POLYGONS. A polygonal line, also called a broken line, join ing the point to the point A. is any finite set of points . . , and the segments A2A3, . . . , In this and the following definition of a polygon the phrase "a point A i" means a point associated with the symbol Ai and the phrase "a segment means the segment, whose ends are the points which are the associates of the symbols A i and A; respectively, associated with the symbol A iAi. A segment is the set of all points of any (straight) line which are between any two points of that line. Each of the latter points is called an end of the seg ment. A polygon is any finite set of points A1A2, . . . , An and the segments A1A2, A2A3 . . . , An-1 An, The points A and segments A A nA 1, i= I, 2, . . . , n, are called respectively the vertices and sides of the polygon; similarly for a polygonal line.
The terms polygonal line and polygon are used also with mean ings which are different from, although closely related to, those given above. A polygon as defined may have one of two senses assigned to it so that the first end and the second end of each side is specified in such a way that the vertex A i is either the first end of the side AjAi_i and the last end of the side Ai_ Ai if i= I, and if i= I, or vice versa. A polygon with such an assign ment of a sense is called an oriented or a sensed polygon. In an obvious way an oriented polygonal line is defined. Thus two sensed polygons or polygonal lines are associated with each polygon or polygonal line. In the sequel the phrase the oriented (or sensed) side AB of a sensed polygon indicates that A is the first end of the oriented side AB of that oriented polygon, and B the second. Polygonal lines and polygons according to the first definition are referred to as unoriented or unsensed polygonal lines and poly gons respectively. In formulating a third meaning of the terms polygonal line and polygon it should be emphasized that the ele ments involved in the above definition are points and segments associated with symbols so that an unoriented or oriented poly gon is neither a set of points and segments nor a set of points. The word polygon also is used to signify either certain sets of points and segments or certain set of points. The distinctions just
pointed out although delicate are logically essential and even practically important. For the purpose of this article the single word "polygon" denotes any set of points which consists of the points which are the associates of the symbols A1, A2, . . .
and the points which belong to the segments which are the asso ciates of the symbols A i=i, 2, . . . n and AnAl of the first definition; i.e., the definition of an unsensed polygon. Simi larly in the case of a polygonal line.
The above definitions are of broad scope and define abstrac tions which are based on the phenomenon of the motion of a parti cle from point to point along intermediate rectilinear stretches. Important specializations of these ideas are the so-called simple polygons or polygonal lines according to any of the definitions given. A simple unoriented polygon is any unoriented polygon which is such that none of its vertices is an end of more than two of its sides and no side of the unoriented polygon contains a vertex or a point which belongs to another side of the unoriented polygon. The definitions of simple unoriented polygonal lines, simple ori ented polygonal lines and polygons as well as those of simple poly gons and polygonal lines in conformity with the third definition are apparent and consequently are not stated formally. Alternative definitions for the several concepts defined or indicated above may be given ; for example a simple polygon may be defined as a finite set of points and segments such that (a) every point of the set is the end of two and only two segments of the set, (b) each end of every segment of the set is a point of the set, (c) no segment of the set contains a point of the set or a point of another segment of the set and (d) no (proper) subset of the given set satisfies (a), (b) and (c). It is easy to show that this definition is equivalent to the one indicated in introducing the third formulation of the idea of a polygon and it is valuable in that it admits of immediate generalization to the idea of a poly hedron in space.