  Let $\mathcal{C}$ be an additive category equipped with an automorphism
$\Sigma$. We show how to obtain $n$-angulations of $(\mathcal{C},\Sigma)$ using
some particular periodic injective resolutions. We give necessary and
sufficient conditions on $(\mathcal{C},\Sigma)$ admitting an $n$-angulation.
Then we apply these characterizations to explain the standard construction of
$n$-angulated categories and the $n$-angulated categories arising from some
local rings. Moreover, we obtain a class of new examples of $n$-angulated
categories from quasi-periodic selfinjective algebras.
