A NEW CONSTRUCTION METHOD FOR CIRCLE CARTOGRAMS
AND ITS APPLICATION
R. Inoue
Department of Civil and Environmental Engineering, School of Engineering, Tohoku University,
6-6-06 Aramaki Aza Aoba, Aoba, Sendai, Miyagi 980-8579, Japan - rinoue@plan.civil.tohoku.ac.jp
KEY WORDS: Circle Cartogram, Construction, Visualization, Distribution of Spatial Data, Non-linear Minimization with
Inequality Constraint
ABSTRACT:
An area cartogram is a transformed map on which areas of regions are proportional to statistical data values; it is considered to be a
powerful tool for the visual representation of statistical data. One of the most familiar cartograms is a circle (or circular) cartogram,
on which regions are represented by circles. Dorling first proposed a circle cartogram construction algorithm in 1996 according to
the requirements ‘avoid overlap of circles’ and ‘keep contiguity of regions as much as possible’. The algorithm first places circles
according to the geographical configuration of regions and then moves them one by one in order to fulfil the requirements. It outputs
results which express a spatial distribution of data; however, the relative positions of circles on cartograms sometimes greatly differ
from the geographical maps. Cartogram readers usually compare the shape of cartograms to that of geographical maps, realize the
shape distortion, and recognize the characteristics of spatial distribution of represented data on cartograms; removing the excess
circle reposition process from the algorithm is preferable to make the relative position of circles on resultant cartograms as similar as
possible to that on the geographical map. In this paper, I propose a new construction method for circle cartograms that attaches a
high value to the preservation of relative position of circles while considering the requirements proposed by Dorling. I formulated a
construction problem for non-linear minimization with inequality constraint conditions and applied the method to the world
population dataset.
1. INTRODUCTION
A cartogram (or area cartogram) is one of the most powerful
visualization tools for spatial data in quantitative geography
(e.g. Monmonier, 1977; Dorling, 1996; Tobler, 2004).
Cartograms are transformed maps on which the areas of regions
are proportional to the data values. Deformation of the shape of
regions and their displacements assist map-readers in intuitively
recognizing the distribution of data represented on cartograms.
Cartograms are classified on the basis of two characteristics:
the shapes and contiguities of regions indicated on the
cartograms. For the shapes of regions, some cartograms use
complex shapes, whereas others use simple shapes such as
circles and rectangles. The ease of comparison between
cartograms with complex shapes of regions and geographical
maps enables map-readers to comprehend the characteristics of
spatial data presented in such cartograms. However,
comparison of cartograms with complex region shapes is
difficult in terms of the size of the regions; in this sense, it is
better to use simple shapes to express data.
Cartograms that express regions in the form of simple shapes
are classified into two types on the basis of contiguities of the
regions illustrated on them. One is contiguous cartograms; a
rectangular cartogram proposed by Rasiz (1934) serves as an
example. Rectangles represent regions, and different rectangles
representing adjacent regions are placed contiguously. A
rectangular cartogram is an effective visualization tool, as the
size of regions is easy to perceive. However, its construction is
difficult because it is impossible to maintain all contiguities of
regions in many cases; it then becomes necessary to omit some
of the contiguities. Therefore, although several solutions have
been proposed (e.g. Heilmann et al., 2004; Speckmann et al.,
2006; van Kreveld and Speckmann, 2007), their applications
are limited. The other type is non-contiguous cartograms;
rectangular cartograms proposed by Upton (1991) and circle (or
circular) cartograms proposed by Dorling (1996) are examples.
They represent regions by rectangles and circles and omit the
contiguities of regions. In particular, a circle cartogram is often
used for visualization because of its simple construction
algorithm.
Dorling (1996) proposed a circle cartogram construction
algorithm according to two requirements for an easily
comprehensible resultant: ‘avoid overlap of circles’ and ‘keep
contiguity of regions as much as possible’. It is also important
to ‘keep the similarity of configuration between circles on
cartograms and regions on geographical maps’; accordingly,
the algorithm first places circles according to the geographical
configuration of regions and then moves circles one by one in
order to fulfil the requirements. It outputs results which express
a spatial distribution of data. However, the relative positions of
circles on cartograms sometimes differ greatly from the
geographical maps; the displacement of circles then causes
difficulty in distinguishing which circles represent which
regions.
A special joint symposium of ISPRS Technical Commission IV & AutoCarto
in conjunction with
ASPRS/CaGIS 2010 Fall Specialty Conference
November 15-19, 2010 Orlando, Florida
I believe that there are two problems with the previous
construction algorithm. One is that the algorithm does not
consider maintaining the relative position of circles explicitly.
The information on regions’ contiguity includes the information
on the relative position of regions; however, I think that it is not
sufficient to keep the similarity of positions between circles on
cartograms and regions on geographic maps. The other problem
is that the previous algorithm moves circles one by one to
search for a circle configuration that satisfies the requirements
for circle cartogram construction. This means that the algorithm
searches the local minimum by changing the x- and ycoordinates of one circle and repeats this procedure until all of
the requirements are satisfied. This step-by-step procedure
outputs results which are dependent on the order of the circles’
movement and may output a local minimum which is far from
the initial data. To find a local minimum near the initial data, it
is better to use a solution that considers all requirements
together and outputs the positions of all circles simultaneously.
In this paper, I propose a new construction method for circle
cartograms that attaches a high value to the preservation of
relative positions of circles while also considering the
requirements proposed by Dorling (1996). I formulated a circle
cartogram construction problem with non-linear minimization
and inequality constraint conditions. I then confirmed the
applicability of the proposed formulation using the world
population dataset.
2. APPROACH TO CIRCLE CARTOGRAM
CONSTRUCTION
2.1 Requirements for resultant cartogram shapes
Here, I confirm the requirements for circle cartogram
construction. The following are considered in the algorithm for
circle cartogram construction as proposed by Dorling (1996),
although the first requirement is not made explicit.
1) Maintain similarity between the position of circles on
cartograms and the position of regions on geographical maps.
2) Place the circles on top of other circles that share their
border on the geographical map if possible.
3) Avoid overlapping of circles.
I agree that fulfilling these requirements enhances the
readability of cartograms. When people read circle cartograms,
they note the differences in the size and position of circles by
comparing those regions on geographical maps and recognize
the characteristics of the spatial distribution for the represented
data on cartograms. Large relocations of circles make data
interpretation difficult; therefore, it is important to retain the
circle alignment in cartograms. Moreover, avoiding any
overlapping of circles is important as overlaps hinder the
recognition of circle sizes.
I consider these requirements to be essential for constructing
visually elegant circle cartograms, and I propose solutions that
satisfy these requirements.
2.2 An approach to circle cartogram construction
Suppose that the data to be expressed on a circle cartogram are
given to every region in the target domain. Let Di denote data
given to region i; then ri, or the radius of circle i, is given by
ri = Di π .
Now, a circle cartogram construction can be considered as a
problem where the positions of the circles’ centres must be
determined. The distance between the centres of neighbouring
circles should be the sum of the radii of those circles, and the
relative positions of the circles’ centres on cartograms should
resemble the corresponding regions on the geographical maps.
In fact, this description of circle cartogram construction is quite
similar to that of distance cartogram construction. Distance
cartograms are diagrams that visualize the proximity indices
between points in a network. Its construction problem is to
determine the location of points on cartograms according to the
given proximity indices between points.
Shimizu and Inoue (2009) formulated the distance cartogram
construction problem as follows. Let pij denote the given
proximity index between points i and j; dij denote the distance
on a cartogram between points i and j; (xi, yi) denote the x- and
y-coordinates of point i on the coordinate system of the
cartogram, respectively; (xGi, yGi) denote the x- and ycoordinates of point i on the geographic coordinate system,
respectively; L denote the set of point pairs that have the
proximity data; θGij denote the bearing of the link that connects
points i and j on the geographical map; and θij denote the
bearing of the link that connects points i and j on the cartogram.
The distance cartogram construction is then formulated as a
least-squares problem:
2
⎧ ⎛d
⎫
⎞
2⎪
⎪
ij
− 1⎟ + (1 − α ) (θij − θiGj ) ⎬
min ∑ ⎨α ⎜
⎜
⎟
x, y
( i , j )∈L ⎪ ⎝ pij
⎠
⎩
⎭⎪
where dij =
(x
− xi ) + ( y j − yi ) ,
2
j
θij = tan −1
(1)
y j − yi
x j − xi
2
, θijG = tan −1
y Gj − yiG
xGj − xiG
,
0 < α < 1.
Equation (1) consists of two objective functions: one minimizes
the difference between the given proximity data and point
distances on cartograms, and the other minimizes the difference
between the direction of links that connect the points on
cartograms and geographical maps. The weight parameter α is
adapted to set the balance between two objective functions.
Shimizu and Inoue (2009) showed that distance cartograms can
be constructed by solving equation (1).
The difference between circle and distance cartogram
constructions is that circle cartogram construction requires the
avoidance of circle overlapping. Thus, I propose to add
A special joint symposium of ISPRS Technical Commission IV & AutoCarto
in conjunction with
ASPRS/CaGIS 2010 Fall Specialty Conference
November 15-19, 2010 Orlando, Florida
constraint conditions to the formulation of distance cartogram
construction in this paper.
3. FORMULATION OF CIRCLE CARTOGRAM
CONSTRUCTIONS
the cartogram and regions on the geographical map; α is the
weight parameter for these two objective functions. The
constraint conditions between all pairs of circles are set so as to
avoid overlapping of circles.
3.2 Expression of formulation by x- and y-coordinates
3.1 Formulation of circle cartogram constructions
Here, I formulate the circle cartogram construction. Let circle i
on a cartogram represent region i on a geographical map; (xi, yi),
the x- and y-coordinates of the centre of circle i on the
coordinate system of the cartogram, respectively; (xGi, yGi), the
x- and y-coordinates of the centroid of region i on the
geographic coordinate system, respectively; ri, the radius of
circle i; dij, the distance between centres of circles i and j; C,
the set of pairs of regions that share borders; θGij, the bearing of
the link that connects the centroids of regions i and j on the
geographical map; and θij, the bearing of the link that connects
centres of circles i and j on the cartogram. Figure 1 shows the
definitions of the notations.
The unknown variables of equation (2) are the coordinates for
the centre of circles xi and yi since the radii of circles, ri, and
geographic coordinates of the centroid of regions, xGi and yGi,
are given. Equation (2) can then be rewritten as follows:
⎧ ⎛
⎪⎪ ⎜
min ∑ ⎨α ⎜
x, y
( i , j )∈C ⎪ ⎜
⎪⎩ ⎝
(x
2
j
2
ri + rj
⎞
⎟
− 1⎟
⎟
⎠
2
(3)
⎛
y − yi
y −y
+ (1 − α ) ⎜ tan −1 j
− tan −1
⎜
−
x
x
x −x
j
i
⎝
G
j
G
j
G
i
G
i
⎞
⎟⎟
⎠
2
⎫
⎪
⎬
⎪⎭
subject to ( xn − xm ) + ( yn − ym ) ≥ rm + rn ∀(m, n) m ≠ n
2
rj
2
where 0 < α < 1.
(xj, yj)
θij
− xi ) + ( y j − yi )
To simplify the second term of the objective function, the
inverse function of the tangent is removed.
dij
ri
(xi, yi)
⎛
y − yi
y G − yiG
min ∑ ⎜ tan −1 j
− tan −1 Gj
⎜
x j − xi
x j − xiG
( i , j )∈C ⎝
⎛ y j − yi y Gj − yiG ⎞
≈ min ∑ ⎜
− G
⎟
⎜
x j − xiG ⎟⎠
( i , j )∈C ⎝ x j − xi
Figure 1. Neighbouring circles on circle cartogram.
The formulation for circle cartogram construction is as follows:
min
x, y
2
⎧ ⎛ d
⎫
⎞
⎪
ij
G 2⎪
α
1
−
⎟⎟ + (1 − α ) (θ ij − θij ) ⎬
⎨ ⎜⎜
∑
( i , j )∈C ⎪ ⎝ ri + rj
⎪⎭
⎠
⎩
( x j − xi ) + ( y j − yi ) ,
θij = tan −1
y j − yi
x j − xi
2
, θijG = tan −1
y Gj − yiG
xGj − xiG
(4)
2
(2)
subject to d mn ≥ rm + rn ∀(m, n) m ≠ n
2
2
To avoid the denominator becoming zero, equation (4) is
transformed as follows:
min
where dij =
⎞
⎟⎟
⎠
,
0 < α < 1.
⎛ y j − yi y Gj − yiG ⎞
− G
⎜⎜
⎟
∑
x j − xiG ⎟⎠
( i , j )∈C ⎝ x j − xi
2
⎧⎪ ( y j − yi ) ( xGj − xiG ) − ( x j − xi ) ( y Gj − yiG ) ⎫⎪
≈ min ∑ ⎨
⎬
d ij d iGj
( i , j )∈C ⎪
⎩
⎭⎪
Finally, the circle cartogram construction
formulated as shown in equation (6):
The formulation for circle cartogram construction (equation
(2)) is quite similar to that for distance cartogram construction
(equation (1)). Equation (2) is also composed of two objective
functions as well as one constraint condition. The first term of
the objective function involves keeping the circles which
represent neighbouring regions closer to be contiguous on the
cartogram; the second term of the objective function involves
maintaining similarity between relative positions of circles on
A special joint symposium of ISPRS Technical Commission IV & AutoCarto
in conjunction with
ASPRS/CaGIS 2010 Fall Specialty Conference
November 15-19, 2010 Orlando, Florida
2
problem
(5)
is
⎡ ⎛
⎢ ⎜
min ∑ ⎢α ⎜
x, y
( i , j )∈C ⎢ ⎜
⎢⎣ ⎝
⎧
⎪
+ (1 − α ) ⎨
⎪
⎩
(x
− xi ) + ( y j − yi )
2
j
2
ri + rj
⎞
⎟
− 1⎟
⎟
⎠
2
( y j − yi ) ( xGj − xiG ) − ( x j − xi ) ( y Gj − yiG )
(x
− xi ) + ( y j − yi )
2
j
2
(x
G
j
− xiG ) + ( y Gj − yiG )
2
2
⎫
⎪
⎬
⎪
⎭
2
⎤
⎥
⎥
⎥
⎥⎦
(6)
0.7. The coloured circles represent countries, and the
abbreviated three-letter codes indicate the country names. The
white circles represent the dummy circles. The sizes of the
dummy circles were decided through a trial-and-error process
to achieve comprehensible outputs.
subject to ( xn − xm ) + ( yn − ym ) ≥ rm + rn ∀(m, n) m ≠ n
2
2
where 0 < α < 1.
Now it is possible to construct circle cartograms by solving
equation (6), a non-linear minimization problem with inequality
constraint conditions. In the following section, I check the
applicability of the proposed formulation.
1 billion
0.3 billion
0.1 billion
Countries
Dummy Circles
4. APPLICATION
4.1 Investigation of applicability of proposed formulation
I applied the proposed formulation to data from the World
Population Prospects by the United Nations Population
Division. I solved the problems through a trust-region interiorpoint method using the mathematical programming software
NUOPT.
NUOPT is a product developed by the software company
Mathematical Systems, Inc., of Japan. It can be used to solve
complex mathematical optimization problems by setting initial
values to the unknown variables (x- and y-coordinates of the
centres of circles) and describing the objective functions and
constraint conditions.
As described previously, getting the configuration of circles on
the cartogram to be similar to that of regions on the
geographical map is preferable; the initial values should then be
set according to the geographic coordinates. Additionally,
setting the initial values to satisfy all the constraint conditions
is important for making the calculations easier; however, it is
also important to set the initial values close to the resultant
coordinates to make the change in coordinate values, i.e., the
change in positions of circles, smaller.
Figure 2. Resultant circle cartogram of the world population in
2010 by proposed formulation (α = 0.7).
It was confirmed that the formulation developed in this study
can be used to construct a circle cartogram. Since the
configuration of circles on the cartogram seems quite similar to
that of regions on a geographical map, it is easy to sense which
circles represent which regions. The population distribution in
the world is well presented in this figure without any overlaps
of circles; the readers of Figure (2) can easily see that the
populations of the Asian countries dominate the 2010 world
population in 2010.
4.2 Evaluation of weight parameter settings
Figure 2 shows the calculation results when the weight
parameter α is equal to 0.7. Without discussing and evaluating
the meaning of setting α, it would be difficult to adjust its value.
In this section, I clarify the meaning of α and explain the
change in figures when it is adjusted by using the 2010 world
population data.
Considering the above conditions, I set the initial values of xand y-coordinates for the centre of circles to touch at least one
pair of circles and avoid overlaps of any other pairs of circles.
There is one additional issue to consider. The construction of
circle cartograms by equation (6) requires that the set of pairs
of contiguous regions C should compose one graph. However,
the world country data do not satisfy this condition, as there are
separate continents and island countries. I thus added dummy
region data to connect all circles in one graph.
Figure 2 is the output figure of the world population circle
cartograms in 2010 obtained when solving for the proposed
formulation. The weight parameter α for this calculation was
A special joint symposium of ISPRS Technical Commission IV & AutoCarto
in conjunction with
ASPRS/CaGIS 2010 Fall Specialty Conference
November 15-19, 2010 Orlando, Florida
GBR
CAN
PRK
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DEU
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VNM
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GTM
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B GD
PAK
VEN
COL
IDN
PNG
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BRA
PER
SOM
UGA
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PRY
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KEN
COD
CHL
RWA
ARG
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NGA
SAU
PAK
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BGD
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THA
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ZWEMOZ
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PHL
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NLD
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USA
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BEL
CAN
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(e) α = 0.9
(a) α = 0.1
Figure 3. Differences in the resultant circle cartograms for
2010 world population by adjusting α
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CA N
PRK
NLD
GBR
HND
SLV
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CIV
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BRA
PER
VNM
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BE L
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IDN
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KEN
RWA
BDI TZA
ZMB
AUS
IND
MWI
ZWE
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MDG
LKA
(b) α = 0.3
SWE
NLD
GBR
CAN
B EL
POL
DEU
FRA
USA
PRT
CUB
ESP
GTM
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(c) α = 0.5
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(d) α = 0.7
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in conjunction with
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As shown in equation (2), when α is close to zero, the
representation of contiguity information is weighted; when α is
close to unity, the similarity to the geographic configuration is
weighted. As a result, a smaller α should place circles closer
together to express the contiguity of regions, although the
deformation of circle positions becomes larger; in contrast, a
larger α should place circles according to the geographic
configuration, although some pairs of circles with contiguity
information are placed separately.
Division, which contains the country population data for every
five years from 1980 to 2050. Figures 4 and 5 show the
population for every 10 years; the colour in the circles
represents the annual population growth of the last five years
for each country.
NOR
FIN
DNK
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NLD
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IRL
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UKR
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FRA
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ITA
USA
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IRQ
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THA
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KEN
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I then checked the differences in the resultant cartograms by
adjusting α. Figure 3 shows the calculation results with α equal
to 0.1, 0.3, 0.5, 0.7, and 0.9. There do not seem to be large
differences between these figures, but the figures calculated
with larger α have larger gaps between circles and keep the
similarity to the geographic configuration of regions.
EGY
TUN
MYS
MDG
MOZ
B OL
ZAF
PRY
CHL
IDN
ARG
PNG
IND
AUS
NZL
LKA
(a) 1980
NOR
LTU
NLD
GBR
IRL
I evaluated the difference in shapes using the two indices: the
percentage of represented contiguity information on cartograms,
and the residual sum of squares for x- and y-coordinates of the
affine transformation from the cartogram coordinates to
geographic coordinates. The former index corresponds to the
first term of the objective functions; it should be large when α
is small. The latter index corresponds to the second term of the
objective functions. The affine transformation is composed of
rotation, scaling, or shear, and if the two configurations of
points are matched by the affine transformation, the similarity
is high; the disparity between the geographic and affinetransformed cartogram coordinates is the index of similarity of
configuration. The residual sum of squares for the affine
transformation should be large when α is small since the
smaller residuals express higher similarity.
CZE
KAZ
MDA
GEO
ROM
IRQ
SDN
UGA
CMR
COD
BRA
PER
AGO
NGA
ZMB
LAO
PHL
KHM
THA
MYS
TZA
MWI
IDN
MDG
ZWE
BOL
VNM
MMR
BGD
KEN
RWA
BDI
ECU
NPL
SOM
TCD
TGO
GHA
ETH
ERI
LBY
NER
BFA
BEN
COL
PAK
YEM
TUN
MLI
CIV
VEN
TWN
SAU
EGY
DZA
GIN
SLE
NIC
JPN
KOR
TJK
AFG
JOR
ISR
MAR
SEN
HTI DOM
HND
SLV
UZB
IRN
GRC
CUB
GTM
TKM
TUR
BGR
ALB
SYR
MEX
AZE
ARM
SBA
ESP
CHN
KGZ
HUN
HRV
BIH
PRT
PRK
AUT
ITA
USA
RUS
UKR
SVK
CHE
FRA
BLR
POL
DEU
BEL
CAN
FIN
SWE
DNK
MOZ
PNG
PRY
CHL
ZAF
IND
ARG
AUS
NZL
0~
~ 10.0
LKA
5
50
(b) 1990
NOR
SWE
DNK
GBR
IRL
CZE
PRK
KAZ
AUT
HUN
MDA
AZE
TUR
BGR
ESP
SYR
IRQ
TWN
LBN
HTI DOM
HND
SLV
GIN
SLE
NIC
VEN
CRI
DZA
NGA
AGO
ZMB
PRY
THA
MYS
SGP
IDN
TZA
PNG
MWI
ZWE
URY
ARG
PHL
KHM
SOM
RWA
BDI
BOL
CHL
LAO
UGA KEN
CAF
CMR
COD
PER
VNM
MMR
BGD
ETH
SDN
TCD
BEN
TGO
BRA
e
NPL
LBY
NER
BFA
GHA
PAK
YEM
TUN
ERI
MLI
CIV
COL
ECU
Table 1 shows the calculation results of two indices. It confirms
that contiguity is well represented when α is small and that the
similarity of configuration is preserved when α is large. These
two favourable characteristics for circle cartograms are the
trade-off; however, users who construct circle cartograms by
this formulation can easily adjust α since its meaning is clear.
EGY
SEN
ARE
SAU
CUB
GTM
JOR
ISR
MAR
MEX
JPN
KOR
TJK
AFG
IRN
GRC
CHN
KGZ
UZB
TKM
SBA
BIH
GEO
ROM
HRV
ITA
USA
RUS
UKR
SVK
CHE
FRA
BLR
POL
DEU
BEL
PRT
FIN
LTU
NLD
CAN
MDG
MOZ
IND
ZAF
AUS
NZL
10.0 ~
5.0 ~ 10.0
2.5 ~ 5.0
LKA
1,000,000,000
(c) 2000
NOR
SWE
DNK
IRL
GBR
POL
DEU
BEL
FIN
LTU
NLD
CAN
Table 1. Tests for representation of regions’ contiguity and
similarity of configuration.
CZE
FRA
USA
CHN
MDA
JPN
KOR
TJK
TKM
BGR
ALB
ESP
PRT
UZB
AZE
TUR
SBA
BIH
GEO
ROM
HRV
ITA
KGZ
KAZ
AUT
HUN
PRK
RUS
UKR
SVK
CHE
BLR
IRN
GRC
SYR
AFG
TWN
IRQ
LBN
α
0.1
0.3
0.5
0.7
0.9
Percentages of
represented contiguity
69.7
67.7
59.6
54.6
53.6
4.3 Visualization
distribution
of
changes
Residual sum of squares
of affine transformation
89.3
37.6
30.0
21.6
19.3
in
world
MEX
ISR
MAR
CUB
ARE
SAU
HTI DOM
GTM
SLV
EGY
SEN
MRT
HND
SLE
NIC
GIN
BFA
COL
BRA
ERI
NPL
VNM
LAO
BGD
PHL
KHM
THA
PRY
GHA
TGO
CMR
SOM
SDN
TCD
MYS
SGP
UGA KEN
CAF
COG
NGA
BOL
CHL
YEM
ETH
TUN
NER
BEN
PER
rate
ar)
MMR
PAK
LBY
CIV
PAN
ECU
DZA
MLI
LBR
VEN
CRI
JOR
COD
ZMB
ARG
PNG
TZA
MWI
MDG
ZWE
ZAF
IDN
RWA
BDI
AGO
URY
IND
AUS
MOZ
NZL
10.0 ~
5.0 ~ 10.0
2.5 ~ 5.0
10
1,000,000,000
LKA
25
(d) 2010
population
I show some examples using the proposed formulation. Figures
4 and 5 show visualization of the change in world population
distribution from 1980 to 2050. The data is from the World
Population Prospects by the United Nations Population
A special joint symposium of ISPRS Technical Commission IV & AutoCarto
in conjunction with
ASPRS/CaGIS 2010 Fall Specialty Conference
November 15-19, 2010 Orlando, Florida
NOR
SWE
DNK
CAN
FIN
GBR
POL
DEU
BEL
CZE
FRA
USA
BLR
MDA
ALB
IRN
GRC
ESP
PRT
SYR
ARE
SAU
HND
SLE
SLV
EGY
SEN
NIC
VEN
CRI
MRT
GIN
LBR
PAN
TGO
CHL
NGA
URY
GTM
SGP
COD
SLV
ARG
BDI
AGO
ZAF
10.0 ~
IDN
NIC
ZMB
LBR
BRA
TZA
IND
AUS
BOL
NZL
MOZ
owth rate
%/year)
NGA
PNG
KEN
AUS
COD
RWA
IND
NZL
BDI
AGO
ZMB
TZA
MWI
ZWE
10.0 ~
5.0 ~ 10.0
IDN
SOM
UGA
URY
ARG
YEM
CAF
COG
PRY
CHL
MWI
MDG
CMR
PHL
MYS
THA
ETH
ERI
SDN
TCD
BEN
TGO
BGD
NPL
SGP
TUN
LBY
BFA NER
GHA
PER
DZA
MLI
CIV
COL
ECU
OMN
EGY
MRT
GIN
VEN
PAN
PNG
LAO
KHM
PAK
ARE
SAU
SEN
SLE
RWA
ZWE
2.5 ~ 5.0
HND
VNM
MMR
KWT
JOR
ISR
MAR
HTI DOM
MYS
CRI
COG
AFG
IRN
IRQ
SYR
UGA KEN
CAF
BOL
wth rate
/year)
THA
SOM
SDN
BEN
CMR
PRY
BGD
ETH
ERI
NER
TCD
GHA
BRA
PER
TUR
BGR
GRC
LBN
CUB
TWN
TKM
ARM
SBA
BIH
ESP
PRT
MEX
PHL
KHM
LBY
BFA
ECU
NPL
VNM
LAO
TJK
UZB
AZE
GEO
ROM
ALB
PAK
YEM
TUN
MLI
CIV
COL
DZA
OMN
JPN
KGZ
MDA
HUN
KOR
MNG
KAZ
HRV
ITA
MMR
JOR
HTI DOM
GTM
AFG
KWT
ISR
CHN
RUS
AUT
TWN
IRQ
LBN
MAR
CUB
CHE
FRA
BLR
UKR
SVK
USA
TKM
ARM
POL
DEU
BEL
CZE
TUR
BGR
GBR
IRL
UZB
AZE
GEO
ROM
SBA
PRK
FIN
NLD
JPN
TJK
HUN
HRV
BIH
NOR
SWE
DNK
KOR
KGZ
KAZ
AUT
ITA
MEX
CHN
RUS
UKR
SVK
CHE
CAN
PRK
NLD
IRL
MDG
MOZ
ZAF
5.0 ~ 10.0
1,000,000,000
2.5 ~ 5.0
LKA
10~ 25
1 Billion
Population Growth rate
(%/year)
LKA
10~ 25
(e) 2020
Population
1,000,000,000
(a) 2030
CAN
PRK
NOR
SWE
DNK
FIN
CHN
NLD
0.3
0.1
GBR
IRL
-5 -2.5 -1 -0.5 0 1 2.5 5 10
POL
DEU
BEL
USA
CZE
CHE
FRA
BLR
RUS
AUT
BIH
AFG
TUR
BGR
GRC
LBN
Figure 4. World population distribution from 1980 to 2020.
SLV
KWT
SEN
SLE
VEN
CRI
DZA
MLI
COL
PER
GHA
CMR
BOL
PRY
CHL
IDN
UGA
CAF
AUS
KEN
IND
URY
COG
Growth rate
(%/year)
ARG
COD
BDI
AGO
TZA
ZWE
2.5 ~ 5.0
1,000,000,000
LKA
25
(b) 2040
CAN
PRK
GBR
IRL
BEL
CHN
FIN
POL
DEU
CZE
CHE
FRA
KGZ
TWN
UZB
AZE
GEO
HUN
SBA
AFG
TUR
BGR
VNM
TKM
ROM
HRV
TJK
KAZ
AUT
ITA
MMR
IRN
ALB
MEX
KHM
GTM
LBN
HND
MAR
GMB
SLV
NIC
VEN
CRI
GNB
SLE
PAN
LBR
ECU
BRA
GHA
BOL
TGO
PRY
CMR
PNG
ETM
ETH
SDN
UGA
CAF
AUS
SOM
KEN
IND
COG
COD
NGA
AGO
RWA
BDI
ZMB
10.0 ~
NAM
TZA
MWI
ZWE
5.0 ~ 10.0
2.5 ~ 5.0
MYS
SGP
IDN
OMN
YEM
ERI
LBY
TCD
BEN
URY
THA
TUN
BFA NER
ARG
BGD
NPL
ARE
SAU
DZA
PAK
KWT
JOR
EGY
MRT
MLI
CIV
PER
CHL
ISR
SEN
GIN
COL
Growth rate
(%/year)
IRQ
SYR
HTI DOM
PHL
LAO
GRC
ESP
PRT
CUB
JPN
KOR
MNG
RUS
BLR
UKR
SVK
Second, Figures 4 and 5 clearly show that the total world
population is increasing rapidly, particularly in Asian and
African countries. The colours in the circles clearly show the
change in population; however, even if the colour is not shown
on these cartograms, visualizing the change in population is
still possible.
NOR
SWE
DNK
NLD
USA
First, circle cartograms were confirmed to be useful for
visualizing the distribution changes of spatial data. By using the
shape of previous circle cartograms as initial data for the
construction of the next circle cartograms, it is possible to
output similarly shaped cartograms; consequently, the change
in data can be clearly visualized.
MDG
MOZ
ZAF
5.0 ~ 10.0
10
MWI
NAM
10.0 ~
NZL
RWA
NGA
ZMB
When comparing cartograms of different years, cartogram
readers first note the difference in figures and then realize the
change in data presented. To make the comparison of figures
easier, I used the coordinate values of circle centres on the
previous year’s cartograms as the initial values for constructing
the next year’s cartogram.
PNG
SOM
SDN
TCD
BEN
TGO
SGP
ETH
ERI
LBY
BFA NER
PHL
MYS
THA
YEM
TUN
CIV
BRA
BGD
NPL
OMN
EGY
MRT
GIN
LBR
PAN
ECU
PAK
ARE
SAU
GMB
NIC
LAO
KHM
JOR
ISR
MAR
MMR
IRQ
SYR
HTI DOM
VNM
IRN
ALB
HND
TWN
UZB
AZE
ROM
SBA
CUB
GTM
TJK
TKM
HRV
ITA
KGZ
GEO
HUN
ESP
PRT
MEX
JPN
KOR
MNG
KAZ
UKR
SVK
ZAF
MOZ
MDG
LKA
1 000 000 000
(c) 2050
Population
1 Billion
0.3
0.1
Population Growth rate
(%/year)
-5 -2.5 -1 -0.5 0 1 2.5 5 10
Figure 5. World population distribution from 2030 to 2050.
A special joint symposium of ISPRS Technical Commission IV & AutoCarto
in conjunction with
ASPRS/CaGIS 2010 Fall Specialty Conference
November 15-19, 2010 Orlando, Florida
NZL
5. CONCLUSIONS
In this paper, I have proposed solutions for circle cartogram
construction by formulating these problems as constrained nonlinear minimization problems. The application of the proposed
solution revealed that the solution for circle cartogram
construction can output cartograms without overlapping circles.
A comparison between the outputs of the proposed solutions
and that of the previous method with the evaluation of outputs
is left for future work.
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ACKNOWLEDGEMENTS
This work was supported by the Ministry of Education, Culture,
Sports, Science and Technology (MEXT) KAKENHI, a
Scientific Research (A) (General) (21246080). I thank Nguyen
Phuong Hieu for research assistance.
A special joint symposium of ISPRS Technical Commission IV & AutoCarto
in conjunction with
ASPRS/CaGIS 2010 Fall Specialty Conference
November 15-19, 2010 Orlando, Florida