### Abstract

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.

Original language | English (US) |
---|---|

Title of host publication | Applying Fuzzy Mathematics to Formal Models in Comparative Politics |

Pages | 65-80 |

Number of pages | 16 |

Volume | 225 |

DOIs | |

State | Published - 2008 |

### Publication series

Name | Studies in Fuzziness and Soft Computing |
---|---|

Volume | 225 |

ISSN (Print) | 14349922 |

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### ASJC Scopus subject areas

- Computer Science (miscellaneous)
- Computational Mathematics

### Cite this

*Applying Fuzzy Mathematics to Formal Models in Comparative Politics*(Vol. 225, pp. 65-80). (Studies in Fuzziness and Soft Computing; Vol. 225). https://doi.org/10.1007/978-3-540-77461-7_3

**Fuzzy geometry.** / Clark, Terry D.; Larson, Jennifer M.; Mordeson, John N.; Potter, Joshua D.; Wierman, Mark J.

Research output: Chapter in Book/Report/Conference proceeding › Chapter

*Applying Fuzzy Mathematics to Formal Models in Comparative Politics.*vol. 225, Studies in Fuzziness and Soft Computing, vol. 225, pp. 65-80. https://doi.org/10.1007/978-3-540-77461-7_3

}

TY - CHAP

T1 - Fuzzy geometry

AU - Clark, Terry D.

AU - Larson, Jennifer M.

AU - Mordeson, John N.

AU - Potter, Joshua D.

AU - Wierman, Mark J.

PY - 2008

Y1 - 2008

N2 - 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.

AB - 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.

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U2 - 10.1007/978-3-540-77461-7_3

DO - 10.1007/978-3-540-77461-7_3

M3 - Chapter

SN - 9783540774600

VL - 225

T3 - Studies in Fuzziness and Soft Computing

SP - 65

EP - 80

BT - Applying Fuzzy Mathematics to Formal Models in Comparative Politics

ER -