Defining shape measures for 3D star-shaped particles: Sphericity, roundness, and dimensions

Truncated spherical harmonic expansions are used to approximate the shape of 3D star-shaped particles including a wide range of axially symmetric ellipsoids, cuboids, and over 40.000 real particles drawn from seven different material sources. This mathematical procedure enables any geometric property to be calculated for these star-shaped particles. Calculations are made of properties such as volume, surface area, triaxial dimensions, the maximum inscribed sphere, and the minimum enclosing sphere, as well as differential geometric properties such as surface normals and principal curvatures, and the values are compared to the analytical values for well-characterized geometric shapes. We find that a particleʹs Krumbein triaxial dimensions, widely used in the sedimentary geology literature, are essentially identical numerically to the length, width, and thickness dimensions that are used to characterize gravel shape in the construction aggregate industry. Of these dimensions, we prove that the length is a lower bound on a particleʹs minimum enclosing sphere diameter and that the thickness is an upper bound on its maximum inscribed sphere diameter. We examine the “true sphericity” and the shape entropy, and we also introduce a new sphericity factor based on the radius ratio of the maximum inscribed sphere to the minimum enclosing sphere. This bounding sphere ratio, which can be calculated numerically or approximated from macroscopic dimensions, has the advantage that it is less sensitive to surface roughness than the true sphericity. For roundness, we extend Wadellʹs classical 2D definition for particle silhouettes to 3D shapes and we also introduce a new roundness factor based on integrating the dot product of the surface position unit vector and the unit normal vector. Limited evidence suggests that the latter roundness factor more faithfully captures the common notion of roundness based on visual perception of particle shapes, and it is significantly simpler to calculate than the classical roundness factor.