  Although conjunctions and oppositions frequently occur in planetary systems,
eclipse-related phenomena are usually described from an Earth-centric
perspective. Space missions to different parts of the Solar system, as well as
the mounting number of known exo-planets in habitable zones and the possibility
of sending featherweight robot spacecraft to them, prompt broader
considerations. Here, we derive the geometry of eclipses, transits and
occultations from a primarily exo-Earth viewpoint, and apply the formulation to
the Solar system and three types of three-body extrasolar planetary systems:
with 1 star and 2 planets (Case I), with 2 stars and 1 planet (Case II), and
with 1 planet, 1 star and 1 moon (Case III). We derive the general conditions
for total, partial and annular eclipses to occur at exo-syzygy, and implement
them in each case in concert with stability criteria. We then apply the
formalism to the TRAPPIST-1, Kepler-444 and Kepler-77 systems -- the first of
which contains multiple potentially habitable planets -- and provide reference
tables of both Solar system and TRAPPIST-1 syzygy properties. We conclude by
detailing a basic algebraic algorithm which can be used to quickly characterize
eclipse properties in any three-body system.
