projector? To answer this ques-
tion one only has to peruse the world maps in popular atlases. Chances
are good that you will find only a half-dozen or so map projections in
common use, including the Eckert IV, Goode Homolosine, Miller Cylindri-
cal, Mercator, Mollweide, Robinson, and Winkel Tripel projection. This
scarcity-amidst-plenty paradox is in part due to the staid preferences of
map publishers who are unwilling to risk sales by exposing readers to un-
familiar world map projections. Educational publishers in the US prefer to
use only one world map projection in a text for consistency and to avoid
confusing students (Bosacki, 2007). Other factors in the lack of diversity
are the many published projections designed exclusively for large and
medium-scale maps, not small-scale world maps; projections created for
purely mathematical reasons and never intended for everyday mapmak-
ing; and projections that are whimsical. For example, the Apple projection,
which depicts the world in the shape of an apple with a bite taken out of it
(Strebe, 1999), will probably never appear in the National Geographic Atlas
of the World. Personal taste is also a major selection criterion; any given
cartographer may or may not like an otherwise appropriate projection
favored by others. For all of the reasons above, the number of acceptable
projections for making world maps for general audiences is small.
The principal goal of Flex Projector is to give cartographers and the
mathematical layperson a means to design new map projections that are
pleasing to look at, functional, and minimize shape and area distortion.
By expanding the pool of people who can design projections, our hope is
for a proliferation of new projections tailored to meet the specific needs
NEW PROJECTIONS NEEDED
“The principal goal of Flex
Projector is to give
cartographers and the
mathematical layperson a
means to design new map
projections that are pleasing to
look at, functional, and
minimize shape and area
distortion.”
cartographic perspectives 15
Number 59, Winter 2008 cartographic perspectives
of cartographers. And from this might emerge the next blockbuster world
map projection.
A freeware application based on Java 1.5, Flex Projector is cross-platform
compatible on Linux, Mac OS X, and Windows. The authors of this article
developed the core of the application, including the graphical interface,
the algorithms for adjusting the projection, and the code for loading, visu-
alizing and exporting geographical data. The source of projections (other
than the Flex Projection created by the user) is Jerry Huxtable’s Java port
of the widely used PROJ.4 library (Huxtable 2007, Evenden 2005). The
user licence allows others to inspect the code, and add their own exten-
sions. Cubic spline interpolation algorithms govern the shape of projec-
tions created in Flex Projector; however, the graphic interface shields the
average user from this underlying technology.
When designing the interface of Flex Projector, it was the authors’ hope
that users would have little need for the manual. Upon opening Flex
Projector for the first time, the user sees a graphic user interface comprised
of three components (Figure 2). The panel in the upper left is a world map
in the familiar Robinson projection. To the right of the map is the Flex
Projector panel with sliders that control the shape of the projection, and
which beckon the user to experiment. Moving any of the sliders results in
an immediate change to the Robinson projection, which then ceases to be
a Robinson projection and starts on its way to becoming an entirely new
projection. Below the map is the Distortion table, which reports in real-
time the amount of distortion contained in the modified “Flex” projection,
including comparisons to common world map projections.
FLEX PROJECTOR 1.0–
OVERVIEW
Figure 2. Action–Reaction: Moving a slider (A) in Flex Projector, in this case to the left, results in commensurate changes to the map (B) and in the
distortion table (C).
16 Number 59, Winter 2008
cartographic perspectives
Flex Projector Panel
The heart of Flex Projector, the sliders in this panel relate to the upper
right (northeast) quadrant of the displayed map projection. Because
projections created in Flex Projector have bi-lateral symmetry, the soft-
ware automatically mirrors the information in the upper right quadrant to
calculate the shape of the rest of the world. The four tabs at the top of the
panel access sliders for adjusting:
• Length of parallels (see Figure 2 above)
• Distance of parallels from the equator
• Convex or concave bending of parallels
• Distribution of meridians
The Linked Sliders option at the bottom of the panel allows the move-
ment of multiple sliders simultaneously. Increasing the number of linked
sliders generally produces projections with smoother, more uniform
shapes. The Move Sliders option works in conjunction with Linked Sliders to
constrain the movement of sliders to peaked, linear, or bell-curve shapes.
An additional slider on the top of the panel, the Proportions slider, offers a
quick method to alter the height-to-width ratio of a projection.
With the basic controls described above a user can adjust a projection
to an almost infinite variety of shapes, including adjusting the position of
the central meridian to any longitude. However, it is not possible to design
every type of projection. The current version of Flex Projector is limited to
map projections that show the entire world, have an equatorial aspect (the
latitude of origin is always the equator), that are symmetrical relative to
the central meridian and the equator, and uninterrupted. (A later version
of the application may address some of these limitations). All projections
designed in Flex Projector use a spherical earth model. A more compli-
cated ellipsoidal model would not significantly enhance the geometry of
small-scale world maps of the type that Flex Projector was conceived to
create.
Because mistakes are invariably made while working on a new projec-
tion, Flex Projector gives the user unlimited undos to go back to a previ-
ous state. If a projection is completely beyond hope, the Options/Reset to
Projection button at the top right allows the user to reload a fresh Robinson
projection, or one of dozens of other projections.
Display Panel
The Display panel contains displayable options that aid in the construction
of a projection. By clicking the Show Second Projection button, the user can
choose a second map projection that appears as a ghosted template behind
the current Flex projection. The second projection serves as a visual refer-
ence for gauging the design of the Flex projection. The user can change the
color of both the Flex and Second projection in the Preferences drop menu.
Turning on Tissot’s Indicatrices, Isolines of Areal Distortion, and Isolines of
Maximum Angular Distortion shows where distortion occurs in a projection
(Figure 3). Also available are controls for setting the graticule density and
choosing a central meridian other than the Greenwich Meridian, which is
the default.
Distortion Table
Flex Projector offers various numerical indices for assessing projection dis-
tortion (see the bottom-left table in Figure 2): (1) the weighted mean error
“. . . a user can adjust a
projection to an almost infinite
variety of shapes . . .”
cartographic perspectives 17
Number 59, Winter 2008 cartographic perspectives
Figure 3: Isolines of areal (left) and angular (right) distortion for the Natural Earth projection.
for overall scale distortion, (2) the weighted mean error for areal distor-
tion, and, (3) the mean angular deformation index. These indices compute
distortion for the entire globe and for terrestrial areas only (Canters, 2002).
The Acceptance index is an additional numerical measure that summa-
rizes overall projection distortion (Capek, 2001). As the user modifies a
Flex projection, the Distortion Table automatically updates every change
and ranks the projection compared to other well-known world projections
from best (top) to worst (bottom).
Associated with the Distortion table are tabs that display Distortion Pro-
files and Flex Curves. The information in these graphs alerts the user about
otherwise unseen distortion irregularities in a projection.
Data
Flex Projector reads and projects vector, raster, and elevation data from
common GIS and raster formats. The Flex Projector website provides pub-
lic domain vector, shaded relief, and Natural Earth II data to begin making
publication-quality maps. Detailed vector data layers import directly into
Flex Projector via the File drop menu. The following data formats can be
imported: ESRI Shape for vector data, ESRI ASCII GRID for gridded raster
elevation and thematic data, and a variety of image formats, such as JPEG,
TIFF, PNG, and BMP. Flex Projector can export to these formats: DXF, ESRI
Shape, Adobe Illustrator, JPEG, PDF, PNG, SVG, TIFF, and Ungenerate.
When reprojecting raster data, users have the choice of using nearest
neighbor or bi-cubic interpolation. The software assumes that any raster
images with a 2:1 aspect ratio are in the Plate Carrée projection and auto-
matically georeferences them. Flex Projector can work with images large
enough for wall maps. For example, making the Natural Earth projection
(described in a later section) involved reprojecting an image measuring
16,200 x 8,100 pixels in size. Reprojected vector data saved in Adobe Illus-
trator (AI) format, include a bounding box indicating the maximum area
extent. These boxes allow the user to register raster art visually to vector
maps in graphical applications.
Map designers can share projections created in Flex Projector by saving
the projection parameters as text files that others using Flex Projector can
read and use. Projections created in Flex Projector are currently not trans-
ferable to other map projection applications.
The steps below outline a sample workflow in Flex Projector that leads
to a customized projection we call the “A4” projection (Figure 4), which
has similarities to the Winkel Tripel projection. Characteristics of the A4
HOW TO DESIGN A NEW MAP
PROJECTION
“The software assumes that any
raster images with a 2:1 aspect
ratio are in the Plate Carrée
projection and automatically
georeferences them.”
^
18 Number 59, Winter 2008
cartographic perspectives
projection include a compact form factor, arcing parallels, a straight pole
line, and meridians that are regularly spaced along the equator. Designing
the A4 projection involved six steps:
1. The procedure started with the Plate Carrée projection opened via
the Reset dialog.
2. Increasing the height-to-width proportion from 0.50 to 0.655 made
the map fit better on an A4 sheet in landscape format with extra
space in the margins.
3. Adjusting the length of parallels (with the Linked Sliders option
selected) curved the meridians and the overall shape of the map.
Using a repetitive trial-and-error method assured that arcs were
smooth and had the desired curvature.
4. Decreasing the distance of parallels from the equator selectively
at high latitudes compressed the polar areas, moderating the areal
distortion found there.
5. Bending of the parallels reduced the north-south elongation of
Africa and South America. The pole lines received no bending.
6. Applying a scale factor of 0.7785 minimized the total areal distortion
of the graticule, reducing the apparent scale of the map.
Creating a new projection like the A4 projection described by the simple
steps above would, in reality, require frequent use of the undo-redo func-
tionality of Flex Projector to evaluate variations. Employing an iterative
process, the user would assess the shape of the graticule and appearance
of major landmasses after each adjustment. Distortion information ob-
tained from isolines, indices of distortion, and Tissot’s indicatrices would
also guide the design decisions.
In a real-world workflow, the recommended way to design a new
projection is usually not to start with the Plate Carrée projection, but with
a predefined projection closer to that of the desired final. For example,
when making the A4 projection, starting with the Winkel Tripel projection
would simplify steps 1–4 as described above, avoiding unnecessary major
adjustments to the shape of the graticule.
Figure 4. Designing the A4 projection.
“Employing an iterative
process, the user would assess
the shape of the graticule and
appearance of major landmasses
after each adjustment.”
cartographic perspectives 19
Number 59, Winter 2008 cartographic perspectives
An illustration of the A4 projection and its parameters can be found in
Appendix A.
Students and teachers are among the intended users of Flex Projector.
With its hands-on interface and interactive feedback, Flex Projector is
fun to use and entices students to explore how to make map projections.
A mathematical background is not required. For teachers, the software
provides a unique environment for devising creative assignments. For
example: ask students to design a world map projection for a hypothetical
publishing company, similar to what Arthur Robinson did for Rand Mc-
Nally. When designing a new projection, students would have no choice
but to think critically about projection characteristics as part of the creative
process. And requiring students to name their creation after themselves, as
is the convention with naming projections, would motivate them to do a
better job.
Other advantages for education include:
• Learning the importance of defining design objectives before begin-
ning work on a new projection.
• By giving students real-time feedback about angular, areal, and scale
distortion, Flex Projector reinforces the idea that every projection
design involves making significant compromises.
• Students can compare the projections they make against published
projections. Besting the distortion rating of a famous map projection
is a worthwhile and achievable objective.
• Universal access: Flex Projector is free, will run on most computers,
and uses common data formats.
• Advanced students with computer programming experience can
modify the source code to extend the capability of Flex Projector.
That Flex Projector is useful for production cartography should be abun-
dantly clear to those who have read this far. In this section, we discuss the
making of a new projection customized to portray the Natural Earth II
dataset, from which the projection takes its name (Figure 5). Natural Earth
II is a raster map dataset of the planet in the Plate Carrée projection that
features natural environment colors, terrestrial shaded relief, and sea floor
shaded relief with depth tints (Patterson, 2007).
The impetus for creating the Natural Earth projection was dissatisfac-
tion with existing world map projections for displaying physical data.
World physical maps typically employ two classes of projections: cylin-
drical and pseudocylindrical. Cylindrical projections are widely used for
maps with sea floor relief, a preference that is perhaps a throwback to the
traditional use of the Mercator projection for ocean navigation. Using the
Mercator projection for a reference map, however, is less than ideal be-
cause of the extreme areal exaggeration in high latitudes. For example, the
World Ocean Floor map published by National Geographic in 1975 uses
the Mercator projection and omits areas beyond 75 degrees of latitude, to
keep the map to a reasonable size. When making a map with raster digital
data the polar problem is even worse because of poor data quality found
in these areas, which degrades even further with enlargement. Even the
more moderate Miller Cylindrical projection with less polar distortion
than the Mercator suffers from this problem. For example, reproject-
ing raster data from the Plate Carée to the Miller Cylindrical projection
stretches the north-south axis of Greenland by nearly 200 percent, damag-
ing image quality in the process. For this reason, and because the world
is not rectangular in shape, we removed cylindrical map projections from
consideration.
EDUCATIONAL USERS
NATURAL EARTH PROJECTION
“When designing a new
projection, students would have
no choice but to think critically
about projection characteristics
as part of the creative process.”
“Using the Mercator projection
for a reference map, however,
is less than ideal because of the
extreme areal exaggeration in
high latitudes.”
20 Number 59, Winter 2008
cartographic perspectives
Pseudocylindrical projections are better suited for presenting raster
data because their arcing meridians converge toward the poles, compress-
ing the size of these areas and tightening image quality. By selectively di-
minishing the distance of parallels from the equator in high latitudes even
more polar compression is possible. An added benefit to curved meridians
is a more rounded shape that hints that the projection is a 2D representa-
tion of a 3D sphere. In selecting a pseudocylindrical projection to display
the Natural Earth II dataset the following requirements were sought:
• No graticule: creating a projection that could “stand alone” without
the supporting framework of a graticule was important. Because
pseudocylindrical projections have straight parallels, readers can
judge the relative latitude of areas without the presence of a grati-
cule. However, some popular pseudocylindrical projections, for
example, the Eckert IV and Mollweide, have ovoid shapes that look
too soft and capsule-like without a graticule. The projection needed
to have a strong shape.
• Wall map: the Winkel Tripel (not a pseudocylindrical projection
because of its curved parallels) and other projections with compact
forms apply to situations where space is limited, such as the printed
page. By contrast, wall maps are largely free of horizontal space con-
straints and can afford to portray the world with greater breadth and
detail.
• Conventional appearance: a projection with pleasing lines and
minimal distortion that would not detract from the Natural Earth II
data presented on it was a high priority. As an example of what was
not desired, the Sinusoidal projection with its sharply pointed poles
and top-like shape would attract unwanted attention to itself. The
ideal projection needed to be both functional and rather familiar in
appearance.
Figure 5. The Natural Earth projection applied to the Natural Earth II dataset. (see page 68 for color version)
cartographic perspectives 21
Number 59, Winter 2008 cartographic perspectives
The Kavraiskiy VII and Robinson projections—both of which are
compromise projections that are neither conformal nor equal area but
are rated well for overall distortion—came closest to fulfilling the above
requirements. However, each projection had at least one undesirable
characteristic. The Kavraiskiy VII, with its 0.5774 height-to-width propor-
tions, depicts tropical and mid latitude areas with minimal distortion, but
exaggerates the size of high latitude areas—Antarctica is enormous. The
Robinson projection, with its 0.5072 height-to-width proportions, suf-
fers from the opposite problem: it is slightly too wide and its sides bulge
outwards. When centered on the Greenwich Meridian, this results in too
much angular distortion in Alaska, Kamchatka, and New Zealand (and
the adjacent ocean floor) near the map edges.
Seeking the best characteristics of each, the Natural Earth projection is
an amalgam of the Kavraiskiy VII and Robinson projections, plus ad-
ditional enhancements (Figure 6). Making the Natural Earth projection
in Flex Projector started with the Robinson projection. In the first step,
the height-to-width proportion was increased from 0.5072 to 0.52, to give
it slightly more height. The Kavraiskiy VII was then loaded as a second
projection in the background and given the same width as the Robinson
projection. Using the Kavraiskiy VII as a template, the parallels on the
Robinson projection were each increased in length to four decimal points
of precision, to match the bounding meridians of the Kavraiskiy VII.
The projection then took on a completely new form, similar to that of a
truncated Kavraiskiy VII projection. The final procedure for creating the
Natural Earth projection was decreasing the length of the pole lines by a
small amount and giving the corners (where the pole lines and bound-
ing meridians meet) a rounded appearance. This involved trial and error
experimentation and hours of contemplative staring at draft projections
before deciding on the final (See Appendix B for Natural Earth projection
parameters).
Designing the Natural Earth projection with rounded corners served
five purposes:
1) They suggest that the projection represents a spherical Earth.
2) Rounding corners and the related action of lessening the length of
pole lines reduced the size of polar areas, thereby making Antarc-
tica appear smaller.
3) At the top and bottom of the projection, meridians converge inward
toward implied poles, suggesting that the poles are in fact points
instead of straight lines.
4) Aesthetics: from iPod music players, to Jaguar automobiles to the
Mona Lisa, curves convey classic elegance.
Figure 6. The Natural Earth projection combines characteristics of the Kavraiskiy VII and Robinson projections.
“Seeking the best
characteristics of each, the
Natural Earth projection is an
amalgam of the Kavraiskiy VII
and Robinson projections . . .”
22 Number 59, Winter 2008
cartographic perspectives
5) The tightly rounded corners of the Natural Earth projection are
unique among commonly used pseudocylindrical projections, help-
ing to differentiate it.
Because the Natural Earth projection derives from two projections with
low overall distortion, its distortion values fall somewhere between those
of the Kavraiskiy VII and Robinson. Distortion values for the Natural
Earth projection could be slightly better. Graphical considerations, rather
than slavish attention to improving distortion, drove the design decisions.
The final result was a projection with cleaner lines and whose distortion
was still well within acceptable limits (See Appendix C for distortion
tables of the Natural Earth projection).
As a compromise projection, the Natural Earth projection is not equal
area and in fact exaggerates the size of high latitude areas. Despite our
egalitarian desire to show all areas on a map at their true relative sizes,
on physical maps exaggerating the size of high latitudes serves a useful
purpose. Most land on Earth lies at high latitudes in the northern hemi-
sphere and these areas also have highly complex coasts. Greenland and
India, for example, are landmasses of roughly comparable shape and area,
one at high latitudes and the other low. The fjorded Greenland coast is
44,000 kilometers in length, while the smooth coast of India measures only
7,000 kilometers. (Taking into account that the northern boundary of India
does not include coast, unlike Greenland, the difference is still consider-
able). Showing Greenland at a slightly greater scale improves the legibility
of its coast. Not that tropical areas lack for attention on the Natural Earth
projection: with its equatorial aspect and when centered on the Greenwich
Meridian, the projection yields Africa-centric maps. The central location
and regular outline of that continent invariably attract the reader’s eyes.
The Natural Earth projection was designed specifically for making
maps centered on the Equator and Greenwich Meridian, 0 latitude, 0
longitude. The distribution of the continents when centered there has a
pleasing balance and symmetry—especially Antarctica. (That the Green-
wich Meridian is an ideal place to graphically divide the world is entirely
good luck). The 180-degree meridians bisect the Ross Sea, indenting the
coast on the left and right map margins to make the Antarctic continent
appear less large. The coast also trends in the same direction as the merid-
ians converging toward the South Pole, emphasizing the projection shape.
Symbolism is also evident in the shape of Antarctica, which appears as a
pair of white-gloved hands holding a precious object—Earth—and directs
the reader’s eyes north across the Southern Ocean toward warmer regions.
The effect is not unlike the skies on Heinrich Berann’s alpine panoramas
with carefully positioned clouds that draw the reader’s eyes toward
landscape features of interest (Patterson, 2000). Moving the Natural Earth
projection center point only 20 degrees to the east or west ruins this effect.
It is the authors’ hope that Flex Projector will democratize the creation of
world map projections and encourage users to develop innovative and
useful new map projections, as Arthur Robinson did nearly 50 years ago.
For the first time ever, a user-friendly tool is available to do this.
While designing the A4 and the Natural Earth projection with Flex
Projector, we identified possible interface enhancements that could further
ease the making of new projections. Placing the sliders directly on the
graticule would eliminate the need for a separate panel alongside the
map. Or the graticule could itself be adjustable—the user could design a
projection by manipulating the graticule by dragging its nodes with the
mouse. Also, enhanced interpolation methods could support irregularly
spaced control points. The user could then freely add points to the bound-
CONCLUSION
“It is the authors’ hope that Flex
Projector will democratize the
creation of world map
projections . . .”
cartographic perspectives 23
Number 59, Winter 2008 cartographic perspectives
ing meridian where needed, similar to how Bézier curves behave in vector
drawing programs.
Exchanging projection files created in Flex Projector with other map-
ping applications would be useful. Since the software is open-source, de-
velopers can extract portions of the code and extend their applications to
read and interpret descriptions of projections designed with Flex Projector.
By doing this, other mapping applications need not provide tools for the
design of new projections.
Having an application like Flex Projector freely available to everyone is
bound to create some problems. For instance, with the proliferation of new
projections it is a certainty that a few will have shoddy designs, includ-
ing peculiar shapes and large amounts of distortion that the reader is not
aware of. The need to document new map projections is another concern.
Without the text file that describes a projection created in Flex Projector,
the projection is not reproducible nor will it register with other projections.
Then there is the challenge of what to call a new projection. Instead of the
convention of naming a projection after oneself, some users will opt for
more descriptive and eclectic names. The authors of this paper broke with
convention when naming the A4 and Natural Earth projections.
The problems mentioned above, however, are minor when weighed
against the benefit that Flex Projector brings to cartography: a simple
means to create new map projections. Over the last two decades sophis-
ticated technology has made other subfields of cartography accessible to
non-specialists, and the profession has adapted as a result. Flex Projector
continues this trend.
The authors wish to thank the anonymous reviewers for their valuable
comments, Richard Furno, (Azimuth Inc.) for generously sharing his ex-
cellent vector data, as well as Daniel Strebe (Mapthematics LTD) and Hans
Walser (University of Basel) for their advice and comments. Thanks go
also to Gerald I. Evenden for making the Proj4 library publically available
and to Jerry Huxtable for porting this library to Java. We also acknowl-
edge the Swiss National Science Foundation for partially financing this
project.
ACKNOWLEDGMENTS
Figure 7. The A4 projection.
See Appendix A for the
Parameters of the A4 Projection
24 Number 59, Winter 2008
cartographic perspectives
APPENDIX A: PARAMETERS OF THE A4 PROJECTION
The following table lists the parameters for the A4 projection for Flex Pro-
jector 1.0. Note: the A4 projection uses a linear distribution of meridians.
These values equal 0 and are not listed here.
Latitude Length of Distance of Bending of
Parallels Parallels Parallels
from (Cosine)
Equator
0 1 0 -0.2218
5 0.998 0.075 -0.2214
10 0.991 0.1496 -0.2198
15 0.98 0.2235 -0.2166
20 0.965 0.2955 -0.2123
25 0.946 0.366 -0.2068
30 0.922 0.435 -0.2
35 0.895 0.502 -0.1919
40 0.864 0.567 -0.1824
45 0.828 0.629 -0.1716
50 0.789 0.6885 -0.1593
55 0.745 0.746 -0.1455
60 0.697 0.801 -0.1301
65 0.647 0.85 -0.113
70 0.596 0.893 -0.0943
75 0.54 0.93 -0.0737
80 0.479 0.959 -0.0512
85 0.415 0.982 -0.0267
90 0.333 1 0
Height / width 0.655
Scale 0.7785
Direction of meridians at poles 62°
cartographic perspectives 25
Number 59, Winter 2008 cartographic perspectives
APPENDIX B: PARAMETERS OF THE NATURAL EARTH
PROJECTION
The following table lists the parameters for the Natural Earth projection
for Flex Projector 1.0. Note: the Natural Earth projection does not bend
parallels and uses a linear distribution of meridians. These values equal 0
and are not listed here.
Latitude Length of Distance of
Parallels Parallels
from
Equator
0 1 0
5 0.9988 0.062
10 0.9953 0.124
15 0.9894 0.186
20 0.9811 0.248
25 0.9703 0.31
30 0.957 0.372
35 0.9409 0.434
40 0.9222 0.4958
45 0.9006 0.5571
50 0.8763 0.6176
55 0.8492 0.6769
60 0.8196 0.7346
65 0.7874 0.7903
70 0.7525 0.8435
75 0.716 0.8936
80 0.6754 0.9394
85 0.627 0.9761
90 0.563 1
Height / width 0.52
Scale 0.8707
Direction of meridians at poles 60°
26 Number 59, Winter 2008
cartographic perspectives
APPENDIX C: DISTORTION TABLES FOR THE A4 AND THE
NATURAL EARTH PROJECTIONS
Below are three tables comparing the A4 and the Natural Earth projec-
tions to other widely used projections. Note: lower distortion values
are better. For details on the computation of these distortion values, see
Canters and Decleir (1989).
Kavraiskiy VII 0.23
Natural Earth 0.25
Winkel Tripel 0.26
Robinson 0.27
Plate Carrée 0.29
A4 0.30
Eckert IV 0.36
Miller Cylindrical 0.39
Mollweide 0.39
Weighted mean error for overall scale distortion
Eckert IV 0
Mollweide 0
A4 0.15
Winkel Tripel 0.18
Robinson 0.19
Natural Earth 0.19
Kavraiskiy VII 0.28
Plate Carrée 0.57
Miller Cylindrical 1.30
Weighted mean error for areal distortion
Miller Cylindrical 7.63
Plate Carrée 16.84
Kavraiskiy VII 19.15
Natural Earth 20.56
Robinson 21.26
Winkel Tripel 23.28
A4 27.38
Eckert IV 28.73
Mollweide 32.28
Mean angular deformation index
cartographic perspectives 27
Number 59, Winter 2008 cartographic perspectives
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