The parameters of the Grain-Fit Shapes are used to compute further grain parameters. We show the parameters graphically with the help of a simple structure, that contains one ellipsoidal grain with dimensions 50 µm x 30 µm x 40 µm in the center of an empty domain with voxel length 1 µm.

The three Grain-Fit Shapes are all described by three values. For the Ellipsoids three diameters are needed, for the Short Ellipsoidal Fibers two diameters and the length of the fiber are required, and the Box is described by the three side lengths.

• to be the length of the longest of the three values,
• to be the length of the medium value,
• to be the length of the shortest value.

• Volume: The volume of the grain is simply determined by the number of voxels the grain contains. To get a volume in metrical units, it is multiplied by the cubed voxel length:

In the example, the grain consists of 31384 voxels, so that the grain volume is 31384 µm3.

• Equivalent Diameter: The equivalent diameter is the diameter of the sphere which has the same volume as the grain. On the right, the volume equivalent sphere for the example (which has a diameter of 39 µm) is shown in red and was added to the initial ellipsoid using the Add command of LayerGeo. The black part represents the overlap.

• Inner Diameter: The inner diameter is the diameter of the largest sphere that can be inscribed into the grain. Below, the inscribed sphere for the example, which has a diameter of 30 µm, is shown in red and was added to the ellipsoid structure with LayerGeo.

• Shortest Fit Diameter: This is the shortest diameter () of the shape fitted into the grain. For the example, the shortest diameter is 30 µm.

• Intermediate Fit Diameter: This is the intermediate diameter () of the shape fitted into the grain. For the example, the intermediate diameter is 40 µm.

• Longest Fit Diameter: This is the longest diameter () of the shape fitted into the grain. For the example, the longest diameter is 50 µm.

• Perimeter: The computed perimeter is the shortest perimeter around the shape fitted into the grain. As an ellipsoid is defined by three diameters,  the perimeter is computed as the perimeter of the ellipse formed from the two smallest of those three diameters. For the example structure the perimeter would be the length of the black line, which is 110 µm.

• Krumbein Sphericity: The Krumbein Sphericity is a measure for the sphericity of the grain, based on the three principal axes (, , ) of the fitted shape. If the Krumbein Sphericity is close to 1, the shape is nearly a sphere. In the example above, the Krumbein Sphericity is 0.78, which means that the diameters differ much.

• Sheppard Sphericity: The Sheppard Sphericity is defined as the diameter of the inscribed sphere divided by the diameter of the volume-equivalent sphere. For the example, the Sheppard Sphericity of the pore is 0.76.

• Aspect Ratio: The shape fitted into the grain is defined by three diameters. The aspect ratio is computed as

where the shortest diameter is divided by the largest diameter , which is 0.6 for the example.

• Surface Area: The surface of the grain is estimated by an algorithm based on MatDict’s Estimate Surface Area command (see also J. Ohser, F. Mücklich). The Area (or Surface) Probability is defined by computing the surface area of each segmented grain (using the same algorithm as in MatDict’s Estimate Surface Area command) and then weighting the grain by that value. The computed surface area does also include the surface between different grain, not only the area between pore and solid material. For the example, the surface is 4985 µm2.

• Surface-to-Volume Ratio: The estimated surface area of the grain is divided by the volume of the grain. For the example a value of 0.159 is obtained.

• Surface Smoothness: This is the surface of the fitted shape divided by the estimated surface of the grain. Usually, the estimated grain surface is larger than the surface of the fitted shape. Hence, the surface smoothness is usually below 1. Only if the shape of the grain is very similar to the fitted shape, then the surface estimation might be a little smaller than the surface of the fitted shape. The latter is the case in the example, where the surface smoothness is 1.002.

• Cut-Surface Ratio: The Cut-Surface Ratio measures how much of the grain surface is part of the domain boundary. The interface of the grain with the domain boundary is the Cut Surface, which is then divided by the remaining surface of the grain. Thus, it is a measure for the quality of boundary grain. For example, a Cut-Surface Ratio of 1 means that the interface of the grain with the domain boundary is as large as the remaining surface of the grain. In the example, the grain has no boundary contacts and thus the cut surface area and the cut-surface ratio are both 0.

• Mass: If a density is assigned to the analyzed material, each grain has a mass which can be calculated from its volume.

• Moment of Inertia: It depends on the mass distribution of the grain and is only computed if a density is assigned.

• Coordination Number: This is the number of contacts of a grain to other grains. In the example the coordination number is 0 as only one grain exists.