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The concept of a centroid is useful for many spatial applications, and the determination of the centroid of a plane polygon is standard functionality in most Geographic Information System (GIS) software. A common reason for determining a centroid is to create a convenient point of reference for a polygon, often for positioning a textual label. For such applications, the rigour with which the centroid is determined is not critical, because in the positioning of a label, for example, the main criteria is that it be within the polygon and reasonably central for easy interpretation.However, there may be applications where the determination of a centroid has, at the very least, an impact on civic pride and quite possibly financial repercussions. We refer here to an administrative or natural region where a nominated centroid has a certain curiosity value with the potential to become a tourist attraction. Such centroids provide economic benefit to those in a sub-region, usually in close proximity to the centroid.Various interpretations of a centroid exist and this paper explores these and the methods of calculation. Variation in position resulting from different interpretations is examined in the context of the centroid of the Australian State of Victoria, and GIS software are evaluated to determine the efficacy of their centroid functions.
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1
THE CENTROID? WHERE WOULD YOU LIKE IT TO BE?
R E Deakin 1
Dr R.I. Grenfell 2
and
S.C. Bird.3
1 Lecturer, Department of Geospatial Science
Email: rod.deakin@rmit.edu.au
2 Business Development Manager, Geospatial Science Initiative
Email: ron.grenfell@rmit.edu.au
3 Student, Department of Geospatial Science
Email: stbird@cs.rmit.edu.au
RMIT University
GPO Box 2476V MELBOURNE VIC 3001
Tel: +61 3 9925 2213 / Fax: +61 3 9663 2517
ABSTRACT
The concept of a centroid is useful for many spatial applications, and the determination of the
centroid of a plane polygon is standard functionality in most Geographic Information System (GIS)
software. A common reason for determining a centroid is to create a convenient point of reference
for a polygon, often for positioning a textual label. For such applications, the rigour with which the
centroid is determined is not critical, because, in the positioning of a label for example, the main
criteria is that it be within the polygon and reasonably central for easy interpretation.
However, there may be applications where the determination of a centroid has, at the very least,
an impact on civic pride and quite possibly financial repercussions. We refer here to an
administrative or natural region where a nominated centroid has a certain curiosity value with the
potential to become a tourist attraction. Such centroids provide economic benefit to those in a sub-
region, usually in close proximity to the centroid.
Various interpretations of a centroid exist and this paper explores these and the methods of
calculation. Variation in position resulting from different interpretations is examined in the context
of the centroid of the Australian State of Victoria, and GIS software are evaluated to determine the
efficacy of their centroid functions.
2
INTRODUCTION
In physics, it is often useful to consider the mass of a body as concentred at a point called the
centre of mass (or the centre of gravity). For a body of homogeneous mass, this point coincides
with its geometric centre or centroid. Thus, we have the commonly accepted meaning that the
centroid is equivalent to the centre of mass. For example, the centre of mass of a homogeneous
sphere (a geometric solid) is coincident with its centre.
In geospatial science, we often deal with relationships between points on the surface of the earth,
but generally, these points are represented as projections onto a plane (a map) and areas of interest
are defined by polygons. The centroid of a polygon, in this case, does not have a tangible
connection with the centre of mass of the object, since it is merely a series of lines (or points) on
paper or a computer image. Therefore, we often equate the centroid to the geometric centre of the
polygon, which in the case of complicated polygons is often impossible to determine. Thus we
resort to mathematical formulae and Cartesian coordinates to calculate centroids.
An age-old joke has an accountant being asked for an opinion, the reply to which is, "what would
you like it to be". Geospatial professionals asked to determine the centre of a complex polygon
might reply in a similar vein. In this paper, in the context of geospatial science, we shall
demonstrate that there are several plausible definitions of a centroid, most leading to relatively
simple means (or averages) of coordinates, but some requiring more advanced methods of
computation. Some examples will be given to demonstrate methods of calculation and highlight
cases where restrictions need to be placed on certain centroid definitions.
In addition, we compare centroids computed by different methods with those of several GIS
software packages in general use. This comparison should shed some light on the computation
methods used by these software products. Finally, some results of the calculation of the centroid(s)
of Victoria, one of the States of the Commonwealth of Australia, and its capital Melbourne are
presented.
DEFINITIONS OF CENTROIDS
The OpenGIS Specification for Feature Geometry (OGC 1999) defines a centroid object without
greatly assisting in its calculation for individual polygons:
The operation "centroid" shall return the mathematical centroid for this
GM_Object. The result is not guaranteed to be on the object. For
heterogeneous collections of primitives, the centroid only takes into
account those of the largest dimension. For example, when calculating
the centroid of surfaces, an average is taken weighted by area. Since
curves have no area they do not contribute to the average.
In this section, a number of centroids are defined and named; the names are only relevant to this
paper. In some cases computational formulae are given. Additional information relevant to the
computation of certain centroids is given in Section 3.
3
The Moment Centroid
The New Shorter Oxford English Dictionary (SOED 1993) defines the centroid as: "A point
defined in relation to a given figure in a manner analogous to the centre of mass of a corresponding
body." Using this definition, and regarding the body as a plane area A of uniformly thin material,
its centroid is
y
M
xA
=and x
M
yA
=(1)
and x
M and y
M are (first) moments with respect to the x- and y-axes respectively (Ayres 1968).
[The moment L
M of a plane area with respect to a line L is the product of the area and the
perpendicular distance of its centroid from the line.] The centroid computed using this method has
a physical characteristic that is intuitively reassuring. That is, if we cut out a shape from uniformly
thin material (say thin cardboard) and suspend it freely on a string connected to its centroid, the
shape will lie horizontal in the earth's gravity field. In this paper, we will call this centroid a
Moment Centroid.
The Area Centroid
In a similar vein, we may divide the uniformly thin shape of area A into two equal portions 1
A
and 2
A about a dividing line BB
. If the shape were symmetrical about this line, it would
balance if it were placed on a knife-edge along BB
. If we choose another line CC
, which
divides the area equally, then the intersection of the dividing lines (or balance lines) defines a point
we call the Area Centroid. This centroid also has an intuitively appealing simplicity but
unfortunately, for a general polygon of area A, different pairs of balance lines intersect at different
points! That is, the method does not yield a unique point unless the direction of the dividing lines
is defined. In addition, unless the figure is symmetrical about both balance lines, this centroid will
not coincide with the moment centroid.
The Volume Centroid
The area centroid could be regarded as the plane analogue of a Volume Centroid defined as
follows. Consider, as a scale model of the earth, a spherical shell with its interior filled with
homogenous mass. On the surface of the sphere, a region of interest (say Australia) appears as a
uniformly thin surface layer. Any great circle plane, which divides the region into equal areas, will
also divide the volume of the earth equally. Two great circle planes, both of which divide the
region into equal areas, will themselves intersect along a diameter of the sphere, which cuts the
surface at a point. We call this point the Volume Centroid.
Is this point some sort of centre of gravity? Consider the spherical shell lying at rest, empty of all
matter, with the region of interest (Australia) in contact with a frictionless level surface. The shell
is in equilibrium and the direction of gravity through the equilibrium point (or contact point) will
pass through the centre of the shell. As will an infinite number of planes, one of which will contain
the great circle dividing the region in two. The Volume Centroid will lie somewhere along this
great circle but not necessarily at the equilibrium point; they will only coincide when the region is
symmetric about the point.
In Australia, a point called the geographical centre of Australia and named the Lambert Centre (in
honour of the former Director of National Mapping, Dr. Bruce P Lambert) has been calculated by
the Queensland Department of Geographic Information. The principle of computation seems to
follow our definition for the Volume Centroid (DGI 1988, ISA 1994).
4
Average Centroids
In the Geodetic Glossary of the US National Geodetic Survey (NGS 1986), a centroid is defined
as: "The point whose coordinates are the average values of the coordinates of all points of the
figure."
This concept of the centroid of a figure as a point having average values of the coordinates (of the
points defining the figure) encompasses three types of averages; the mean, the median and the
mode (Reichmann 1961). All three are measures of central tendency.
The first type of average, the mean, can be subdivided into arithmetic mean, root mean square
mean, harmonic mean and geometric mean, all of which can be defined by using two equations
from Apostol (1967). The pth-power mean p
M of a set of real numbers 12
,,,
n
xx x
! is
1/
12
p
pp p
n
pxx x
Mn

+++
=

"(2)
where the number 1
M is the arithmetic mean, the number 2
M is the root mean square and 1
M is
the harmonic mean. The geometric mean G of a set of real numbers 12
,,,
n
xx x
! is
1/
12
()
n
n
Gxxx="(3)
The second type of average, the median, is the central value, in terms of magnitude, of a set of n
values ordered from smallest to largest. If n is odd, the median is the middle value and if n is even,
it is the arithmetic mean of the middle pair of values. In both cases, there will be k values less than
or equal to the median and k values greater than or equal to the median.
The third type of average, the mode, is the value that occurs most frequently in a set of numbers.
It is used in statistics to indicate the value of a substantial part of a data set. As a measure of a
central point of a geometric figure, the mode has little or no practical use. It will not be discussed
further.
In addition, we can define a centroid as the arithmetic mean of the maximum and minimum values
in a set of real numbers. This point, as we show below, is the centre of a rectangle that encloses the
figure (the Minimum Bounding Rectangle).
Six different "average" centroids are therefore defined below.
The Arithmetic Mean Centroid
Using (2) with p = 1, we define the Arithmetic Mean Centroid (,)
xy as
1
n
k
kx
xn
=
=and 1
n
k
ky
yn
=
=(4)
The Root Mean Square Centroid
Using (2) with p = 2, we define the Root Mean Square Centroid (,)
xy as
2
1
n
k
kx
xn
=
=and
2
1
n
k
ky
yn
=
=(5)
5
The Harmonic Mean Centroid
Using (2) with p = 1, we define the Harmonic Mean Centroid (,)
xy as
1
1
1
n
kk
x
xn
=
=and 1
1
1
n
kk
y
yn
=
=(6)
The Geometric Mean Centroid
Using (3) and taking natural logarithms to overcome numerical problems encountered with long
products, we define the Geometric Mean Centroid (,)
xy as
1ln
ln
n
k
kx
xn
=
=and 1ln
ln
n
k
ky
yn
=
=(7)
The Median Centroid
If the x and y coordinates of the n points defining the figure, are each ordered, from smallest to
largest, into two arrays
[]
123 n
xxx x=x" and
[]
123 n
yyy y=y" then the Median
Centroid (,)
xy is
for n odd: k
xx
=and k
yy
=where 1
2
n
k+
=(8a)
and
for n even: 1
2
kk
xx
x+
+
=and 1
2
kk
yy
y+
+
=where 2
n
k=(8b)
The Minimum Bounding Rectangle Centroid
The Minimum Bounding Rectangle Centroid (,)
xy is defined as
2
MIN MAX
xx
x+
=and 2
MIN MAX
yy
y+
=(9)
where ,
MIN MIN
xy
and ,
MAX MAX
xy
are the minimum and maximum values of the x and y
coordinates respectively. This centroid will lie at the centre of a rectangle, whose sides are parallel
with the coordinate axes, and which completely encloses the figure. The dimensions of the
rectangle are width= MAX MIN
xx
and height= MAX MIN
yy
.
The Minimum Distance Centroid
We define a Minimum Distance Centroid (,)
xy, as the point where the sum of the distances k
d
from the centroid to every point defining the polygon is a minimum. That is, the minimum value of
the function
22
11
(,) ( ) ( )
nn
kk k
kk
fxy d x x y y
==
== −+
∑∑ (10)
Minimum Distance centroids have a connection with spatial analysis where a gravity function
may be used to model relationships between points or regions. Such functions, often arbitrarily
defined, usually have an inverse relationship with distance (or distance squared); thus minimising
6
distances maximises the gravity function. The earliest reference to such a gravity function (as a
concept in studies of human interaction) is attributed to H.C.Carey, who reportedly stated in the
early 1800's; "The greater the number collected in a given space, the greater is the attractive force
that is there exerted … Gravitation is here, as everywhere, in the direct ratio of the mass, and the
inverse one of distance." (Carrothers 1956, p.94).
The Negative Buffer Centroid
A polygon of n sides has an internal polygon of m sides (mn) where each side is parallel to, but
offset by a constant distance from, its complimentary side in the original polygon. This process of
reduction of size (and shape) is known (in GIS parlance) as negative buffering. Positive buffering
creates zones of constant width around polygons (Prescott 1995). By repeated negative buffering, a
polygon of many sides can be reduced to a simpler polygon of far fewer sides. In many cases,
negative buffering leads to a triangular polygon with an easily derived geometric centre. We define
a Negative Buffer Centroid as the geometric centre of the polygon having the least number of sides
resulting from repeated negative buffering of an original polygon.
The Circle Centroid
Closely allied to the Negative Buffer centroid, we define a Circle Centroid as the centre of the
inscribed circle of a polygon. The Circle centroid can be obtained by using negative buffering to
reduce the original polygon to a simpler internal polygon of (generally) far fewer sides. The centre
of the inscribed circle of this simpler shape will also be the centre of the inscribed circle of the
original polygon. In most cases, negative buffering will reduce an n-sided polygon to a triangle
whose inscribed circle can be easily calculated. It should be noted that the Circle centroid (the
centre of the inscribed circle) is not, in general, coincident with the Negative Buffer centroid.
CALCULATION OF CENTROIDS
To demonstrate some of the methods used to compute centroids, and to show the variation between
different centroids, the simple polygon in Figure 1 will be used. The coordinates (metres) of the
polygon corners are given in Table 1 and the area of the polygon is 2643.17 m2.
Point
1 103.450 287.760
2 151.860 315.990
3 141.030 289.410
4 167.830 255.920
5 114.130 235.840
xy
Table 1
Figure 1
y
x
1
2
3
4
5
7
Since the calculation of some centroids requires areas of polygons, the following algorithm will
be useful, noting that it yields positive areas proceeding clockwise around polygons and negative
areas anticlockwise.
{}
11
1
2( ) ( )
n
kk k
k
Area x y y
−+
=
=−
(11)
The Moment Centroid of Figure 1
The Moment centroid is calculated from (1) where the moments y
M and x
M are found by
regarding the polygon of area A as being composed of a network of triangles of area k
A each
having a centroid (,)
kk
xy. We can then employ the rule for calculating moments of composite
areas: the moment of a composite figure with respect to a line is the sum of the moments of the
individual areas with respect to the line. With each triangle having moments k
ykk
MAx= and
k
xkk
MAy= the centroid of the composite figure is given by
12 1
n
n
kk
yy y
ykAx
MM M
M
xAA A
=
+++
== =
" and similarly, 1
n
kk
kAy
yA
=
=(12)
Figure 2 shows the component triangles in the polygon and Table 2 shows their centroid
coordinates and areas. Note that the centroid of a triangle is located at the average values of the x
and y coordinates of the triangle vertices.
2
22
32
4
Triangle Area
132.1133 297.7200 490.5035m
137.4367 277.6967 651.3871m
128.4700 259.8400 1501.2792m
kk k
xy A
A
A
A
Table 2
The Moment Centroid of the composite figure is
3
1
3
1
131.356m
271.270m
kk
k
kk
k
Ax
xA
Ay
yA
=
=
==
==
Figure 2
The values in Table 2 (component areas k
A and centroids ,
kk
xy
) have been computed from the
following algorithms
1
211
kk
kkiki
ii
Axyyx
==

=∆ ∆∆ ∆


∑∑ (13)
1
131
2k
kik
i
xx x x
=

=+ ∆


and 1
131
2k
kik
i
yy y y
=

=+ ∆


(14)
1
2
3
4
5
A
2
3
A
A
4
8
where x
and y are coordinate differences of each side of the polygon and it should be noted
that for a polygon having n sides, 10
n
AA
==
. The algorithms (13) and (14) can be used to
compute the area and centroid location of any polygon. The authors have not been able to find any
references to these formulae, although it is unlikely that they are original; a derivation is given in
Appendix A.
The Area Centroid of Figure 1
Area centroids of Figure 1, determined by the intersection of balance lines do not give a unique
point. This is demonstrated by the two cases shown below. In the first case, Figure 3 shows the
balance lines B-B' and C-C' in the cardinal directions. In the second case, Figure 4 shows another
pair of balance lines B-B' and C-C', still perpendicular to each other, but no longer in cardinal
directions. The centroids for both cases differ by an appreciable amount.
Area Centroid: Case 1 (Cardinal directions)
2
12
22
32
4
Point Area
107.163 269.710 685.28m
156.795 269.710 636.33m
131.006 303.829 685.26m
131.006 242.156 636.32m
xy
BA
BA
CA
CA
=
=
=
=
Table 3
The Area Centroid of the figure is
131.006m
269.710m
x
y=
=
Figure 3
Area Centroid: Case 2 (Non-Cardinal directions)
2
12
22
32
4
Point Area
104.336 288.277 793.47m
160.327 253.114 528.12m
112.467 243.926 793.47m
141.030 289.410 528.13m
xy
BA
BA
CA
CA
=
=
=
=
Table 4
The Area Centroid of the figure is
131.142m
272.071m
x
y=
=
Figure 4
1
2
3
4
5
BB'
C'
C
AA
A
A
1
2
3
4
5
B
B'
C'
C
A
A
A
A
9
The "Average" Centroids of Figure 1
These centroids are calculated using equations (4) to (9) and tabulated below.
Average Centroid
Arithmetic Mean 135.660 276.984
Root Mean Square 137.728 278.399
Harmonic Mean 131.363 274.082
Geometric Mean 133.521 275.542
Median 141.030 287.760
Minimum Bounding Rectangle 135.640 275.915
xy
Table 5. Average Centroids of Figure 1
It should be noted that all of these centroids, except the Minimum Bounding Rectangle, are
functions of the number of points that make up the figure, not necessarily its shape. For example,
in Figure 5 three points have been added, which divide the line 5 to 1 into four equal parts. The
shape of the figure has not altered but the centroids will change substantially.
1
2
3
4
5
6
7
8
.
.
.
.
.
C
C'
Figure 5
In Figure 5, C is the Arithmetic Mean Centroid of points 1 to 5 and C' is the Arithmetic Mean
Centroid using the additional three points. This movement from C to C', caused by the additional
points along the line 1-5, is representative of changes in location of the Root Mean Square,
Harmonic Mean and the Geometric Mean Centroids.
The Minimum Distance Centroid of Figure 1
The Minimum Distance Centroid of Figure 1 is obtained by minimising (10). This cannot be done
by simple arithmetic but instead requires sophisticated function minimisation techniques.
Microsoft's Excel solver has been used to obtain the minimum value of function (10) as
141.019m
289.397m
x
y=
=
The solution that the Excel solver obtains may be described in the following way. Imagine a fine
mesh grid placed over the figure with every grid intersection a computation point with coordinates
10
,
jk
xy
. From each computation point the distance to every point in the figure is computed and
summed, yielding a single value, say ,
jk
z. As j and k vary from 1 to n a grid of z values will be
obtained and a three dimensional plot would reveal a surface with a low point. The x, y coordinates
of this low point will be the Minimum Distance Centroid of the figure.
The Negative Buffer Centroid and Circle Centroid of Figure 1
Figure 6 shows the original five-sided polygon of Figure 1 and two internal polygons, ABCDE of
five sides and PQR of three. The internal polygons have been created by negative buffering; ie
zones of constant width inside a polygon define a smaller polygon. ABCDE is defined by a 10
metre wide zone within the original figure and PQR is defined by a 10-metre buffer zone within
ABCDE. The Negative Buffer Centroid in this case is the geometric centre (and centroid) of the
triangle PQR.
Point Point
114.692 282.740 126.337 275.761
131.510 292.547 134.962 264.982
129.554 287.746 129.012 262.757
151.396 260.451
121.571 249.299
xy xy
AP
BQ
CR
D
E
Table 6. Coordinates of polygons ABCDE and PQR
The Negative Buffer Centroid of the figure is
130.104m
267.833m
x
y=
=
Figure 6
The Circle Centroid is the centre of the inscribed circle of triangle PQR. The radius r of the
inscribed circle is given by ()()()
r ssasbscs
=− where a, b and c are the sides of the
triangle and
()
2
sabc=++ =
is the semi-perimeter (Spiegal 1986). Now, since sides PQ, QR
and RP are parallel to sides 3-4, 4-5 and 5-1 respectively, then 20
r+ will be the radius of the
largest circle that can be drawn inside the original polygon. This inscribed circle is shown in
Figure 6, its radius is 22.492 metres and its centre is located at 130.867 m
c
x= and
266.111 m
c
y=. Note that the Circle Centroid and the Negative Buffer Centroid are not the same
point.
COMMENTARY ON CENTROID DEFINITIONS
The Arithmetic Mean, Root Mean Square, Harmonic Mean, Geometric Mean and Median
Centroids are all legitimate measures of spatial central tendency. While they have the benefit of
being easily calculated, we have shown that they are a function of the vertices of a polygon but are
not sensitive to the order of the vertices and therefore the shape of the polygon. They are, therefore
considered to be useful descriptors of point data sets but not of polygons. Similarly, while the
Minimum Bounding Rectangle Centroid is easy to calculate and is representative of the four
extreme vertices of the polygon, it is not sensitive to the entire shape.
1
2
3
4
5
A
B
C
D
E
P
Q
R
11
The Area Centroid suffers from not providing a unique result for a given polygon unless the
balance lines are constrained to particular directions, such as for the Minimum Bounding Rectangle
where the cardinal directions are assumed.
The Negative Buffer and Circle Centroids are not sensitive to large poorly conditioned portions of
a polygon (best described as spikes). The computation of these centroids is difficult to automate,
particularly to deal with highly irregular polygons.
The Moment and Minimum Distance Centroids both provide a logical and intuitively appealing
result, but the latter requires sophisticated function minimisation software for calculation. This
computational drawback, and deficiencies mentioned above of the other centroids, leads the authors
to prefer the Moment Centroid as the best measure of the centre of a complex polygon.
GIS CENTROID FUNCTIONS
Determinations of a polygon centroid is a standard requirement in Geographic Information
System (GIS) software, primarily for providing a representative point within a polygon. This point
provides a focus for the positioning of attribute labels. Because, in general, a centroid cannot be
guaranteed to fall within the polygon to which it relates, the representative point is often a
paracentroid, this satisfies this criterion at the expense of centrality. The definition of centroid and
method of calculation adopted by a particular GIS product is rarely documented for the user.
Three commonly used GIS packages were tested with the simple polygon in Figure 1 to determine
the likely definition of centroid which they adopt. These packages are MapInfo Professional V5.5
(MapInfo Corporation), ARCView V3.1 (ESRI) and ARCInfo V7.2.1 (ESRI).
The MapInfo centroidX and centroidY functions produce results consistent with the Minimum
Bounding Rectangle (MBR) definition of centroid.
According to the ARCView help documentation, the ReturnCenter method also returns the MBR
Centroid, unless it falls outside the polygon, in which case it is moved in the x direction the
minimum amount to place it within the polygon. In practice, ReturnCenter was found to return the
MBR Centroid regardless of whether it fell within the polygon or not.
ARCInfo uses the createlabels command to create a paracentroid within a polygon. The
documentation clearly states that this point will not necessarily be at the centroid (without defining
centroid). Once paracentroids (label points) have been created, they can be moved to the centroid
using the centroidlabels command. Although the definition of centroid adopted is not documented,
the point produced by centroidlabels is consistent with the Moment Centroid.
CENTROIDS OF VICTORIA
To calculate the various centroids of Victoria a digital outline of the state was derived from a
coordinate data set known as VIC500-2000, which is part of the L500 library of Natural Resources
and Environment (NRE) Corporate Geospatial Data Library. The VIC500-2000 data set are
VICMAP_TM coordinates which are related to a grid superimposed over a Transverse Mercator
projection of latitudes and longitudes on the Australian Geodetic Datum 1966 (AGD66). The
projection has a central meridian of 145° 00' 00" with a scale factor of unity. The north and east
axes of the grid are parallel to the central meridian and equator respectively and the origin of
coordinates is 500,000 metres west and 10,000,000 metres south of the intersection of the central
meridian and the equator. The GIS software product ARCInfo was used to transform the VIC500-
2000 coordinates to AGD66 latitudes and longitudes ( ,
φ
λ
). The route analysis function of
ARCInfo was then used to generate a subset of 1648 consecutive points at 2 km intervals around the
12
Victorian borders and coastline. The ,
φ
λ
coordinates of the 1648 points were then transformed to
east and north (E,N) coordinates (kilometres) related to an Equal Area Cylindrical projection of a
sphere of radius R using the following formulae
()
0
sin
XR
YR
λλ
φ
=−
=(15)
where 6372.800
R= km, 0
λ
= 145° 00' 00" and
5,000 km
5,000 km
EX
NY
=+
=+ (16)
An Equal Area projection was chosen in this study because two of the centroids require the
computation of areas of plane figures and this projection preserves area scale. That is, an element
of area on the spherical Earth dA is projected as an element of area on the map da and the ratio
da/dA = 1.
The positional accuracy of the original data (VICMAP_TM) is stated in the NRE documentation
as ±0.5 km and the derived digital outline (1648 points) does not include any offshore islands and
is a relatively coarse approximation of Victoria's boundaries. No tests were conducted on the
accuracy of the ARCInfo transformation formulae (VICMAP_TM to ellipsoid) and the
transformation to E,N coordinates assumes the points are located on a sphere of radius equal to the
mean radius of curvature of the AGD ellipsoid appropriate for the latitude of Victoria. All centroid
values for Victoria, computed from the derived digital outline, are rounded to the nearest 0.5 km
(E,N). These values are then regarded as exact for the purposes of transformation to latitudes and
longitudes, using inverse relationships obtained from (15 and (16), which are then rounded to the
nearest 15 seconds of arc. These results are shown in Table 9.
Centroid EN
Latitude Longitude
Moment 4921.0 1180.0 -36° 49' 45" 144° 17' 30"
Area (Cardinal directions) 4874.5 1168.5 -36° 57' 30" 143° 52' 15"
Arithmetic Mean 4972.0 1170.0 -36° 56' 30" 144° 45' 00"
Root Mean Square 4981.0 1178.0 -36° 51' 00" 144° 49' 45"
Harmonic Mean 4954.0 1154.5 -37° 07' 00" 144° 35' 15"
Geometric Mean 4963.0 1162.0 -37° 02' 00" 144° 40' 00"
Median 4975.5 1128.5 -37° 24' 30" 144° 46' 45"
Minimum Bounding Rectangle 5051.5 1207.5 -36° 31' 15" 145° 27' 45"
Minimum Distance 4815.5 1122.0 -37° 29' 00" 143° 20' 30"
Negative Buffer 4711.5 1205.5 -36° 32' 30" 142° 24' 30"
Circle (r = 160.1 km) 4711.5 1206.5 -36° 31' 00" 142° 24' 30"
Table 9 Coordinates of Centroids
Figure 7 shows an Equal Area Cylindrical projection of Victoria and parts of New South Wales and
South Australia, and three centroids (i) the Negative Buffer Centroid, (ii) the Minimum Bounding
Rectangle (MBR) Centroid and (iii) the Moment Centroid. The Negative Buffer Centroid is at the
centroid of a triangle, the result of successive negative buffering of the digital outline of Victoria.
The Circle Centroid (the centre of the inscribed circle of this triangle) is also the centre of the
largest circle (radius 160.1 km) that can be inscribed within this projection of Victoria. The Circle
Centroid and Negative Buffer Centroid differ by only 1 km but the size and location of the
inscribed circle is dependent on the type of projection. The MBR Centroid (the simplest to
compute) and the Negative Buffer Centroid (the most difficult to compute) appear to be evenly
balanced about the Moment Centroid, located at Mandurang, approximately 6 km south-east of
Bendigo, a major regional city.
13
Figure 8 shows an enlargement of the region near Bendigo with the major roads and regional
towns. The Moment Centroid is shown south of Bendigo with the other centroids (Area, Minimum
Distance and the Average centroids), dispersed across the region. The map covers an area of
approximately 180 km by 90 km.
Figure 7
Negative Buffer Centroid, Minimum Bounding Rectangle (MBR) Centroid and
Moment Centroid
moment rms
arithmetic
geometric
harmonic
Kyneton Lancefield
Broadford Yea
Wallan
Kilmore
Rushworth
Inglewood
Nagambie
Romsey
median
Macedon
Woodend
Trentham
DaylesfordDaylesford
Creswick
Clunes
Lexton
Avoca Maryborough
Beaufort
mindist
Maldon
Dunolly
area Puckapunyal
Castlemaine
Bendigo
Figure 8
The Moment, E-W Area and Minimum Distance Centroids
and the five Average Centroids
CENTROID OF MELBOURNE
Melbourne, the capital of Victoria, is a large urban metropolis with a population of approximately
3.5 million. In May, 2001, officers of the Department of Natural Resources and Environment,
using Victoria's digital cadastral database (VicMap Digital) created a digital outline of Urban
Melbourne. This polygon, consisting of 5733 points (containing an area of 1718 km2 ) defines a
boundary between urban Melbourne and surrounding rural areas. Its delineation depended on
criteria such as the extent of drainage and sewerage schemes, electricity and water supply schemes,
housing density and land allotment areas. Its complex shape describes Melbourne's urban sprawl,
which is generally in an easterly and south-easterly direction from the central business district and
in places is 35-40 km from Melbourne central.
+++
Negative Buffer
Centroid
Centroid
Centroid
Moment
NEW SOUTH WALES
VICTORIA
SOUTH
AUSTRALIA
MBR
.
.
Melbourne
Bendigo
14
The data set was provided in Australian Map Grid 1966 (AMG66) coordinates. These values
were converted to AGD66 values ( ,
φ
λ
) and then to east and north (E,N) Cartesian coordinates
(kilometres) related to an equal area cylindrical projection of a sphere of radius R using formulae
(15) and (16) and the associated parameters
The Moment Centroid for Melbourne computed from the derived E,N coordinates is shown in
Table 10 rounded to the nearest 0.5 km. The latitude and longitude values are computed by using
the inverse relationships obtained from (15) and (16), which are then rounded to the nearest 15
seconds of arc.
Centroid E (km) N (km) Latitude Longitude
Moment 5007.5 1089.5 -37° 51' 00" 145° 04' 30"
Table 10 Coordinates of Moment Centroid of Melbourne
Figure 9
Moment Centroid of Urban Melbourne
+
Port Phillip Bay
.
Melbourne
Centroid
15
CONCLUSION
This paper has provided ten working definitions for the centroid of a plane polygon. Others could
no doubt be constructed, but in the opinion of the authors, the Moment Centroid, which is
intuitively appealing, is the most appropriate.
In the context of requiring a centroid for the traditional cartographic purpose of labelling a
polygon, consideration of how the centroid is defined is largely academic. This is particularly so
when the computed point may have to be relocated to ensure it is within the polygon object or for
cartographic license in producing a pleasing output.
In the context of defining the 'official' centre of an administrative or physical region, the outcome
is far from purely academic interest. The work associated with this paper in determining the
centroids of Victoria and Melbourne attracted the attention of the popular media in at least one
national television program, two statewide newspapers, a regional newspaper, two radio interviews
and a government department publication. In the case of the (Moment) centroid of Victoria at
Mandurang south of Bendigo, the local council and tourism authority are considering the erection
of a substantial monument, signposting and other promotional material. The owner of the property
near which the centroid falls is considering how best to capitalise on the windfall, with a Bed and
Breakfast accommodation facility likely. In the case of the centroid of Melbourne (located at
Ferndale Park in the suburb of Glen Iris), a plaque has been placed and its dedication was attended
by a Government Minister, Departmental Heads, local government officials and the media.
Clearly, the centroid of a State or a Capital city has a high curiosity value and potential as a tourist
attraction providing significant economic benefit and civic pride.
In the case of the Victoria, the ten centroids canvassed, span a region 275 km East-West by 100 km
North-South and are included in seven different municipalities. Had Map Info or ARCView been
used to calculate the centroid without consideration of the definition they implement, the location
would be nearly 100 km from the position provided by ARCInfo. If you were a representative of
one of those municipalities, where would you like the centroid to be?
16
REFERENCES
Apostol, Tom M., 1967. Calculus, Vol.1, 2nd edn, Blaisdell Publishing Co., London.
Ayres, F., 1968. Differential and Integral Calculus, Shaum's Outline Series, McGraw-Hill Book
Company, New York.
Carrothers, G.A.P., 1956. 'An historical review of the gravity and potential concepts of human
interaction.' Journal of the American Institute of Planners, Spring 1956, pp.94-102.
DGI (Department of Geographic Information), 1988. 'The Centre of Australia', in: From the DGI,
Queensland Surveyors Bulletin, Vol.4, August 1988, pp.32-3.
ISA (Institution of Surveyors, Australia), 1994. 'The Lambert Centre', The Australian Surveyor,
Vol.39, No.3, September 1994, p.215.
NGS (National Geodetic Survey), 1986. Geodetic Glossary, United States National Geodetic
Survey, Rockville, Maryland, USA.
OGC, 1999. The OpenGISTM Abstract Specifications, Topic 1: Feature Geometry, Version 4.
OpenGISTM Project Document Number 99-101.doc, Open GIS Consortium,
http://www.opengis.org
Prescott, George W., 1995. A Practioner's Guide to GIS Terminology: A Glossary of Geographic
Information System Terms, Data West Research Agency, Washington, USA.
Reichmann, W.J., 1961. Use and Abuse of Statistics, Penguin Books, Middlesex, England.
SOED, 1993. The New Shorter Oxford English Dictionary, ed Lesley Brown, Clarendon Press,
Oxford.
Spiegel, M.R., 1968. Mathematical Handbook of Formulas and Tables, Schaum's Outline Series,
McGraw-Hill Book Company, New York.
SoV (State of Victoria), 1982. Atlas of Victoria, ed J.S. Duncan, Victorian Government Printing
Office, Melbourne, Victoria, Australia.
17
APPENDIX A
The algorithm for computing the area of a polygon, given as equation (13), can be derived by
considering Figure A1, where the area is the sum of the trapeziums bBCc, cCDd and dDEe less the
triangles bBA and AEe.
The area can be expressed as
()()()
()()()
()()()
()()
()()
21 31 2 3
31 4 1 3 4
41 51 45
212 1
5115
2Axxxx yy
xx xx yy
xx xx yy
xxyy
xxyy
=−+− −


+−+− −


+−+− −


−− −
−− −
(A1)
Expanding (A1) then cancelling and
re-arranging terms gives
()
()
()
()
()
15 2
21 3
32 4
43 5
54 1
2Axy y
xy y
xy y
xy y
xy y
=−
+−
+−
+−
+−
which can be expressed as
{}
11
1
2()
n
kk k
k
Axyy
−+
=
=−
(A2)
In Figure A2, the coordinate origin is shifted to A
where 11
0
xy
′′
==
and the area, using (A2), is
23 34 32 45 43 54
2Ayxyxyxyxyxyx
′′ ′′ ′′ ′ ′′ ′′
=+−+−− (A3)
Considering each side of the polygon to have components
,
kk
xy
∆∆
for 1to5
k=, (A3) can be written as
()
()( )
()()
()( )
()()
()()
11 2
12123
121
1231234
12312
1234123
2Ayx x
yyxxx
yyx
yyyxxxx
yyyxx
yyyyxxx
=∆ ∆ +
+∆ + ∆ + +
−∆ +
+ ∆ +∆ +∆ +∆ +∆ +∆
− ∆ +∆ +∆ +∆
− ∆ +∆ +∆ +∆ +∆ +∆
Figure A2
A
B
C
D
E
y
x
b
c
d
e
x
xx
x
x
y
yy
y
y
2
13
4
5
2
13
4
5
Figure A1
A
B
C
D
E
y'
x'
x'
x' x'
x'
x'
y'
y' y'
y'
y'
2
13
4
5
2
13
4
5
12
3
4
5
18
Expanding and gathering terms gives
()()
()()
()()
()
11 2 34 11 23
21 2 34 21 23
31 2 3 4 3 1 2 3
41 2 3
2332 32
222 32
22
Ay x x x x y x x x
yx x xx yx xx
yx x x x y x x x
yx x x
=∆ ∆ + + ∆ +∆ −∆ ∆ + ∆ +∆
+∆ + ∆ + ∆ + + ∆ +
+∆ ∆ + + + ∆ + ∆ +
−∆ ∆ +∆ +∆
and cancelling terms and re-ordering gives
()
()
()
()
1234
21 34
312 4
4123
20
0
0
0
Ay x x x
yx xx
yxx x
yxxx
=∆ +∆ +∆ +∆
+∆ −∆ + + +
+∆ −∆ + +
+∆ −∆ − +
(A4)
This equation for the area can also be expressed as a matrix equation
[]
1234 1
2
3
4
20111
10 11
1101
1110
Ayyyy x
x
x
x
=∆ ∆ ∆ ∆







− − 

−−−


(A5)
By studying the form of equations (A4) and (A5), the following algorithm for calculating the
1
kn
=−
area components k
A for a polygon of n sides may be deduced as
1
211
kk
kkiki
ii
Axyyx
==

=∆ ∆∆ ∆


∑∑
where 1, 2, 3, 1
kn=−
!(A6)
Equation (A6) is an efficient way to accumulate the area of a polygon given the coordinate
components of the sides. By studying the algorithm, it can be seen that 10
n
AA
==
and hence the
area of a polygon is accumulated without having to deal with the last side. In addition, it can be
seen that each area component k
A is a triangle with one vertex at the starting point and the line k,
with components ,
kk
xy
∆∆
, the opposite side. This leads to the formulae for the centroids of each
triangle
1
131
2k
kik
i
xx x x
=

=+ ∆


and 1
131
2k
kik
i
yy y y
=

=+ ∆


(A7)
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Introduction: The spatial distribution of Aedes aegypti is heterogeneous, and the interaction between positive and potential breeding sites located both inside and outside homes is one of the most difficult aspects to characterize in vector control programs. Objective: To describe the spatial relationship between potential and positive breeding sites of A. aegypti inside and outside homes in Cali, Colombia. Materials and methods: We conducted an entomological survey to collect data from both indoor and outdoor breeding sites. The exploratory analysis of spatial data included location, spatial trends, local spatial autocorrelation, spatial continuity and spatial correlation of positive and potential breeding sites according to habitat. Results: Spatial trends were identified, as well as clusters of potential and positive breeding sites outdoors using local spatial autocorrelation analysis. A positive correlation was found between potential and positive breeding sites, and a negative correlation existed between indoor and outdoor sites. Conclusions: The spatial relationship between positive and potential A. aegypti breeding sites both indoors and outdoors is dynamic and highly sensitive to the characteristics of each territory. Knowing how positive and potential breeding sites are distributed contributes to the prioritization of resources and actions in vector control programs.
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