  In this article, we study the geometric realizations of $m$-cluster
categories of Dynkin types A, D, $\tilde{A}$ and $\tilde{D}$. We show, in those
four cases, that there is a bijection between $(m+2)$-angulations and
isoclasses of basic $m$-cluster tilting objects. Under these bijections, flips
of $(m+2)$-angulations correspond to mutations of $m$-cluster tilting objects.
Our strategy consists in showing that certain Iyama-Yoshino reductions of the
$m$-cluster categories under consideration can be described in terms of cutting
along an arc the corresponding geometric realizations. This allows to infer
results from small cases to the general ones.
