Riker is aboard a research station Botanica Four checking on the progress of Dr. Nel Apgar, a Tanugan scientist who is working on the invention of Krieger waves, a new source of energy. Just as Riker beams back to the ship, however, the station explodes, killing Apgar. A Tanugan detective, Krag, arrives on the ship to arrest Riker for murder.
Picard is caught between his responsibility to impartially respect the laws of other cultures and his duty to protect his crew. He convinces the detective to conduct his investigation aboard the ship, using the holodeck to recreate the station exactly. Krag, Picard, Riker, and Troi watch a series of recreations of Riker's time on the station, each portraying his stay from the perspective of a different witness.
Riker's version has him and Geordi beam aboard to meet a stand-offish Apgar and his accommodating wife Manua. Apgar is upset that the Enterprise has arrived early to check on his work, but Riker assures him that it's just a friendly visit. The Enterprise is about to leave to conduct other business, so Manua offers to let Riker stay on the station (even though he and Geordi have reservations planetside). When Riker and Manua get a moment alone, there's some sexual chemistry that Manua tries to exploit and Riker politely tries to resist, but Apgar interrupts them and accuses Manua of cheating on him. He tries to punch Riker, but Riker easily dodges the attack and tries to calm Apgar down. The humiliated Apgar vows revenge, but Riker leaves the station the next day without incident.
Manua's version, however, differs significantly. Riker behaves lasciviously, leering at Manua throughout his introduction and insisting that Manua set him up for the night. When they're alone, Riker tries to force himself on Manua only to be interrupted by Apgar, whom Riker pummels mercilessly. Apgar's research assistant provides the story that Apgar himself told her, in which Apgar beats up Riker, who is the one to vow revenge. Through it all, Troi maintains that each person is telling the truth as he or she remembers it.
Meanwhile, Data, Geordi and Wesley investigate a series of strange radiation spikes throughout the ship. They eventually conclude what the source is, which gives Picard the answers he needs to announce that he's discovered who killed Apgar. He assembles all of the parties involved and explains what happened.
Apgar had secretly completed his work on Krieger waves. The radiation spikes around the ship were actually Krieger waves being reflected off the holodeck's version of Apgar's completed machine. Apgar was planning to sell it for use as a weapon rather than give it to the Federation, which is why he was nervous about Riker's early visit. After seeing Riker with his wife, Apgar decided to use his Krieger waves to kill Riker and make it look like a transporter accident. But the Krieger wave that Apgar fired at Riker bounced off the transporter beam and reflected back into the machine, causing it to explode.
Krag is satisfied with this explanation and withdraws his request for Riker's arrest. Picard welcome Riker back to the bridge and allows him to set the next course.
Tropes featured in "A Matter of Perspective":
- Attempted Rape: According to Mrs. Apgar, Riker tried to force himself on her until her husband intervened. Riker denies it and claims that he instead declined her advances, while Dr. Apgar's second-hand account claims that it was mutually consensual.
- Badass Boast: Apgar's line in each of the three scenarios turns into a boast in his own account, when he says it after beating Riker up.Dr. Apgar: I'm not the fool you take me for.
- Chekhov's Gun: The recurring radiation bursts turn out to be Krieger waves, which Apgar claims he hasn't yet created.
- Curb-Stomp Battle: Apgar's version of events has him effortlessly beat up Riker. His wife's version, on the other hand, has Riker beat up Apgar, which is much more plausible. Riker's version has no battle, with him trying (and failing) to calm Apgar down.
- Damned by Faint Praise: Data tries to compliment Picard's painting as a synthesis of styles, but he can't help himself but describe how discordantly it all comes together. Picard's ego is obviously bruised by the less-than-flattering critique.
- Facepalm: Riker facepalms during a holorecreation of the murder.
- Hoist by His Own Petard: Picard and La Forge are able to demonstrate that Dr. Apgar attempted to fire a Krieger wave beam at Riker while he was transporting, to frame his murder as a Teleporter Accident. However, the beam bounced off the transporter beam that Riker used and hit the reactor, destroying the station and killing Apgar.
- Holodeck Malfunction: Yes and no. The holodeck functions exactly as intended, recreating Dr. Apgar's orbiting lab down to the last detail. However, no one thought to turn off his generator down on the planet, and the holo-lab, which Apgar had claimed was non-functional, succeeds in converting its otherwise harmless emissions into Krieger waves, which threaten the ship. But it's these same waves that exonerate Riker, by showing that Apgar was lying about his progress.
- Homage: The whole episode is an homage to Rashomon. The investigation is conducted "Rashomon"-Style, and the circumstances of the crime also match the formula of the original film: a man is murdered, and the stories come from a sexually desirable man, the murdered man's wife who may or may not have had an affair with him, and the murdered man himself.
- Kangaroo Court: Krag blithely states that on his world, suspects are guilty until proven innocent. Picard does everything he can to prevent Riker from getting sucked into a kangaroo court on the planet.
- Karma Houdini: Manua doesn't even attempt to apologise for falsely accusing Riker of trying to rape her let alone give an explanation for saying something so far from what actually happened. That said, it's entirely possible that whatever happened between the two of them met somewhere in the 'middle' of both their testimonies, and Riker's infallible charm collided with her own desires and created a situation where both were neither entirely blameless or at ultimate fault.
- Magical Computer: Realistically, there is no way in hell the holodeck simulations could be that precise when based just on peoples recollections.
- Murder the Hypotenuse: Manua and Apgar's versions of the events blame Riker for murdering Apgar to be with Manua.
- Pink Means Feminine: Two out of three recreations show Mrs. Apgar wearing a pink underdress while trying to seduce Riker. The exception is her own, in which she's a blameless and victimized woman in white.
- Posthumous Character: Dr. Apgar is never seen while he's alive, only as a holographic recreation.
- Punch-Clock Villain: Krag isn't a bad guy, just a detective doing his job. He even apologizes to Riker after viewing the exonerating evidence.
- "Rashomon"-Style: Achieved via the holodeck, so that Captain Picard and Inspector Krag can view the events on the destroyed station step-by-step. Witness depositions from Riker, Mrs. Apgar, and Dr. Apgar's assistant Tayna (representing Apgar's side of the story as he told it to her) are each programmed into a holo-recreation of the station and viewed in turn. Each story contains both commonalities and differences. Troi confirms that none of the witnesses are knowingly lying, and yet it takes additional evidence for Picard to piece together what really happened. Even then, some events are left ambiguous.Riker: We can't both be telling the truth.
Troi: It is the truth... as you each remember it.
Riker: Yes, but her version puts a noose around my neck.
- Reasonable Authority Figure: This episode shows the downside of it for the protagonists. Picard is absolutely certain of Riker's innocence, but the evidence just isn't in his favor, and he's prepared to grant extradition. It takes Data, Geordi, and Wesley uncovering new information to save Riker at the last minute.
- Riddle for the Ages: We never know exactly what happened between Riker and Apgar's wife, and if it was an attempt at an affair, who started it nor how willing each participant was in its events.
- Say My Name: A purposefully over-the-top example. In Tayna's retelling, Riker trumpets, "You're a dead man, Apgar! A dead man!
- Shifting the Burden of Proof: This appears to be Tanuga 4's Planetary Hat; in their justice system, criminal suspects are "guilty until proven innocent." Taken even further when they allow a "witness" to testify to events she did not witness (Picard even calls this out as hearsay, but their rationale is that she's describing events the victim told her about, since he's not around to testify for himself), and it's up to Riker to prove her story false—which, since he wasn't there at the time, he can't do.
- Terrible Artist: Apparently, Picard, since the best Data can do is damn his painting with faint praise, much to his annoyance.
- Unreliable Narrator: Each person is telling the truth based on how they remember it. We'll never get a completely objective rendition of the events, though considering that Riker is such a stand-up guy, we can assume that his version is the closest to the real events.