Content uploaded by Bernhard Jenny
Author content
All content in this area was uploaded by Bernhard Jenny on Dec 28, 2015
Content may be subject to copyright.
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 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.
Keywords: projection design; projection blending; Flex Projector; Pacific projection
Introduction
Flex Projector (www.flexprojector.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
simplifies 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 user’s 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 Robinson’s 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 first 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, 151–152). 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
Downloaded by [Oregon State University] at 11:09 07 May 2013
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 fidelity 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 Projector’sother 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 Pacific projection, before
we conclude with a few final 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 five degrees of latitude
(A), which in turn changes the projection shape (B) and the distortion ranking (C).
2B. Jenny and T. Patterson
Downloaded by [Oregon State University] at 11:09 07 May 2013
indices, which reports in real-time the amount of distor-
tion contained in the modified 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 modification
of projection parameters, including Wagner’s 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 first 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° 44′12′′; 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, 217–220, for over-
views). A trait of most non-continuous combined projec-
tions is a discontinuity in the first 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 Goode’s 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 “starter”projections 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, 231–232); 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 first 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 influence of each projection.
As the user experiments, changes appear instantaneously
on the large composite world map (Figures 2–4).
Cartography and Geographic Information Science 3
Downloaded by [Oregon State University] at 11:09 07 May 2013
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 defines 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 “DNA”of 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
Downloaded by [Oregon State University] at 11:09 07 May 2013
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 difficult to grasp for novice
users and because the effect is difficult 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 first 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 final 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 first step, Flex
Projector “deconstructs”the two selected source projec-
tions by converting them into tabular form. A projection is
commonly defined 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 first 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-defined 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 final 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 Pacific 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
define the hybrid Pacific projection.
Cartography and Geographic Information Science 5
Downloaded by [Oregon State University] at 11:09 07 May 2013
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 fine-tune the new projec-
tion by adjusting individual values of one of the four
tables with Flex Projector’sgraphical 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 first 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 Pacific projection
As a practical example on how to use the Mixer, we
created the hybrid Pacificprojection 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 Pacific Ocean and with relatively little areal
distortion. We chose the Mollweide projection because it
is oval in shape, which complements the roundness of the
Pacific 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
Pacific in relation to other parts of the world. Creating
the Pacific 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 Pacific 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 Pacific 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 Pacific Ocean because the
meridians near the center point are more widely spaced
than those at the map margins.
Once work in the Mixer is finished, the user can further
enhance the combined projection using the other tools in
Flex Projector. In the case of the Pacific 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 Pacific Ocean (Figure 5). Additionally, we
scaled the entire map by a factor of 0.8 to minimize area
distortion for the Pacific region. By saving the final Pacific
projection as a small text file, we could use it again in Flex
Projector for producing publishable-quality maps with
imported shapefiles and raster geodata. However, opera-
tional use of the Pacific projection and all custom
Figure 5. The Pacific projection (black) overlaid on the Robinson projection (gray). The Robinson is scaled to the same width.
The taller Pacific projection devotes relatively more area to the Pacific Ocean than does the Robinson.
6B. Jenny and T. Patterson
Downloaded by [Oregon State University] at 11:09 07 May 2013
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 Pacific 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 influence of the non-equal-area Ginzburg
VIII. The Flex Projector distortion tables (Figure 6) show
how the Pacific 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 Pacific 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 Pacific
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 Pacific
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 simplifies 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 fine-
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 Pacific projection is
the highlighted row.
Figure 7. The tint indicates the area of acceptable angular and
areal distortion for the Pacific projection. Most of the Pacific 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 Pacific projection.
Cartography and Geographic Information Science 7
Downloaded by [Oregon State University] at 11:09 07 May 2013
Repeating this process can yield an almost infinite 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.
References
Anderson, P. B., and W. R. Tobler. (s.d.). “Blended Map
Projections are Splendid Projections.”Accessed August 3,
2011. http://www.geog.ucsb.edu/~tobler/publications/pdf_docs/
inprog/BlendProj.pdf.
Boggs, S. 1929. “A New Equal-Area Projection for World
Maps.”Geographical Journal 73 (3): 241–245.
Canters, F. 2002. Small-Scale Map Projection Design. London:
Taylor & Francis.
Eckert, M. 1906. “Neue Entwürfe für Weltkarten.”Petermanns
Mitteilungen 52 (5): 97–109.
Érdi-Krausz, G. 1968. Combined Equal-Area Projection for
World Maps, Hungarian Cartographical Studies. 44–49.
Budapest: Földmérési Intézet.
Foucaut, H. C. 1862. Notice sur la Construction de Nouvelles
mappemondes et de Nouveaux Atlas de Géographie. France:
Arras.
Garver, J. B. 1988. “New Perspective on the World.”National
Geographic 174: 910–913.
Gede, M. 2011. “Optimising the Distortions of Sinusoidal-
Elliptical Composite Projections.”In Advances in
Cartography and GIScience. Volume 2: Selection from
ICC 2011, Paris, Lecture Notes in Geoinformation and
Cartography 6, doi;10.1007/978-3-642-19214-2_14, edited
by A. Ruas, 209–225. Berlin: Springer-Verlag.
Goode, J. P. 1925. “The Homolosine Projection: A New
Device for Portraying the Earth’s Surface Entire.”Annals
of the Association of American Geographers 15 (3):
119–125.
Hammer, E. 1900. “Unechtcylindrische and Unechtkonische
flächentreue Abbildungen.”Petermanns Geographische
Mitteilungen 46: 42–46.
Hatano, M. 1972. “Consideration on the Projection Suitable
for Asia-Pacific Type World Map and the Construction
of Elliptical Projection Diagram.”Geographical Review of
Japan 45 (9): 637–647.
Jenny, B., and T. Patterson. 2007. “Flex Projector.”Accessed
August 3, 2011. http://www.flexprojector.com.
Jenny, B., T. Patterson, and L. Hurni. 2008. “Flex
Projector –Interactive Software for Designing World
Map Projections.”Cartographic Perspectives 59:
12–27.
Jenny, B., T., Patterson, and L. Hurni. 2010. “Graphical Design
of World Map Projections.”International Journal of
Geographic Information Science 24 (11): 1687–1702.
McBryde, F. W. 1978. “A New Series of Composite Equal-Area
World Maps Projections.”International Cartographic
Association, 9th International Conference on Cartography,
College Park, Maryland, Abstracts, 76–77.
Nell, A. M. 1929. “Äquivalente Kartenprojektionen.”
Petermanns Geographische Mitteilungen 36: 93–98.
Putniņš, R. V. 1934. “Jaunas projekci jas pasaules kartēm.”
[Latvian with extensive French résumé.] GeografiskiRaksti,
Folia Geographica 3 and 4: 180–209.
Robinson, A. 1974. “A New Map Projection: Its Development
and Characteristics.”In International Yearbook of
Cartography, edited by G.M. Kirschbaum, and K.-H.
Meine, 145–155. Bonn-Bad Godesberg: Kirschbaum.
Šavrič, B., B. Jenny, T. Patterson, D. Petrovič, and L. Hurni.
2011. “A Polynomial Equation for the Natural Earth projec-
tion.”Cartography and Geographic Information Science 38
(4): 363–372.
Snyder, J. P. 1977. “A Comparison of Pseudocylindrical Map
Projections.”The American Cartographer 4 (1): 59–81.
Snyder, J. P. 1993. Flattening the Earth: Two Thousand Years of
Map Projections. Chicago, IL: University of Chicago Press.
Tobler, W. R. 1973. “The Hyperelliptical and Other New Pseudo
Cylindrical Equal Area Map Projections.”Journal of
Geophysical Research 78 (11): 1753–1759.
Wagner, K. 1949. Kartographische Netzentwürfe. Leipzig:
BibliographischesInstitut.
Winkel, O. 1921. “Neue Gradnetzkombinationen.”Petermanns
Mitteilungen 67: 248–252.
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
Downloaded by [Oregon State University] at 11:09 07 May 2013