** **

**THE CENTRE OF AUSTRALIA**

(Originally appeared as *From the DGI-The Centre of
Australia*, Queensland Surveyors Bulletin, No.4, August 1988, pp.32-33.)

The Department of Geographic Information has calculated
the *centre of gravity* of Australia.

This
new geographical concept was calculated at the request of Les Isdale, a member
of the Royal Geographical Society. A Royal Geographical Society trip to the *centre*,
is planned for September this year.

Before
the calculations could begin, a definition was needed. This was provided by the
Macquarie Dictionary, which defines a *centre of gravity* as, *that
point of a body from which it could be suspended, or on which it could be
supported and be in equilibrium in any position, in a uniform gravitational
field*.

Further parameters had to be set.

- It was necessary to assume a curved surface with no distortions, as the distribution of weight, thickness of crust, distortions of gravity and so on, are not available.
- Calculations
were carried out on the sphere, while using
*Robbins Reverse*to calculate distances. This was necessary because of the problems of gathering accurate coastline data. Coastline, rivers and lakes are constantly changing and it is difficult to assess tidal movements and arrive at a definition of mean high water mark.

Step one in the process of calculating the point was deciding which method/methods could be used. From the definition, the centre of gravity of a body is that point where two lines dividing the volume of the object into equal halves intersect. Another interpretation, notes that this point lies where the sum of all vectors in the object equals zero. After adopting the first interpretation (which seems more widely accepted), two methods were decided upon. The first method utilised a polygon routine to calculate the areas whilst the second uses spherical trigonometry. Both methods adjusted the data to ensure that it was in a suitable format for equal area calculations.

The second step was to get the data into a suitable
format for calculation. The data used in the calculation was supplied by the
Commonwealth Surveying and Land Information Group, which had digitised data in *stream
mode* from the original repromat used for the 1: 5,000,000 *Australian
General Reference Map* in unjoined segments. It was necessary to order these
segments into one continuous loop to allow area calculations to be made. This
was achieved using the Queensland Centre for Surveying and Mapping Studies' *Autocad*
system to manipulate the data. This task proved to be a time consuming one as locating
the beginning and end of each segment and editing them was essentially a trial
and error process. Our thanks go to the Centre for their time, skill and
expertise.

**Method 1**

Method 1 utilises the formula :

A
^{ψ}_{ψ} = 4b²ϖ (Bsin y' cos δ' Csin 3y' cos 3δ'
+ Dsin 5y' cos 5δ' - Esin 7y' cos 7δ'
.)

B, C, D, E... are constants; ψ - ψ = 2γ and ψ + ψ = 2δ to find the area of the quadrilateral formed by the two points on the spheroid.

The Queensland Centre for Surveying and Mapping Studies (QCSMS) utilised a high speed personal computer to calculate the point. This computer took around half an hour to arrive at a result after processing some 24,000 odd points. The programming for this task was relatively simple and utilised a centroid routine.

**Method 2**

The DGI's Computer Services Sub Program employed the
Departments mainframe computer system. This method required considerable
computing power as it calculated distances using the *Robbins Reverse method*
and the formulae to calculate the area of spherical triangles :‑

tan Όk = √ [tan ½/s tan ½(s-a) tan ½(s-b) tan ½(s-c)]; where

s= (a + b + c)/2; and
k = a/R^{2} sin 1".

The procedure followed the following steps :

a) |
Calculated all spheroidal distances in the triangle formed by the south pole and two adjacent points on the Australian Coastline; |

b) |
Calculated the area of the triangle and summed these areas (assigning a positive value when heading in an easterly direction). The total area of Australia, once found, was then halved. |

c) |
Two points on opposite sides of the coastline were arbitrarily selected and the area of the resultant figure calculated using the process outlined in (b). |

d) |
The process outlined in (c) is repeated iteratively until the area formed by the figure equals half the total area of Australia. The great circle formed by this calculation forms the first intersection line. |

e) |
Steps (c) and (d) are repeated to find a second great circle. Where these two great circles intersect is the centre of gravity by definition. The process is repeated several more times in order to check the original results. |

The Royal Geographical Society will announce the latitude and longitude of this point in late August and then visit the site in September. The Northern Territory Lands Department and Royal Australian Survey Corps are jointly marking the point prior to the Royal Geographical Society's journey.

The Department takes this opportunity to wish the Society every success in its voyage to the Centre of Australia.