The roughness of a shape can be characterised by the distribution of the local shape curvatures: a perfectly smooth surface (ie a sphere) will have constant curvature, while a rough surface will have a variety of local curvatures. Curvature implies the calculation of second order derivatives, or alternatively, the calculation of the changes in the normal vector from one point on the surface to the next. We can quantify this change in the normal vector by calculating the angle between two normal vectors. Suppose we consider only chords of the surface of length d and we find the orientation difference between the normal vectors that correspond to the end points of each chord. If the surface were spherical, this angle would be a constant, no matter which chord we chose. For any other surface, a distribution of values will result. By varying the values of d we can create a representation of the surface that contains information for various scales. The larger d is the more global the information. Thus, for local roughness characterisation, only small values of d should be used.
Maria Petrou