A Rectilinearity Measure

Motivated by the properties of the function $\max\limits_{\alpha\in [0, 2\pi]} \frac{{\cal P}_2(P)}{{\cal P}_1(P, \alpha)}$ we give the following definition for the new rectilinearity measurement of polygons.

Definition 1. For an arbitrary polygon $P$ we define its rectilinearity ${\cal R}(P)$ as

$${\cal R}(P) \; = \; \frac{4}{4-\pi}\cdot \left( \max\limit... ...al P}_2(P)}{{\cal P}_1(P, \alpha)} - \frac{\pi}{4} \right)\; .$$

The following theorem summarises the properties of the polygon rectilinearity measure proposed here. Theorem 3. For any polygon $P$, we have:

i)

${\cal R}(P)$ is well defined and ${\cal R}(P) \in (0, 1]$;

ii)

${\cal R}(P) = 1$    if and only if     $P$ is rectilinear;

iii)

$\inf\limits_{P\in\Pi} ({\cal R}(P)) = 0$;

iv)

${\cal R}(P)$ is invariant under similarity transformations.

where $\Pi$ denotes the set of all polygons.