Fuzzy Plane Geometry: Triangles

Abstract

(Also cross-referenced as CAR-TR-694) A fuzzy triangle T (with a discrete-valued membership function)

can be regarded as a nest of parallel-sided triangles Ti with successively higher membership values. Such a nest is determined by its max projections on any two of its "sides". The area (per imeter) of T is a weighted sum of the areas (perimeters) of the Ti's. The side lengths and altitudes of T can also be defined as weighted sums obtained from projections; using these definitions, the perimeter of T is the sum of the side lengths, and the s ide lengths are related to the vertex angles by the Law of Sines, but there is no simple relationship between the area of T and the products of the side lengths and altitudes.