While there are many definitions of sphericity, the two sphericity indices given in the GrainFind output are those defined by Sheppard (2006) (optional) and Krumbein (1941).

Sheppard Sphericity

Based on Sheppard, the sphericity of a grain is defined as the ratio of the inner radius of a grain to its equivalent radius .

The equivalent radius is the radius of a sphere with the same volume as the grain. The inner radius is the radius of the largest sphere which fits into the grain. The inner radius can be computed based on the Euclidean Distance Transform (EDT).

Sphericity values range between 0 and 1. The more the grain resembles a sphere, the higher is its sphericity value. A value of 1 marks a perfect spherical grain. In the example below, the sphericity of the square shaped grain is .

Krumbein Sphericity

To calculate sphericity based on Krumbein, three principal axes (, , ) of the grains fit-shape are determined. The Krumbein sphericity is then calculated using the length of these axes as

where is length of the longest axis, while and are the lengths of the two shorter axes.

Analogously to the Sheppard sphericity, the values for the Krumbein sphericity range between 0 and 1, with 1 characterizing perfectly spherical grains.