THE THEORY OF PLANE POLYGONS The Interior, Exterior and Peripheral Angles of a Plane Polygon.—We now proceed to the special theory of plane poly gons. As remarked above, this theory concerns itself largely with unoriented and oriented polygons. In the case of a simple poly gon the meaning of "an interior angle of a polygon" is immediate in virtue of the theorem concerning the separation of the plane by the polygon. This meaning leads to an interesting generali zation in the case of an oriented polygon. In proceeding to this generalization and to related ideas it should be stated that all of the terms used are not defined with the mathematical completeness that would be possible with a greater allowance of space, but it has been aimed at least to indicate clearly the way to that com pleteness. Now it can be proved that if P is a simple oriented polygon and A any vertex of P then for the positive (counter clockwise) rotation of any side AB of P about its first end A, in accordance with the sense of P, which transforms the side AB into the other side of P having A as an end and which has a magnitude not exceeding 27n radians, then the points on the inter mediate positions of AB which are within a certain distance of A are all in the interior of the polygon determined by P if the sense of P is the same as that of the rotation and all in the exterior in the contrary case. (Note that senses of oriented polygons are compared only in the case of simple oriented polygons.) Accord ingly the interior angle at any vertex Ai of any oriented polygon P is defined as the positive angle (rotation) which has Ai as its vertex and whose initial side contains the side of P which has A as its first end, say the side A whose terminal side contains the side with the ends Ai and A if and if i= 1, and whose magnitude does not exceed 27r. (See fig. I.) Further if is any vertex of an oriented polygon P and the first end of the oriented side A of P then there is a rotation about Ai of the half-line having Ai as its initial point and having the same direction as the oriented side A (i— if i= I), that is the half-line of the line through and Ai, which has Ai as its initial point and which does not contain into the half-line which contains the side A iA which has a magnitude greater than -7 and less than or equal to 7r and there is an other such rotation which is posi tive and has a magnitude not exceeding air. The former angle (rotation) is called the exterior angle and the latter the peripheral angle of the oriented polygon P at the vertex Ai. If ai, 7i are respectively the magnitudes
of the interior, exterior and peripheral angles of the oriented polygon P at the vertex A i then r and r or 37r according as is or is not less than 7r. If ai is less than r then 7i is and conversely. If the sum of all the Oi is set equal to 2ar and the sum of all the 7i to 2a'lr then a and a' are integers or zero as a simple consideration shows and a' —a is the number of interior angles of P whose magnitudes are greater than or equal to ir. If P' and P" are two oriented polygons which differ only in orientation and if ai', f3i', and ai", gi", are respectively the magnitudes of the interior, exterior and pe ripheral angles of P' and P" at the vertex then 27r, = o and 7i/-1- 7i" = 2 ir. Hence the value of a for P" is the negative of its value for P' while the value of a' for P" is the number of vertices of P' (or P") minus the value of a' for P'. Classification of Plane Polygons.—In the classification of oriented polygons the numbers a and a' have been used to define the so-called types of such polygons. It is obvious how they may be used in the classification of unoriented polygons. There exist oriented polygons of any number n of vertices for which the value of a is any number whose absolute value does not exceed i(n— I) except that for a triangle a cannot be zero. A more detailed scheme of classification is according to the values of a and q=a'—a. If all of the interior angles of an oriented polygon have magnitudes which do not exceed r then the polygon is called a convex oriented polygon. The unoriented polygon P is convex if and only if P with a sense assigned to it is convex. Both of these definitions are in conformity with the important notions of a convex simple polygon and a convex polygonal region. A con vex region is a region such that all of the points of any segment whose ends belong to the region belong to the region also. A simple polygon is said to be convex if it is the boundary of a convex region, which is then a convex polygonal region. As theorems we have : A line which does not contain a side of a convex simple polygon contains not more than two points of the polygon and conversely. Also, no point of a convex simple polygon is on a particular one of the two sides of the line which contains any side of the polygon and conversely.