  The main purpose of the note is to explore the invariant properties of
sphericalization and flattening and their applications in quasi-metric spaces.
We show that sphericalization and flattening procedures on a quasimetric spaces
preserving properties such as Ahlfors regular and doubling property. By using
these properties, we generalize a recent result in \cite{WZ}. We also show that
the Loewner condition can be preserved under quasim\"obius mapping between two
$Q$-Ahlfors regular spaces. Finally, we prove that the $Q$-regularity of
$Q$-dimensional Hausdorff measure of Bourdon metric are coincided with
Hausdorff measure of Hamenst\"adt metric defined on the boundary at infinity of
a Gromov hyperbolic space.
