One difficulty with convex hulls is that they are slightly awkward shapes to work with, and in many cases it is simpler to deal with a boundary that is a regular shape, such as a circle or rectangle. Rectangles that align with the coordinate system and enclose selected features compactly are known as Minimum Bounding Rectangles, or Minimum Enclosing (or Envelope) Rectangles (MBRs or MERs).

In most instances it is very fast to determine whether the MBR for one feature or set of features is completely contained within a study region or within the MBR of another set of features. Identifying where the centre of an MBR lies is also very simple and thus they are useful for first-stage identification of simple overlays and membership.

MBRs and convex hulls provide an indication of the limits to the area over which you can reliably interpolate values and many GIS packages take one or other of these boundaries as the limit for procedures such as interpolation, generation of contour lines or the creation of triangulated irregular networks (TINs). Figure 4‑28 shows the result of interpolating income per capita data to a fine grid using the census map shown earlier in Figure 4‑9 (darker areas have higher per capita income). In this case the ArcGIS Spatial Analyst facility has been used, and as can be seen the interpolation process extends to the MBR of census tract centres and no further. We can also see, in this example: (i) the interpolation process replaces census tracts with a grid surface of values, comprising around 250x500 cells, which we have classified into 9 groups for display purposes; (ii) the generated surface pattern is closely tied to the designated centroid locations; and (iii) interpolation has occurred for our rogue locations, including those that should have been screened out as being lakes (no data) prior to interpolation, and two small polygons which are classified as land but for which no census data are provided.

Another problem with hulls (MBRs, convex polygons or more general hull forms) is that by definition they enclose the points for which you have data, and you may be aware of other points, which could or should be taken into consideration, but which lie outside this region.

MBRs and other simple shapes are provided as options for feature enclosure in many GIS packages and in more specialised toolsets such as Crimestat (which is concerned with point events). Frequently the aim is to select a boundary which is reasonable and meaningful for subsequent analysis, often analyses that involve statistical calculations that rely on knowing the point density in advance. As boundaries are altered, the area, A, is changed and this can have a dramatic influence on the analytical results. This may distort analysis, for example giving poor interpolation results near to the region boundary or under-estimating the average distances between sampled points. For example, the nearest tree to one on the boundary may actually be in the adjacent field and not within the pre-defined boundary (Figure 4‑29).

 

Figure 4‑29 Point locations inside and outside bounding polygon