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Blending world map projections with Flex Projector

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The idea of designing a new map projection via combination of two projections is well established. Some of the most popular world map projections in use today were devised in this manner. One construction method is to combine two source projections along a common parallel; a second method calculates the arithmetic means of two projections. These two methods for creating new world map projections are included in the latest version of Flex Projector. Flex Projector, afreeware mapping application, offers a graphical approach for customizing existing projections and creating new projections. The Mixer is a new feature in the latest version that allows the user to blend two existing projections to create a new hybrid projection. In addition to the two established combination methods, the software includes a new method for blending projections specific to its visual design approach. With this new method, a unique trait of one projection is transferable to a second projection. Flex Projector allows for the blending of four different projection traits separately or in combination: (1) the horizontal length of parallels, (2) the vertical distance of parallels from the equator, (3) the distribution of meridians, and (4) the bending of parallels. This article briefly describes the main characteristics of Flex Projector and then documents the new approaches to projection blending. The integration of the three methods into Flex Projector makes creating new projections simple and easy to control and allows the user to evaluate distortion characteristics of new projections. As an applied example, the article also introduces the new Pacific projection that is a blend of the Ginzburg VIII and Mollweide projections.
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Blending world map projections with Flex Projector
Bernhard Jenny
a
* and Tom Patterson
b
a
College of Earth, Ocean, and Atmospheric Sciences, Oregon State University, Corvallis, OR 97331-5503, USA;
b
US National Park
Service, Harpers Ferry Center, Harpers Ferry, West Virginia 25425-0050, USA
(Received 12 October 2012; accepted 15 February 2013)
The idea of designing a new map projection via combination of two projections is well established. Some of the most
popular world map projections in use today were devised in this manner. One construction method is to combine two source
projections along a common parallel; a second method calculates the arithmetic means of two projections. These two
methods for creating new world map projections are included in the latest version of Flex Projector.Flex Projector,
a freeware mapping application, offers a graphical approach for customizing existing projections and creating new
projections. The Mixer is a new feature in the latest version that allows the user to blend two existing projections to create
a new hybrid projection. In addition to the two established combination methods, the software includes a new method for
blending projections specic to its visual design approach. With this new method, a unique trait of one projection is
transferable to a second projection. Flex Projector allows for the blending of four different projection traits separately or in
combination: (1) the horizontal length of parallels, (2) the vertical distance of parallels from the equator, (3) the distribution
of meridians, and (4) the bending of parallels. This article briey describes the main characteristics of Flex Projector and
then documents the new approaches to projection blending. The integration of the three methods into Flex Projector makes
creating new projections simple and easy to control and allows the user to evaluate distortion characteristics of new
projections. As an applied example, the article also introduces the new Pacic projection that is a blend of the Ginzburg
VIII and Mollweide projections.
Keywords: projection design; projection blending; Flex Projector; Pacic projection
Introduction
Flex Projector (www.exprojector.com) is a free, open-
source, cross-platform application with a graphical user
interface for designing world map projections. This article
discusses the Mixer, a feature in Flex Projector for selec-
tively combining two projections. The Mixer complements
the other design tools found in Flex Projector and further
simplies the creation of new world map projections.
When developing Flex Projector, the goal was to
give users without expert knowledge in mathematics an
accessible tool for designing world map projections.
The application creates pseudocylindrical and cylindrical
projections, as well as projections with curved parallels.
It allows users to shape the graticule and provides visual
and numerical feedback for assessing distortion properties.
The design of the graphical user interface was done from
the end users perspective ease-of-use and encouraging
experimentation were priorities (Jenny and Patterson
2007; Jenny, Patterson, and Hurni 2008). The intended
users of Flex Projector are practicing mapmakers
and cartography students. Details for the mathematics
and algorithms that convert user settings to formulae for
projecting digital data are covered in Jenny, Patterson, and
Hurni (2010).
The inspiration for developing Flex Projector was
Arthur Robinsons graphical approach to projection
design. In 1961, working on a commission for the
Rand McNally publishing house, Robinson created his
eponymous world map projection, originally dubbed the
orthophanic, meaning correct-looking (Robinson 1974).
Robinson proceeded through an iterative process to create
his pseudocylindrical projection, graphically evaluating
the appearance and relative relationships of landmasses.
He rst estimated the values for parallel lengths and
spacing, then the projection was drawn and the continents
plotted. When he found early drafts less than satisfactory,
compensating adjustments to the graticule were made
and the continents replotted. This iterative process, a
sort of graphic successive approximation, was repeated
until it became obvious that further adjustments would
produce no improvement, at least to the eyes of the
author (Robinson 1974, 151152). Others agreed with
Robinson. His projection has since become widely popu-
lar for making world maps, used by National Geographic
Society (Garver 1988) and other respected cartographic
establishments.
The appeal of the Robinson projection is due in large
part to the pleasant appearance of the graticule and the
*Corresponding author. Email: jennyb@geo.oregonstate.edu
Cartography and Geographic Information Science, 2013
http://dx.doi.org/10.1080/15230406.2013.795002
© 2013 Cartography and Geographic Information Society
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major landmasses. It presents the world in a handsome,
partially oval container; the continents within it look cor-
rect in size and shape to most readers. The success of
the Robinson projection is due largely to the fact that it is
a compromise projection, that is, it preserves neither
angles nor areas. Because designing projections always
involves compromises, a projection by necessity must
distort geographic shapes, often grossly. Conformality
(the preservation of angles) is a property ill-suited to
general world maps (Canters 2002). In contrast, most
cartographers value the equal-area property, as the com-
parison of areal extents is made easier. Additionally, some
cartographic methods require an equal-area base, for
example, choropleth maps (showing values usually nor-
malized by area by differently shaded areas) or dot maps
(where the relative density of dots changes with areal
distortion). However, when strict adherence to the equal-
area property is not required, a compromise projection
often shows the shapes of continents with a more pleasant
appearance than equal-area projections (Canters 2002).
Flex Projector and its Mixer feature allow cartographers
to design compromise projections that balance the com-
peting priorities of equal-area delity and pleasing
appearance.
This article extends this graphical approach and intro-
duces graphical tools for blending existing projections to
create new world projections. Three different methods
for blending projections are included that offer comple-
mentary approaches to the design of world map projec-
tions and are often faster and easier to control than the
original method. Flex Projectorsother functionality and
its graphical user interface are described in more details
by Jenny, Patterson, and Hurni (2008), whereas Jenny,
Patterson, and Hurni (2010) documents the mathemati-
cal background and the visualization of distortion
characteristics.
The discussion is structured as follows: First, we
examine existing methods for combining projections
applied in the past to create a variety of projections. We
identify three groups: combining along lines of latitude,
arithmetic means, and interpolating with varying weights.
The second section discusses three methods implemented
in Flex Projector, including a new approach allowing
the user to combine selected traits of two projections to
create a new projection. The last section then discusses the
design and characteristics of the Pacic projection, before
we conclude with a few nal remarks.
Flex projector for customizing projections
Upon opening Flex Projector, the user sees a graphical
user interface comprised of three components (Figure 1).
The panel in the upper left is a world map in the Robinson
projection, the default. To the right of the map is a panel
with sliders that control the shape of the projection and
tempting 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. Four
groups of sliders exist, for adjusting the length of parallels,
their vertical distribution, their bending, and the distribu-
tion of meridians. Below the map is a table with distortion
Figure 1. Screenshot of Flex Projector: moving sliders change the length of parallels based on increments of ve degrees of latitude
(A), which in turn changes the projection shape (B) and the distortion ranking (C).
2B. Jenny and T. Patterson
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indices, which reports in real-time the amount of distor-
tion contained in the modied projection, including
comparisons to common world map projections. These
distortion indices, as well as specialized distortion visua-
lization techniques, have been documented before (Jenny,
Patterson, and Hurni 2010). The tools in Flex Projector
provide a means to design projections in the same manner
as Robinson did nearly 50 years ago more accurately,
quickly, and with much less tedium.
Combining projections
The idea of designing a new map projection by combin-
ing two existing projections is well established. Some of
the most popular world map projections in use today were
devised in this manner. In general, the goal is to merge
the desired characteristics of two projections, while elim-
inating disadvantages. For example, the pointed poles of
the sinusoidal projection add considerable angular distor-
tion to polar areas, while projections with a polar line,
such as the Robinson projection, introduce less shape
distortion at poles. To date, cartographers have developed
various techniques for combining world map projections,
which can be grouped in three categories, as described
below.
It remains to be mentioned that a variety of alterna-
tive methods for modifying a single projection exist.
Canters (2002, 115ff) distinguishes between polynomial
transformations for projections and the modication
of projection parameters, including Wagners powerful
Unbeziffern (or re-numbering) method (Wagner 1949).
These methods modify a single source projection to
create a new projection and do not combine two source
projections.
Combining projections along lines of latitude
Projections in the rst group are hybrid projections made
by fusing together parts of other projections. For world
map projections, this typically involves joining two pseu-
docylindrical projections along a common parallel. For
example, Goode (1925) combined the Sanson sinusoidal
and the Mollweide projection at 40° 4412′′; north and
south latitude, which is the latitude of equal scale. The
resulting Goode homolosine projection is most common in
its interrupted form. Others proposing non-continuous
combined projections include, for example, Érdi-Krausz
(1968), Hatano (1972), and McBryde (1978) (also see
Canters 2002, 154 and Snyder 1993, 217220, for over-
views). A trait of most non-continuous combined projec-
tions is a discontinuity in the rst derivatives at the
latitude where the two projections join. This typically
appears as a sharp crease where the meridians meet. This
discontinuity can be visually disturbing, especially when it
is concave, as is the case for Goodes homolosine
projection. For some source projections, mathematical
methods exist for eliminating the discontinuities in the
destination projection. For example, Gede (2011) elimi-
nates the visual join of the Érdi-Krausz projection.
Arithmetic means of two projections
Calculating the arithmetic means of two different projec-
tions is the technique for devising a large number of
projections. The two starterprojections are often a
cylindrical projection, such as the plate carrée and a pseu-
docylindrical projection with meridians converging at pole
points. Examples include projections devised by Eckert
(1906), Putniņš (1934), and Winkel (1921). For example,
the Winkel Tripel projection is the arithmetic mean of an
equirectangular projection and the Aitoff projection
(Winkel 1921; Snyder 1993, 231232); Eckert V is an
average of the plate carrée and the sinusoidal (Eckert
1906). Foucaut (1862), Hammer (1900), Nell (1929),
and Tobler (1973) have averaged the cylindrical equal
area and the sinusoidal (after Snyder 1977). Some of
these projections are equal area, which is possible if either
the xor ycoordinate is obtained by averaging the two
source projections and the other coordinate is mathemati-
cally derived from the equal-area condition (Tobler 1973).
This is also the technique used by Boggs (1929) for his
combination of the sinusoidal and the Mollweide.
Interpolating projections with varying weight
Interpolating projections using variable weighting is an
extension of the previous technique. In examples dis-
cussed by Anderson and Tobler (s.d.), the imposed
weighting varies with the latitude, decreasing from one
at the equator to zero at the poles. Tobler (1973) has
applied this technique to create various equal-area
projections.
Combining projections with the Flex Projector mixer
Flex Projector offers three methods to combine projec-
tions. The rst method joins two projections along a
selected latitude, the second method computes an arith-
metic means of two projections, and the third method is a
new approach to combine selected characteristics of two
projections. As described in the introduction of this article,
Flex Projector aims at providing a graphical approach to
the design of map projections. For all three methods, the
user loads two map projections from pop-up lists on
the right side of the main window (Figure 2). Moving
the sliders at the top right interactively combines the
loaded projections. Moving a slider to the left or right
proportionally controls the inuence of each projection.
As the user experiments, changes appear instantaneously
on the large composite world map (Figures 24).
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Combining projections along lines of latitude
With the Latitude Mixer, the user can choose the latitude
along which the two projections are combined. The shared
parallel generally does not have the same length with both
projections, which requires one of the projections to be
scaled. We chose to scale the projection showing higher
latitudes. The scale factor is normally computed automa-
tically or can be adjusted manually (although this option is
probably only useful for didactical purposes). A second
slider denes a latitude band for linearly interpolating
around the shared parallel, which can smooth the crease
along the parallel where the two projections join. A third
slider adjusts the height-to-width ratio of the combined
projection (Figure 2).
Arithmetic means of two projections
The Simple Mixer computes the means of two source
projections. For example, the DNAof the new projec-
tion depicted in Figure 3 is 35% Ginzburg VIII and 65%
Eckert IV. The Ginzburg VIII projection was chosen
because of its appealing depiction of landmasses at mod-
erate latitudes. The equal-area Eckert IV was chosen to
compensate the overly large polar areas of the Ginzburg
Figure 2. Latitude Mixer panel in Flex Projector combining the Miller Cylindrical I (center right) and Mollweide (bottom right)
projections at 45° latitude. (The resulting projection is a dramatic example of the technique and not intended for actual mapping.)
Figure 3. A projection created by computing the weighted means of the Ginzburg VIII (35%) and the Eckert IV (65%) projections and
scaling vertical coordinates by 0.9.
4B. Jenny and T. Patterson
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VIII. Additionally, scaling the height-to-width ratio to 0.9
depicts major landmasses with more graphically pleasing
proportions (Figure 3).
The authors also experimented with options for gra-
phically adjusting blending weights with latitude, the third
idea for combining projections discussed in the previous
section. Interactive spline curves similar to the Curve
Adjustments panel in Adobe Photoshop were added to a
prototype version of Flex Projector. The curves allowed
users to adjust the weight with latitude by adding spline
segments to the curve and by adjusting the position of the
knots between the spline segments. Because the function-
ality of this tool is relatively difcult to grasp for novice
users and because the effect is difcult to control, the
current version of Flex Projector does not include this
option.
Selective combinations
The Flex Mixer selectively transfers a unique trait from
one projection to another, such as replacing the straight
parallels of the Eckert IV projection with the arced paral-
lels of the Winkel Tripel projection. No other character-
istics of the Eckert IV would change.
Four different projection properties can be combined:
(1) the horizontal length of parallels, (2) the vertical dis-
tance of parallels from the equator, (3) the distribution of
meridians, and (4) the bending of parallels. The user can
adjust weights using four sliders (Figure 4, top right). For
example, when setting the weight for the horizontal length
of parallels to 30%, the parallel lengths of the mixed
projection are a combination of 30% of the parallel length
of the rst source projection and 70% of the second
projection. The same principle applies to the other three
properties. If neither of the two source projections has bent
parallels or irregularly distributed meridians, mixing these
attributes would not change the nal projection and the
corresponding sliders are accordingly deactivated.
The simple graphical interface hides an algorithm from
the user that proceeds in three steps. In the rst step, Flex
Projector deconstructsthe two selected source projec-
tions by converting them into tabular form. A projection is
commonly dened by a pair of transformation formulae
with the form X=f(φ,λ) and Y=g(φ,λ) that convert
longitude λand latitude φinto projected Cartesian X/Y
coordinates. The rst step creates four tables of numerical
values for each projection using the corresponding pair of
transformation formulae. Three of the resulting tables
contain values for every 5° of increasing latitude (the
length, vertical distribution, and bending of parallels)
and one table contains values for every 15° of increasing
latitude (the horizontal distribution of meridians). Both
source projections are converted to these tabular forms,
resulting in 2 ×4 tables.
In a second step, the four pairs of tables are blended
using the four user-dened weights. The four pairs of
tables are merged to four blended tables by computing a
weighted average of each pair of corresponding tabular
values.
The nal third step converts geographical longitude/
latitude coordinates to Cartesian X/Ycoordinates. The
conversion interpolates spline curves through the values
stored in the four tables and then applies an extended
version of the method presented by Robinson (1974) for
Figure 4. Flex Mixer panel in Flex Projector: The Pacic projection (left) is a blend of the Ginzburg VIII (center right) and the
Mollweide (bottom right) projections. Sliders at the top right control the blending of the three active parameters (with blue buttons) that
dene the hybrid Pacic projection.
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projecting geographic coordinates to Cartesian coordinates
(see Jenny, Patterson, and Hurni (2010) for details).
After mixing the four different projection traits with
this method, the user may then ne-tune the new projec-
tion by adjusting individual values of one of the four
tables with Flex Projectorsgraphical user interface, as
shown in Figure 1.
It must be mentioned that the described technique does
not work perfectly for all projections. The reason is that
the rst step in the algorithm (the transformation from
formulae to tabular values) sometimes does not accurately
replicate the original projection. While cylindrical and
pseudocylindrical projections with regularly distributed
meridians are matched perfectly, only approximate trans-
formations are possible for projections with arcing paral-
lels or projections with irregularly spaced meridians.
The Pacic projection
As a practical example on how to use the Mixer, we
created the hybrid Pacicprojection by combining the
Ginzburg VIII and Mollweide projections (Figure 4).
The design intent was a world map with a rather conven-
tional appearance centered on 160° west longitude to
focus on the Pacic Ocean and with relatively little areal
distortion. We chose the Mollweide projection because it
is oval in shape, which complements the roundness of the
Pacic basin, and equal-area. The Ginzburg VIII contri-
butes unevenly distributed meridians that are widely
spaced at the projection center and compressed at the
map margins, a useful feature for emphasizing the
Pacic in relation to other parts of the world. Creating
the Pacic projection involved adjustments to three para-
meters controlled by the sliders at the top right of the
Mixer panel (Figure 4). Because neither the Ginzburg
VIII nor Mollweide projection has parallels that bend,
this parameter is disabled in the graphical user interface.
Adjustments to the sliders included the following:
Length of Parallels. Setting the slider in the middle
at 50% gives equal weight to the Ginzburg VIII and
Mollweide projections for this parameter. This combi-
nation gives the Pacic projection highly rounded pole
lines that merge smoothly into the lateral meridians.
Distance of Parallels. Setting the slider at 100%
by dragging it all the way to the right toward the
Mollweide projection weighted this parameter entirely
from that projection. The Pacic projection, asa result,
has a Mollweide-like vertical distribution of parallels.
Distribution of Meridians. Setting the slider at 0% by
dragging it all the way left weights this parameter
entirely toward the Ginzburg VIII projection, thus
increasing the area of the Pacic Ocean because the
meridians near the center point are more widely spaced
than those at the map margins.
Once work in the Mixer is nished, the user can further
enhance the combined projection using the other tools in
Flex Projector. In the case of the Pacic projection, we
increased the height-to-width proportions from 0.507 to
0.55 to make the map taller and accentuate the round
shape of the Pacic Ocean (Figure 5). Additionally, we
scaled the entire map by a factor of 0.8 to minimize area
distortion for the Pacic region. By saving the nal Pacic
projection as a small text le, we could use it again in Flex
Projector for producing publishable-quality maps with
imported shapeles and raster geodata. However, opera-
tional use of the Pacic projection and all custom
Figure 5. The Pacic projection (black) overlaid on the Robinson projection (gray). The Robinson is scaled to the same width.
The taller Pacic projection devotes relatively more area to the Pacic Ocean than does the Robinson.
6B. Jenny and T. Patterson
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projections created with Flex Projector is not possible
outside of the Flex Projector environment. Doing this
would require programming of custom code or, for some
cases, development of the approximating mathematical
formulas (Šavričet al. 2011).
Projections created with the Mixer exhibit distortion
values similar to the projections from which they derive.
For example, because the Pacic projection is part
Mollweide projection, which is equal-area, it ranks favorably
for areal distortion. It is not free of areal distortion, however,
because of the inuence of the non-equal-area Ginzburg
VIII. The Flex Projector distortion tables (Figure 6) show
how the Pacic projection ranks against common world map
projections for the other distortion categories it is unexcep-
tional. When comparing the distortion values in Figure 6,
lower values are better for all categories. The tint in Figure 7
shows the extent of acceptable area on the Pacic projection.
The acceptable area in this example is the area with an
angular distortion of less than 30° and areal distortion
between 90% and 111%. The isoline maps in Figure 8
show patterns of maximum angular distortion and areal dis-
tortion. For both metrics, the amount of distortion is mini-
mized over the one-third of Earth occupied by the Pacic
Ocean, including relatively high latitudes. The area distortion
isolines in Figure 8 show the compression and expansion of
area, with the thick isoline indicating a line without area
distortion. The center of the map at 160°W and 0°N has an
area distortion of 97.2%, which means that the central part of
the map is compressed by less than 3%. The entire Pacic
Ocean, up to the latitude of approximately 50° N and S, is
displayed with less than 5% area distortion. For a full expla-
nation of the distortion tables and indices, as well as the
distortion isolines, refer to Jenny, Patterson, and Hurni
(2010).
Conclusion
Flex Projector simplies the design of new hybrid projec-
tions. It can selectively blend individual characteristics of
existing projections. If necessary, the blended projection
can serve as a good starting point for additional ne-
tuning using the graphical user interface for adjusting the
curves and other options (as shown in Figure 1). It is also
possible to load a projection created in the Mixer back into
the Mixer to combine it again with other projections.
Figure 6. Flex Projector distortion table with the weighted
mean errors for scale and area distortion and mean angular
deformation indices (Canters 2002). The Pacic projection is
the highlighted row.
Figure 7. The tint indicates the area of acceptable angular and
areal distortion for the Pacic projection. Most of the Pacic with
the exception of waters adjacent to Antarctica show angular
distortion of less than 30° and areal distortion between 90%
and 111%.
Figure 8. Isolines of maximum angular distortion (top) and
areal distortion (bottom) for the Pacic projection.
Cartography and Geographic Information Science 7
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Repeating this process can yield an almost innite variety
of new projections.
In the Mixer, it is easy to design new projections that not
only are visually pleasing, but also have excellent distortion
characteristics. For example, the two equal-area projections
blended in Figure 9 yield a new pseudocylindrical projection
with an overall shape similar to the Robinson projection and
with less areal distortion (0.05 vs. 0.19, 0.0 is equal-area),
albeit at the expense of additional deformation for continen-
tal shapes. According to Anderson and Tobler (s.d.),
blended map projections are splendid projections.We
think that users of the Flex Projector Mixer will come to
the same conclusion.
Acknowledgments
The authors thank the anonymous reviewers for their valuable
comments.
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Mitteilungen 67: 248252.
Figure 9. Combining two equal-area projections (left and middle) produces a blended hybrid (right) that is nearly equal-area.
8B. Jenny and T. Patterson
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... Over the last two thousand years, several map projections have been proposed. Map construction methods involve geometry projections, mathematical constructions [6][7][8][9], transformation and combinations [10][11][12][13][14][15], and approximation and optimization [16][17][18][19]. Modern computer software and libraries are also available for cartography and coordinate transformations, e.g., NASA's G.Projector and Open Source Geospatial Foundation's PROJ library [20,21]. ...
... The calculation of the GCA-based metric is dependent on ϕ(x, y) and λ(x, y), the inverse equations of the map projection (see Equation (13)). To avoid the influence of inverse equations, the FWD-GCA metric, an interpolated version of the GCA-based metric, is introduced in this section. ...
Article
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We studied the numerical approximation problem of distortion in map projections. Most widely used differential methods calculate area distortion and maximum angular distortion using partial derivatives of forward equations of map projections. However, in certain map projections, partial derivatives are difficult to calculate because of the complicated forms of forward equations, e.g., equations with iterations, integrations, or multi-way branches. As an alternative, the spherical great circle arcs–based metric employs the inverse equations of map projections to transform sample points from the projection plane to the spherical surface, and then calculates a differential-independent distortion metric for the map projections. We introduce a novel forward interpolated version of the previous spherical great circle arcs–based metric, solely dependent on the forward equations of map projections. In our proposed numerical solution, a rational function–based regression is also devised and applied to our metric to obtain an approximate metric of angular distortion. The statistical and graphical results indicate that the errors of the proposed metric are fairly low, and a good numerical estimation with high correlation to the differential-based metric can be achieved.
... When combining two source projections to create a new projection, the goal is to replace an unfavorable trait of one projection with the characteristics of another projection. Jenny and Patterson (2013) introduce an alternative approach to the blending and compositing techniques discussed above that aims at selectively combining the traits of two projections. This software-based technique first extracts the geometric characteristics of the two source projections and builds tables with according values, then lets the user selectively mix these tables, and finally converts the mixed tables back to a new projection. ...
... With a height-to-width aspect set to 0.54, a compact graticule results with distinctly curved meridians near pole lines. The reader is referred to Jenny and Patterson (2013) and Jenny et al. (2010) for additional examples and more details on the algorithmic procedure. ...
Chapter
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Cartographers have developed various techniques for deriving new projections from existing projections. The goal of these techniques is to substitute a disadvantageous trait of one of the source projections with the second source projection. This chapter discusses creating new projections by the juxtaposition and blending of two existing projections. It also presents a new approach for selectively combining projection characteristics. The emphasis in this chapter is on projections for world maps , as the described techniques are most useful for this scale.
... Con respecto al estudio de las deformaciones, los trabajos se encaminan a desarrollar métodos de cálculo de los valores de deformación de las proyecciones y la forma gráfica de su representación (Kirtiloglu, 2010;Battersby y Kessler, 2012;Oztug, 2015). Por su parte, las perspectivas dedicadas a la propuesta de nuevas proyecciones buscan reducir en la medida de lo posible las deformaciones; la tendencia es desarrollar proyecciones de manera visual (visually designed projections) con el uso de softwares, esta práctica permite que usuarios no profesionales en la materia puedan crear sus propias proyecciones; las proyecciones que se formulan se fundamentan en las desarrolladas matemáticamente y, generalmente, son resultado de la combinación de dos proyecciones (Alashaikh, et al., 2014;Jenny y Patterson, 2013), aunque siguen surgiendo nuevas con parámetros distintos (Šavrič, et al., 2011;Jenny, et al., 2011;Safari y Ardalan, 2007;Paterson, et al., 2015). Finalmente, otros estudios se están encaminando en proponer nuevas proyecciones para mapas desplegados en los visualizadores dinámicos de Internet; el argumento se centra en que la proyección de Mercator, empleada en la web, presenta desventajas en el manejo de la información a diferentes escalas (Šavrič y Jenny, 2014;Streben, 2016). ...
Article
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La cartografía nacional de México emplea de manera oficial tres proyecciones: la Universal Transversa de Mercator, utilizada en las cartas topográficas para escalas medianas y grandes; la Cónica Conforme de Lambert, adaptada a las condiciones del país para su representación en escalas pequeñas; y la Normal de Mercator, empleada con fines de navegación marítima. El uso por largo tiempo de estas proyecciones ha sido asimilado por la comunidad científica y se emplean de forma regular; no obstante, la investigación cartográfica relacionada con esta temática se ha detenido y no se ha buscado la aplicación de otras proyecciones cartográficas en el afán de alcanzar una mejor representación de los diversos temas de investigación, lo que impacta también en la precisión de los datos obtenidos. El presente trabajo tiene por objetivo mostrar el empleo de la proyección Cónica Equivalente de Albers con las adaptaciones de parámetros necesarios; en este sentido, se presentan las formulaciones que posibilitan la proyección de las coordenadas, el análisis de las deformaciones y la concepción una cartografía nacional a escala 1:500,000; subsecuentemente, el trabajo también pretende estimular la reflexión en torno a las posibilidades que tiene esta proyección que no preserva el ortomorfismo, considerado hasta ahora como única propiedad en la cartografía nacional mexicana.
... Con respecto al estudio de las deformaciones, los trabajos se encaminan a desarrollar métodos de cálculo de los valores de deformación de las proyecciones y la forma gráfica de su representación (Kirtiloglu, 2010;Battersby y Kessler, 2012;Oztug, 2015). Por su parte, las perspectivas dedicadas a la propuesta de nuevas proyecciones buscan reducir en la medida de lo posible las deformaciones; la tendencia es desarrollar proyecciones de manera visual (visually designed projections) con el uso de softwares, esta práctica permite que usuarios no profesionales en la materia puedan crear sus propias proyecciones; las proyecciones que se formulan se fundamentan en las desarrolladas matemáticamente y, generalmente, son resultado de la combinación de dos proyecciones (Alashaikh, et al., 2014;Jenny y Patterson, 2013), aunque siguen surgiendo nuevas con parámetros distintos (Šavrič, et al., 2011;Jenny, et al., 2011;Safari y Ardalan, 2007;Paterson, et al., 2015). Finalmente, otros estudios se están encaminando en proponer nuevas proyecciones para mapas desplegados en los visualizadores dinámicos de Internet; el argumento se centra en que la proyección de Mercator, empleada en la web, presenta desventajas en el manejo de la información a diferentes escalas (Šavrič y Jenny, 2014;Streben, 2016). ...
Article
Full-text available
La cartografía nacional de México emplea de manera oficial tres proyecciones: la Universal Transversa de Mercator, utilizada en las cartas topográficas para escalas medianas y grandes; la Cónica Conforme de Lambert, adaptada a las condiciones del país para su representación en escalas pequeñas; y la Normal de Mercator, empleada con fines de navegación marítima. El uso por largo tiempo de estas proyecciones ha sido asimilado por la comunidad científica y se emplean de forma regular; no obstante, la investigación cartográfica relacionada con esta temática se ha detenido y no se ha buscado la aplicación de otras proyecciones cartográficas en el afán de alcanzar una mejor representación de los diversos temas de investigación, lo que impacta también en la precisión de los datos obtenidos. El presente trabajo tiene por objetivo mostrar el empleo de la proyección Cónica Equivalente de Albers con las adaptaciones de parámetros necesarios; en este sentido, se presentan las formulaciones que posibilitan la proyección de las coordenadas, el análisis de las deformaciones y la concepción una cartografía nacional a escala 1:500,000; subsecuentemente, el trabajo también pretende estimular la reflexión en torno a las posibilidades que tiene esta proyección que no preserva el ortomorfismo, considerado hasta ahora como única propiedad en la cartografía nacional mexicana.
... Finally, I recommend that textbooks consider referencing interactive web-based resources that students can use to explore distortion patterns on projections, such as Flex Projector (Jenny and Patterson 2013), which is freely available. It allows students to view and explore 30 pre-existing projections (although its different options also allow an infinite number of customizable projections). ...
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by the author(s). This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by-nc-nd/4.0. As developments in the field of map projections occur (e.g., the deriving of a new map projection), it would be reasonable to expect that those developments that are important from a teaching standpoint would be included in cartography textbooks. However, researchers have not examined whether map projection material presented in cartography textbooks is keeping pace with developments in the field and whether that material is important for cartography students to learn. To provide such an assessment, I present the results of a content analysis of projection material discussed in 24 cartogra-phy textbooks published during the 20 th and early 21 st centuries. Results suggest that some material, such as projection properties, was discussed in all textbooks across the study period. Other material, such as methods used to illustrate distortion patterns, and the importance of datums, was either inconsistently presented or rarely mentioned. Comparing recent developments in projections to the results of the content analysis, I offer three recommendations that future cartography textbooks should follow when considering what projection material is important. First, textbooks should discuss the importance that defining a coordinate system has in the digital environment. Second, textbooks should summarize the results from experimental studies that provide insights into how map readers understand projections and how to choose appropriate map projections. Third, textbooks should review the impacts of technology on projections, such as the web Mercator projection, programming languages, and the challenges of projecting raster data. K E Y W O R D S : map projection; datum; content analysis; cartographic education; history of cartography
... The development of the Equal Earth projection started by blending various pairs of existing equal-area projections with Flex Projector (Jenny et al. 2008(Jenny et al. , 2010. The projections 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 technique that Strebe (2017Strebe ( , 2018) introduced recently. ...
Article
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The Equal Earth map projection is a new equal-area pseudocylindrical 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.
... Flex Projector, a freeware application for the interactive design and evaluation of map projections (Jenny & Patterson, 2014), was used to design Natural Earth II. For developing projections, Flex Projector takes a graphic approach that was first introduced by Arthur H. Robinson during the design of his well-known, eponymous projection (Jenny & Patterson, 2013;Jenny et al., 2008;Jenny, Patterson, & Hurni, 2010;Robinson, 1974). In Flex Projector, the user adjusts the length, shape, and spacing of parallels and meridians for every 5°of latitude and longitude. ...
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The Natural Earth II projection is a new compromise pseudocylindrical projection for world maps. The Natural Earth II projection has a unique shape compared to most other pseudocylindrical projections. At high latitudes, meridians bend steeply toward short pole lines resulting in a map with highly rounded corners that resembles an elongated globe. Its distortion properties are similar to most other established world map projections. Equations consist of simple polynomials. A user study evaluated whether map-readers prefer Natural Earth II to similar compromise projections. The 355 participating general map-readers rated the Natural Earth II projection lower than the Robinson and Natural Earth projections, but higher than the Wagner VI, Kavrayskiy VII and Wagner II projections.
Article
By focusing on the properties used to measure map projections and utilizing them to manipulate map projection parameters, we have developed a new interface for manipulating such map projections. With our interface, a user can create his or her own map projections by manipulating the weight of each property via sliders. Specifically, we employ equidistancy, equiareality, and conformality as property elements and allow the user to balance these property elements according to their perceived levels of importance. Additionally, along with this interface, we also have developed a technique for reducing the number of projection parameters by assuming that latitude lines are horizontal and that longitude widths are identical in each latitude, which improves execution speed. Furthermore, as an additional interface, we have implemented a method for creating interrupted maps. This allows users to make interruptions on a map easily by clicking the interruption panel. Our system also provides a new manipulation method with which the user can traverse a number of existing cylindrical projections, such as those created with the Mercator and Lambert cylindrical projection methods.
Article
There are several techniques to obtain new map projections by combining two known projections. The aim of these techniques is to achieve a defined distribution of distortion, type of distortion, or certain forms of graticules. The obtained projection may or may not retain the properties of the two initial projections. In this study, a technique was developed to combine two conformal projections, namely the conical and Gauss–Kruger projections. This technique retained the properties of the two initial projections, i.e., it preserved conformality. Furthermore, the developed technique could be described as flexible because it contained two different weighted parameters, which defined as the percentage of each projection in the combination process. It was observed that the value of the weight differed from case to case depending on the geographic location, shape, and extension of the mapped area.
Thesis
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The selection process for map projections is a mystery to many mapmakers and GIS users. Map projections ought to be selected based on the map’s geographic extent and the required distortion properties, with the goal of minimizing the distortion of the mapped area. Despite some available selection guidelines, the selection of map projections is not yet automated. Automated selection would help mapmakers and GIS users to better select a projection for their map. The overall goal of this dissertation is to take a step towards this automation and explore user preferences with an objective to provide additional criteria for selecting world map projections. An additional goal is to optimize automatic map projection selection for web maps. The results presented in this work are mathematical models (new map projections for world maps, polynomial equations for selecting standard parallels) and new selection criteria for world maps. They improve our knowledge about map projection selection for world maps and web maps. As a result of the research presented in this doctoral dissertation, we know more about the map projection preferences of map-readers and have improved techniques for adapting map projections for scalable web maps and GIS software. Altogether, four concrete research questions were addressed. The first research question explores user preferences for world map projections. Many small-scale map projections exist and they have different shapes and distortion characteristics. World map projections are mainly chosen based on their distortion properties and the personal preferences of cartographers. Very little is known about the map projection preferences of map-readers; only two studies have addressed this question so far. This dissertation presents a user study among map-readers and trained cartographers that tests their preferences for world map projections. The paired comparison test of nine commonly used map projections reveals that the map-readers in our study prefer the Robinson and Plate Carrée projections, followed by the Winkel Tripel, Eckert IV, and Mollweide projections. The Mercator and Wagner VII projections come in sixth and seventh place, and the least preferred are two interrupted projections, the interrupted Mollweide and the interrupted Goode Homolosine. Separate binominal tests indicate that map-readers involved in the study seem to like projections with straight rather than curved parallels, and meridians with elliptical rather than sinusoidal shapes. The results indicate that map-readers prefer projections that represent poles as lines to projections that show poles as protruding edges, but there is no clear preference for pole lines in general. The trained cartographers involved in this study have similar preferences, but they prefer pole lines to represent the poles, and they select the Plate Carrée and Mercator projections less frequently than the other participants. The second research question introduces the polynomial equations for the Natural Earth II projection and tests user preferences for its graticule characteristics. The Natural Earth II projection is a new compromise pseudocylindrical projection for world maps. It has a unique shape compared to most other pseudocylindrical projections. At high latitudes, the meridians bend steeply toward short pole lines resulting in a map with highly rounded corners that resembles an elongated globe. Its distortion properties are similar to most other established world map projections. The projection equation consists of simple polynomials. A user study evaluated whether map-readers prefer Natural Earth II to similar compromise projections. The 355 participating general map-readers rated the Natural Earth II projection lower than the Robinson and Natural Earth projections, but higher than the Wagner VI, Kavrayskiy VII, and Wagner II projections. The third question examines how Wagner’s transformation method can be used for improving map projections for scalable web maps, and its integration into the adaptive composite map projections schema. The adaptive composite map projections schema, invented by Bernhard Jenny, changes the projection to the geographic area shown on a map. It is meant as a replacement for the commonly used web Mercator projection, which grossly distorts areas when representing the entire world. The original equal-area version of the adaptive composite map projections schema uses the Lambert azimuthal projection for regional maps, and three alternative projections for world maps. In this dissertation, it is explored how the adaptive composite map projections schema can include a variety of other equal-area projections when the transformation between the Lambert azimuthal and the world projections uses Wagner’s method. In order to select the most suitable pseudocylindrical projection, the distortion characteristics of a pseudocylindrical projection family were analyzed, and a user study among experts in the area of map projections was carried out. Based on the results of the distortion analysis and the user study, a new pseudocylindrical projection is recommended for extending the adaptive composite map projections schema. The new projection is equal-area throughout the transformation to the Lambert azimuthal projection, has better distortion characteristics than small-scale projections currently included in the original adaptive composite map projections schema, and aligns with map-readers’ preferences for world map projections. The last research question explores how the selection of the standard parallels of conic projections can be automated. Conic map projections are appropriate for mapping regions at medium and large scales with east-west extents at intermediate latitudes. Conic projections are appropriate for these cases because they show the mapped area with less distortion than other projections. In order to minimize the distortion of the mapped area, the two standard parallels of conic projections need to be selected carefully. Rules of thumb exist for placing the standard parallels based on the width-to-height ratio of the map. These rules of thumb are simple to apply, but do not result in maps with minimum distortion. There also exist more sophisticated methods that determine standard parallels such that distortion in the mapped area is minimized. These methods are computationally expensive and cannot be used for real-time web mapping and GIS applications where the projection is adjusted automatically to the displayed area. This article presents a polynomial model that quickly provides the standard parallels for the three most common conic map projections: the Albers equal-area, the Lambert conformal, and the equidistant conic projection. The model defines the standard parallels with polynomial expressions based on the spatial extent of the mapped area. The spatial extent is defined by the length of the mapped central meridian segment, the central latitude of the displayed area, and the width-to-height ratio of the map. The polynomial model was derived from 3825 maps—each with a different spatial extent and computationally determined standard parallels that minimize the mean scale distortion index. The resulting model is computationally simple and can be used for the automatic selection of the standard parallels of conic map projections in GIS software and web mapping applications.
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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 cartographers 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.
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The Natural Earth projection is a new projection for representing the entire Earth on small-scale maps. It was designed in Flex Projector, a specialized software application that offers a graphical approach for the creation of new projections. The original Natural Earth projection defines the length and spacing of parallels in tabular form for every five degrees of increasing latitude. It is a pseudocylindrical projection, and is neither conformal nor equal-area. In the original definition, piece-wise cubic spline interpolation is used to project intermediate values that do not align with the five-degree grid. This paper introduces alternative polynomial equations that closely approximate the original projection. The polynomial equations are considerably simpler to compute and program, and require fewer parameters, which should facilitate the implementation of the Natural Earth projection in geospatial software. The polynomial expression also improves the smoothness of the rounded corners where the meridians meet the horizontal pole lines, a distinguishing trait of the Natural Earth projection that suggests to readers that the Earth is spherical in shape. Details on the least squares adjustment for obtaining the polynomial formulas are provided, including constraints for preserving the geometry of the graticule. This technique is applicable to similar projections that are defined by tabular parameters. For inverting the polynomial projection the Newton-Raphson root finding algorithm is suggested.
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The sinusoidal and the elliptical projections are two large groups of pseudocylindrical projections. A sinusoidal-elliptical composite projection combines the members of these groups by applying one projection for the polar regions and another for the equatorial area. Although several attempts were made to create equal-area projections of this kind, none of these solutions were perfect. The main issues were the break of meridian lines at the connection latitude and the different scale of the two parts. This paper’s aim is to fill this gap by deducing two different, parametrisable sinusoidal-elliptical combined projections.
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There are at least 80 published pseudocylindrical map projections, with straight parallels and curved meridians. Of these, about 40 are equal area (a few of which are not significantly different from others), and about 20 have equidistant parallels. Many are mentioned only qualitatively if at all in references other than the original sources. Some 40 inventors are involved. Seven significantly different projections using sinusoids offer alternatives, such as flat or pointed poles, equal area, or equidistant parallels. A dozen elliptical pseudocylindricals offer these variations, perhaps more attractively, with added possibilities from full versus partial ellipses. Five parabolics are rounder and probably preferable aesthetically to their sinusoidal counterparts. Nine projections with hyperbolas or converging straight lines for meridians are almost useless. Over two dozen projections use less well-known curves in attempts to give maps less real or apparent distortion. They include seven pseudocylindricals that are at least as prevalent in published atlases as the utilized handful of classically-curved projections. The projections are described here with definitive formulas and comparisons of appearance, theory, and standard parallels to facilitate choices and calculations.