Consider the union of region A,B,C where
$A$ is a set of $(\theta_1,\theta_2,\theta_3)$ satisfying
(1) $ |\theta_1-\theta_2|\leq\alpha$ and
(2) $\{|\theta_1-\beta|>\alpha\text{ or } |\theta_3-\beta|>\alpha\}$ and
(3) $ \{|\theta_2-\beta^*|>\alpha\text{ or } |\theta_3-\beta^*|>\alpha\}\Big\}$
$B$ is a set of $(\theta_1,\theta_2,\theta_3)$ satisfying
(1) $ |\theta_2-\theta_3|\leq\alpha$ and
(2) $\{|\theta_2-\beta|>\alpha\text{ or } |\theta_1-\beta|>\alpha\}$ and
(3) $ \{|\theta_3-\beta^*|>\alpha\text{ or } |\theta_1-\beta^*|>\alpha\}\Big\}$
$C$ is a set of $(\theta_1,\theta_2,\theta_3)$ satisfying
(1) $ |\theta_1-\theta_3|\leq\alpha$ and
(2) $\{|\theta_1-\beta|>\alpha\text{ or } |\theta_2-\beta|>\alpha\}$ and
(3) $ \{|\theta_3-\beta^*|>\alpha\text{ or } |\theta_2-\beta^*|>\alpha\}\Big\}$
where $\beta:=-\pi/2 + 2 * \alpha * m$, $\beta^*:=-\pi/2 + 2 * \alpha * m$, $\theta_i\in [-\pi/2,\pi/2]$, $\alpha<\pi/2$.
My question is, does the union of set $A,B,C$ equal to or cover the following region?
$$\{(\theta_1,\theta_2,\theta_3)\big| |\theta_1-\theta_2|\leq \alpha \text{ or }|\theta_1-\theta_3|\leq \alpha \text{ or }|\theta_3-\theta_2|\leq \alpha\}$$