Polygon centres have many important functions in GIS: they are sometimes used as “handle points”, by which an object may be selected, moved or rotated; they may be the default position for labelling; and for analytical purposes they are often used to “represent” the polygon, for example in distance calculations (zone to zone) and when assigning zone attribute values to a point location (e.g. income per capita, disease incidence, soil composition). The effect of such assignment is to assume that the variable of interest is well approximated by assigning values to a single point, which essentially involves loss of information. This is satisfactory for some problems, especially where the polygons are small, homogeneous and relatively compact (e.g. unit ZIP-code or postcode areas, residential land parcels), but in other cases warrants closer examination.

With a number of raster-based GIS packages the process of vector file importing may utilise polygon centres as an alternative to the polygons themselves, applying interpolation to the values at these centroids in order to create a continuously varying grid rather than the common procedure of creating a gridded version of discretely classified polygonal areas (see further Chapter 6). Similar procedures may be applied in vector-based systems as part of their vector-to-raster conversion algorithms.

If (x_i,y_i) are the coordinate pairs of a point set or defining a single polygon, we have for the Mean Centre, M1:

The Mean Centre is not the same as the centre of gravity for a polygon (although it is for a point set, see subsection 4.2.5.2). A weighted version of this expression is sometimes used:

Mean Centre (weighted), M1*:

The RMS variation of the point set {x_i,y_i} about the mean centre is known as the standard distance. It is computed using the expressions:

As noted above, the term centroid for a polygon is widely used to refer to its assumed centre of gravity. By this is meant the point about which the polygon would balance if it was made of a uniform thin sheet of material with a constant density, such as a sheet of steel or cardboard. This point, M2, can be computed directly from the coordinates of the polygon vertices in a similar manner to that of calculating the polygon area, A, provided in Section 4.2.1. Indeed, it requires computation of A as an input. If the polygon is a triangle, which is a widely used form in spatial analysis, the centre of gravity lies at the mean of the vertex coordinates, which is located at the intersection of straight lines drawn from the vertices to the mid-points of the opposite sides (Figure 4‑7).

Figure 4‑7 Triangle centroid

For general polygons, using the same notation as before, the formulas required for the x- and y-components are (x>=0, y>=0):

The formula arises as the weighted average of the centroids of triangles in a standard triangularization of the polygon, where the weights correspond to the triangle areas. Figure 4‑8 shows a sample polygon with 6 nodes or vertices, A-F, together with the computed locations of the mean centre (M1), centroid (M2) as defined by the centre of gravity, and the centre (M3), of the Minimum Bounding Rectangle (or MBR) which is shown in grey.

Figure 4‑8 Polygon centroid (M2) and alternative polygon centres

Each of these three points fall within the polygon boundary in this example, and are fairly closely spaced. The MBR centre is clearly the fastest to compute, but the most subject to outliers — e.g. a single vertex location that is very different from the majority of the elements being considered. For example, if point B had (valid or invalid) coordinates of B(34,3) then we would find M1=(10.67,6.5), M2=(8.67,5.36) and M3=(17,6.5). M3 is now well outside of the polygon, M1 is close to the polygon boundary and M2 remains firmly inside. None of these points minimises the sum of distances to polygon vertices.

Despite the apparent robustness of M2, with a polygon of complex shape the centroid may well lie outside of its boundary. In the example illustrated in Figure 4‑9 we have generated centroids for a set of polygons using the X‑Tools add-in for ArcGIS, and these produce slightly different results from those created by ArcGIS itself. ArcGIS includes a function, Features to Points, which will create polygon centres that are guaranteed to lie inside the polygon if the option INSIDE is included in the command. Manifold includes a Centroids|Inner command which performs a similar function.

An alternative to using the MBR is to find the smallest circle that completely encloses the polygon, taking the centre of the circle as the polygon centre (Figure 4‑10, M4). This is the default in the Manifold GIS (viewable simply by selecting the menu option to View|Structure|Centroids) and/or using the Transform function “Centroids”. This procedure suffers similar problems to that of using the MBR. Yet another alternative is to find the largest inscribed circle, and take its centre as the polygon centre (Figure 4‑10, M5). This approach has the advantage that the centre will definitely be located inside the polygon, although possibly in an odd location, depending on the polygon shape. If the polygon is a triangle then M5 and M2 will coincide. Manifold supports location of centres of types M2, M3 and M4.

As we can also see in Figure 4‑9 multiple polygons may be selected and assigned combined centres. In this example the combined centre appears to have been computed from the MBR of the two selected census tracts, but different packages and toolsets may provide alternative procedures and thus positions. In such cases the centre may be nowhere near any of the selected features, and if it is important to avoid such a circumstance the GIS may facilitate selection of the most central feature (e.g. the most central point) if one exists. In some instances the combined centre calculation may be weighted by the value of an attribute associated with each polygon. This procedure will result in a weighted centre whose location will be pulled towards regions with larger weights. However, in such cases the distribution of polygons used in the calculation may produce unrepresentative results.

This is apparent again in Figure 4‑9 where a polygon centre calculation for all census tracts in the western half of the region, weighted by farm revenues, would be pulled strongly to the east of the sub-region where many small urban tracts are found, even though these may have relatively low weights associated with them.

Figure 4‑10 Polygon centre selection