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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)