  This work offers a didactical introduction to the calculations and
geometrical properties of a static, spherically symmetric spacetime foliated by
hyperboloidal time surfaces. We discuss the various degrees of freedom
involved, namely the height function, responsible for introducing the
hyperboloidal time coordinate, and a radial compactification function. A
central outcome is the expression of the Trautman-Bondi mass in terms of the
hyperboloidal metric functions. Moreover, we apply this formalism to a class of
wave equations commonly used in black-hole perturbation theory. Additionally,
we provide a comprehensive derivation of the hyperboloidal minimal gauge,
introducing two alternative approaches within this conceptual framework: the
in-out and out-in strategies. Specifically, we demonstrate that the height
function in the in-out strategy follows from the well-known tortoise coordinate
by changing the sign of the terms that become singular at future null infinity.
Similarly, for the out-in strategy, a sign change also occurs in the tortoise
coordinate's regular terms. We apply the methodology to the following
spacetimes: Singularity-approaching slices in Schwarzschild, higher-dimensional
black holes, black hole with matter halo, and Reissner- Nordstr\"om-de Sitter.
From this heuristic study, we conjecture that the out-in strategy is best
adapted for black hole geometries that account for environmental or effective
quantum effects.
