Fuzzy Geometry

Contents

In this chapter we discuss a version of fuzzy plane geometry developed by Buckley and Eslami (1997a,b). The approach presented here is one in which the area, heights, width, diameter, and perimeter of fuzzy subsets are fuzzy numbers. The chapter lays the basic groundwork for the models that we develop in the ensuing chapters, particularly chapter five. Those readers not interested in the formalism behind the geometry of these models will find the discussions in the chapters sufficient.

We begin by reformulating the definition of a fuzzy number in a manner better suited for the geometry that follows. Understanding that fuzzy points in two dimensional space can be visualized as surfaces in three dimensions is the key to understanding fuzzy geometry. It is this property that will help model uncertainty in ways that a single crisp point could not.

Spatial models are useful because relationships can be visualized. Hence, we need a concept of distance between points and a concept of regions bounded by points. For the former, we build on the preliminary discussion of fuzzy distance at the end of Chapter 2. For the latter, we move toward defining fuzzy shapes with the definition of a fuzzy line. Just as a crisp line can be understood as a collection of points, a fuzzy line can be thought of as a collection of fuzzy points. As a result, a fuzzy line not only has length, but can be thick as well. We also define a measure of parallelness that indicates the extent to which two fuzzy lines can be said to be parallel.

From fuzzy lines, we move to fuzzy circles and their properties, and then to line segments. With these tools we can finally define generic fuzzy polygons. This chapter concludes with some geometry and trigonometry of fuzzy polygons and a note on the distinction between crisp and fuzzy shapes.