In this section we discuss geometric properties of grains that are related to the convex hull of the grains. In the follwing we illustarte these properties with some artificial grain shapes

Smoothing of grains is necessary since otherwise many grain segments are either vertical or horizontal (for a square grid) and perimeters rather measure the "cityblock" distance. See also https://t.co/1vQ3SR8noy?amp=1 for examples. Note, that for very small grains, the error between the smoothed grains and their convex hull may lead to unsatisfactory results.

The convex hull of a list of grains can be computed by the command `hull`

. The result is a list of convex grains which can be analyzed almost analogously like the original list of grains.

One major difference is that grains may now overlap but their convex hulls usually do. Accordingly, the boundaries of the convex hull grains are not a boundaries between adjecent grains and, therefore, the second phase in all convex hull boundary segments is set to `'notIndexed'`

.

## Deviation from fully convex shapes

There are various measures to describe the deviation from fully convex shapes, i.e. the lobateness of grains. Many of these are based on the differences between the convex hull of the grain and the grain itself. Depending on the type of deviation from the fully convex shape, some measures might be more appropriate over others.

One measure is the relative difference between the grain perimeter and the perimeter of the convex hull. It most strongly discriminizes grains with thin, narrow indenting parts, e.g. fracture which not entirely dissect a grain.

A similar measure is the paris which stands for Percentile Average Relative Indented Surface and gives the difference between the actual perimeter and the perimeter of the convex hull, relative to the convex hull.

The relative difference between the grain area and the area within the convex hull is more indicative for a broad lobateness of grains

The total deviation from the fully convex shape can be expressed by

Lets visualize all these shape parameters in one plot