The Scientific World Journal

Volume 2016 (2016), Article ID 9057263, 18 pages

http://dx.doi.org/10.1155/2016/9057263

## Fuzzy Logic for Incidence Geometry

Dassault Systemes, 175 Wyman Street, Waltham, MA 02451, USA

Received 3 November 2015; Revised 2 June 2016; Accepted 26 June 2016

Academic Editor: Oleg H. Huseynov

Copyright © 2016 Alex Tserkovny. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The paper presents a mathematical framework for approximate geometric reasoning with extended objects in the context of Geography, in which all entities and their relationships are described by human language. These entities could be labelled by commonly used names of landmarks, water areas, and so forth. Unlike single points that are given in Cartesian coordinates, these geographic entities are extended in space and often loosely defined, but people easily perform spatial reasoning with extended geographic objects “as if they were points.” Unfortunately, up to date, geographic information systems (GIS) miss the capability of geometric reasoning with extended objects. The aim of the paper is to present a mathematical apparatus for approximate geometric reasoning with extended objects that is usable in GIS. In the paper we discuss the fuzzy logic (Aliev and Tserkovny, 2011) as a reasoning system for geometry of extended objects, as well as a basis for fuzzification of the axioms of incidence geometry. The same fuzzy logic was used for fuzzification of Euclid’s first postulate. Fuzzy equivalence relation “extended lines sameness” is introduced. For its approximation we also utilize a fuzzy conditional inference, which is based on proposed fuzzy “degree of indiscernibility” and “discernibility measure” of extended points.

#### 1. Introduction

In [1–4] it was mentioned that there are numerous approaches by mathematicians to restore Euclidean Geometry from a different set of axioms, based on primitives that have extension in space. An approach, aimed at augmenting existent axiomatization of Euclidean geometry with grades of validity for axioms (fuzzification), is also presented in [1–4]. But in contrast with [1–4], where the* Lukasiewicz* logic was only proposed as the basis for “fuzzification” of axioms and no proofs were presented for both fuzzy predicates and fuzzy axiomatization of incidence geometry, we use fuzzy logic from [5] for all necessary mathematical purposes to fill up above-mentioned “gap.”

#### 2. Axiomatic Geometry and Extended Objects

##### 2.1. Geometric Primitives and Incidence

Similarly to [1–3, 6–8] we will use the following axioms from [9]. These axioms formalize the behaviour of points and lines in incident geometry, as it was defined in [1]:(*I*1)*For every two distinct points p* and* q*,* at least one line l exists that is incident with p* and* q*.(*I*2)*Such a line is unique*.(*I*3)*Every line is incident with at least two points*.(*I*4)*At least three points exist that are not incident with the same line*.

The uniqueness axiom (*I*2) ensures that geometrical constructions are possible. Geometric constructions are sequential applications of construction operators. An example of a construction operator is the following: *Connect*:* point* ×* point* →* line*.

Take two points as an input and return the line through them. For* connect* to be a well-defined mathematical function, the resulting line needs always to exist and needs to be unique. Other examples of geometric construction operators of 2D incidence geometry are the following: *Intersect*:* line* ×* line* →* point*. *Parallel through point*:* line* ×* point* →* line.*

The axioms of incidence geometry form a proper subset of the axioms of Euclidean geometry. Incidence geometry allows for defining the notion of parallelism of two lines as a derived concept but does not permit expressing betweenness or congruency relations, which are assumed primitives in Hilbert’s system [9]. The complete axiom set of Euclidean geometry provides a greater number of construction operators than incidence geometry. Incidence geometry has very limited expressive power when compared with the full axiom system.

The combined incidence axioms (*I*1) and (*I*2) state that it is always possible to connect two distinct points by a unique line. In case of coordinate points* a* and* b*, Cartesian geometry provides a formula for constructing this unique line:

As it was shown in [1–4], when we want to connect two extended geographic objects in a similar way, there is no canonical way of doing so. We cannot refer to an existing model like the Cartesian algebra. Instead, a new way of interpreting geometric primitives must be found, such that the interpretation of the incidence relation respects the uniqueness property (*I*2).

Similarly to [1–4] we will refer to extended objects that play the geometric role of points and lines by* extended points* and* extended lines*, respectively. Section 3 gives a brief introduction on proposed fuzzy logic and discusses possible interpretations of fuzzy predicates for extended geometric primitives. The fuzzy logic from [5] is introduced as possible formalism for approximate geometric reasoning with extended objects and based on extended geometric primitives fuzzification of the incidence axioms (*I*1)*–*(*I*4) is investigated.

#### 3. Fuzzification of Incidence Geometry

##### 3.1. Proposed Fuzzy Logic

Let, and continuous function , which defines a distance between and . Notice that , where and . When normalized, the value of is defined as follows:

It is clear that. This function represents the value of “closeness” between two values (potentially* antecedent* and* consequent*), defined within single interval, which therefore could play significant role in formulation of an implication operator in a fuzzy logic. Before proving that is defined asand is from (2), let us show some basic operations in proposed fuzzy logic. Let us designate the truth values of logical* antecedent * and* consequent * as and , respectively. Then relevant set of proposed fuzzy logic operators is shown in Table 1. To get the truth values of these definitions we use well-known logical properties such as and the like.