A new measure for the rectilinearity of polygons

A polygon P is said to be rectilinear if all interior angles of P belong to the set {pi/2, 3pi/2}. In this paper we establish the mapping R(P) = (pi)/(pi - 2 . root2) (.) ((max)(alphais an element of[0,2pi]) (P)((P,alpha)(1))/(root2 .P2(P) -) (2 . root2)/(pi) where P is an arbitrary polygon, P-2(P) denotes the Euclidean perimeter of P, while P-1(P, alpha) is the perimeter in the sense of l(1) metrics of the polygon obtained by the rotation of P by angle a with the origin as the center of the applied rotation. It turns out that R(P) can be used as an estimate for the rectilinearity of P. Precisely, R(P) has the following desirable properties: any polygon P has the estimated rectilinearity R(P) which is a number from [0, 1]; R(P)=1 if and only if P is a rectilinear polygon; inf R(P) (Pis an element ofII) = 0, where II denotes the set of all polygons; a polygon's rectilinearity measure is invariant under similarity transformations. The proposed rectilinearity measure can be an alternative for the recently described measure R-1(P).(1) Those rectilinearity measures are essentially different since there is no monotonic function f, such that f (R-1(P)) = R(P), that holds for all P is an element of Pi. A simple procedure for computing R(P) for a given polygon P is described as well.