A number of the sections in this Guide have touched on the issue of polygon shape: how this affects where the center lies; how compactness of polygon form may be related to standard shapes such as a circle or rectangle; and how polygons may be decomposed into simpler shapes, such as triangles. In practice defining some form of shape measure that provides an adequate description is difficult and many indices and procedures have been proposed. Shape measures may be applied to polygon forms or to grid patches (sets of contiguous grid cells that have common or similar attributes or are regarded as being a patch for analytical purposes). Measures relating to grid structures are described in Section 5.3.4, Landscape Metrics. The comments in this latter section apply equally well to shape measures for polygonal (vector) regions:
“The most common measures of shape complexity are based on the relative amount of perimeter per unit area, usually indexed in terms of a perimeter-to-area ratio, or as a fractal dimension, and often standardized to a simple Euclidean shape (e.g., circle or square). The interpretation varies among the various shape metrics, but in general, higher values mean greater shape complexity or greater departure from simple Euclidean geometry.”
Vector-based GIS software packages do not normally provide shape index measures directly — it is up to the user to create index values. In many instances such values are functions of the area and perimeter of the polygon, and may be generated as a new calculated field from these (intrinsic or explicit) attributes. If we define Ai as the Area of polygon i, and Li as its perimeter length, and Bi as the area of a circle with perimeter Li, then example measures include:
Perimeter/Area ratio (P1A):
Note that this ratio is unsatisfactory for most applications as its value changes with the size of the figure (i.e. irrespective of resolution issues).
Perimeter2/Area ratio (P2A):
This second ratio was used in the UK 2001 census Output Area generation exercise (see further, Section 4.2.11, Districting and re-districting). If preferred, this index, or its square root may be adjusted so that its value equals 1 when applied to a circle of the same area as Ai.
Shape Index or Compactness ratio (C):
Note that this ratio is dimensionless (i.e. not affected by the size of the polygon) and has a value of 1 for a circular region and a range of [0,1] for all plane shapes. Note that sometimes this index is computed as (1‑Ci]. Grid-based packages (e.g. Idrisi, CRATIO command) and specialized pattern-analysis software (e.g. Fragstats) may compute ratios of this type as inbuilt functions.
Related bounding figure (RBF):
where Fi is the area of a bounding figure. Typically the bounding figure will be the minimum bounding circle, but could equally well be the MBR or convex hull. In each case the ratio lies in the range [0,1], with 0 being the value if the polygon matches the bounding figure. When index values of this type are computed for multiple polygons (e.g. as a new column in an attribute table) a shape distribution is created, that may be examined with conventional univariate statistics (e.g. mean, median, standard deviation, range etc.) and/or graphically examined as a histogram.
More complex shape measures, such as those based on multiple characteristics of polygon form (e.g. the length of the major and minor internal axes, or the set of possible straight lines computable within the polygon) are more difficult to compute. At present all such measures typically require separate programmed computation, either within a GIS environment or externally.
Shape measures may also be computed for other vector objects, such as sets of points or lines. In the case of point sets the common procedure is to compute an RBF-type measure, i.e. essentially generating a polygon such as the convex hull of a cluster of points and analyzing its form. The standard deviational ellipse (see further, Section 4.5.3, Directional analysis of point datasets, and Section 5.4, Point Sets and Distance Statistics) also provides an option for point-set shape analysis, utilizing the ellipse area and major and minor axes.
For linear objects or collections of linear objects, many options exist, a number of which are directly supported within standard GIS packages, particularly those with hydrological analysis components (e.g. providing ratios such as stream length/basin area). Many such measures are indirect measures of linear component shapes. Perhaps the simplest measure for individual polylines is to compute the ratio of polyline length to the Euclidean (or spherical) distance between start and end points. This kind of measure is highly dependent on scale, line representation, generalization and point pair selection.
A circuity index (devised by Kansky) for transport networks based on this kind of idea involves comparing the inter-vertex network distance, N, with the Euclidean distance, E:
where V is the number of vertices in the network and n is the number of distinct pairwise connections, i.e. n=V(V‑1)/2. This overall index can be disaggregated into separate indices for each vertex, providing separate measures of vertex based circuity. If information is available on user preferences for travel between vertex pairs, then so-called “desire lines” can be utilized in index computations of this kind (e.g. as weightings). As with many of the other measures described in this subsection the computation of index components and totals must utilize generic calculation facilities within the GIS or be computed externally.