The Basic Idea

Theorem 1 gives the basic idea for the polygon rectilinearity measurement described in this paper. In the first stage, Theorem 1 together with $${\cal P}_2(P) \leq {\cal P}_1(P)$$ suggests that the ratio $\frac{{\cal P}_2(P)}{{\cal P}_1(P)}$ can be used as a rectilinearity measure for the polygon $P$.

To make it invariant under similarity transformations $\max\limits_{\alpha\in [0, 2\pi]} \frac{{\cal P}_2(P)}{{\cal P}_1(P, \alpha)}$ is used instead.

Since ${\cal P}_2(P) \leq {\cal P}_1(P, \alpha)$ it follows that ${\cal Q}(P, \alpha) = \frac{{\cal P}_2(P)}{{\cal P}_1(P, \alpha)} \leq 1$. However, the infimum for the set of values of $\frac{{\cal P}_2(P)}{{\cal P}_1(P, \alpha)}$ is not zero. In the paper we prove the existence of the following lower bound.

Theorem 2. The inequality

$$\displaystyle \max\limits_{\alpha\in[0,2\pi]} \frac{{\cal P}_2 (P)}{{\cal P}_1 (P, \alpha)} \ge \frac{\pi}{4}$$

holds for any polygon $P$.