## Quantitative Description of Planar Shape

Two-dimensional shape parameters provide a means to monitor shape changes occurring in microstructures of P/M products due to variations in processing parameters and a means to compare the particle shape of powders obtained with different materials and production techniques. For this method, Fischmeister (Ref 39) uses the term "fingerprinting," because if applied properly, it yields indirect but accurate information on three-dimensional shape. The basic requirement outlined previously, however, must also be observed for two-dimensional shape factors.

Hausner's shape parameters, described earlier in this article in the section on conventional shape parameters, are an example of "fingerprinting," as are other methods of planar shape description, such as Fourier coefficients and fractals of the sectioned or projected particle circumferences. In sectioned or projected particle circumferences, parameters are described that are sensitive to particular shape aspects and are easily determined with semiautomatic or fully automatic image analyzers. In the Hausner method, the fitting of the minimum area rectangle requires repeated fitting, which in some cases leads to ambiguous results.

By far the most frequently used planar shape parameter (implemented in most image analyzing devices) combines the area and the perimeter of a planar feature into a dimensionless number normalized to 1 for circles of the same size:

where a and b are the area and the perimeter of the planar features (intersects or projections), respectively. Averaging can be completed easily if a and b are measured individually for each feature:

where n is the total number of planar features. However, if a and b are averaged, or if total area and perimeter length are used (as provided by field analysis), the average shape parameters obtained depend on size distribution, with values deviating from unity for a system of circles of varying diameter.

If Eq 6 is used, deviations from unity are interpreted as deviations from circularity. Elongation (elliptical deformation), as well as concave deformations of the perimeter ("rugged" outlines), yields smaller values approaching zero for highly elongated features or highly rugged perimeters.

Thus, F\ combines different aspects of shape. To differentiate between these aspects, simple combinations of moments of inertia, Feret diameters, or curvature have been derived by Schwarz (Ref 42). Exner and Hougardy (Ref 34) propose the following parameters for individual features, which also can be averaged for any number of features in the same way as fpL (see Eq 6):

where i1 and /2 are the two principal moments of inertia, bc is the convex perimeter, and b is the actual perimeter.

These parameters are sensitive to the respective shape aspects, are easily measured by computer-aided image analysis, and are independent of other geometric properties; their interpretation through visualization is easily accomplished.

Consideration should be given to the method of averaging, however. Averaging on a volume basis, rather than on a number basis as in Eq 6, may be appropriate in some practical applications. If three-dimensional measurements are performed, similar shape parameters can be derived for the various aspects of spatial shape.

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