Two-dimensional shape analysis of toner may be extended to three dimensions by combining shape information from typical image analysis of oriented toner with information from instruments that measure toner volume such as the Coulter Multisizer. A mean aspect ratio for the hidden dimension perpendicular to the plane of the image can be estimated by assuming a model shape such as an ellipsoid. Furthermore, if the shape is independent of the size, a relationship between the aspect ratio of the hidden dimension and the aspect ratio of the visible dimensions may be developed. Shape is independent of size when brittle fracture dominates the grinding process in melt-pulverized toners (MPT) and is independent by design in the limited coalescence manufacturing method used to make chemically prepared toner (CPT). The relationship between the different aspect ratios is useful to distinguish toners that have flat disk-like particles from toners with spherical particles within a broader distribution of shapes that result in different performance in the electrophotographic process.
Kevin Lofftus, "Distributed 3D Aspect Ratios of Toner" in Proc. IS&T Int'l Conf. on Digital Printing Technologies and Digital Fabrication (NIP24), 2008, pp 90 - 93, https://doi.org/10.2352/ISSN.2169-4451.2008.24.1.art00023_1