Metrics for vague spatial objects based on the concept of mass

Many spatial phenomena exhibit vagueness. Representation of such phenomena requires vague objects. In previous work, we provided definitions for vague objects: vague points, vague lines, and vague regions. Each of these objects is presented as a fuzzy set in ℝ
²
that satisfies well-defined properties. In this paper, we propose a number of geometric measures for vague objects, using the concept of mass distribution. The membership function of a vague object can be seen as a mass distribution. According to this view, a crisp object is a body with constant density, and a vague object is a body of varying density. We provide mathematical definitions for length of a vague line, area of a vague region, centroid of a vague object, as well as a measure for the vagueness of an object. The length of a vague line and the area of a vague region are indeed the mass of the vague line and of the vague region, respectively. Both metrics give an average of the values of the corresponding crisp metric on the α-cuts of the vague object. The centroid of a vague object is its centre of mass associated with a membership degree. The last metric functions as a measure of the degree of vagueness for a vague object.