(covers information from several alternate timelines)
Rigel was the primary of the Rigel system. This system was located in the Kandari sector. This was a region of space in the Alpha Quadrant. (Star Trek VI: The Undiscovered Country, production art; TNG: "Conspiracy" display graphic)
The location of Rigel in the Milky Way Galaxy was first depicted on a Federation star chart in 2293. (Star Trek VI: The Undiscovered Country) In 2367, the star appeared on a chart in Captain Benjamin Maxwell's ready room on the USS Phoenix. (TNG: "The Wounded" display graphic) One star chart depicted Rigel near Vega. This chart was often displayed in astrometrics on USS Voyager. (VOY: "Child's Play")
|Rigel II • Rigel III • Rigel IV • Rigel V • Rigel VII • Rigel X • Rigel XII|
In "The Explored Galaxy" star chart, Rigel was located between the Tholian Assembly and Romulus. Both of which were identified in DS9: "Call to Arms" as being located in the Alpha Quadrant. This chart was also seen in several Star Trek: The Next Generation and Star Trek: Deep Space Nine episodes.
In the real world, the star Rigel was a star located 773 light years from the Earth in the constellation Orion While this constellation was seen on screen in TNG: "Family" and on a constellation star chart in the TOS: "The Corbomite Maneuver". The Rigel in Star Trek was apparently not a star named by Humans, since Vulcans and Klingons used this name in their native languages and the location of Rigel was not known to Humans before it was learned from the Vulcan star charts in "Broken Bow".
According to The Worlds of the Federation (pg. 36), the Rigel system and the Orion system were one and the same. According to the reference book Star Trek: Star Charts (pg. 36), there were in fact two stars called Rigel. These were Beta Rigel and Beta Orionis. The Rigel system was in Beta Rigel. According to Star Trek Maps (Chart D), the Rigel system was in Beta Orionis.