What are character analyses of Batman and the Joker in the movie The Dark Knight?

Under the direction of Christopher Nolan, the screenplay he and his brother Jonathan Nolan wrote, The Dark Knight, transcends the usual tropes of the superhero action film to deeply explore themes of law and order, justice, morality, and the forces of chaos.
The Dark Knight's Batman (portrayed by Christian Bale) approaches but does not quite become an anti-hero.  His outlook is darker, perhaps more ruthless, than other incarnations of Batman, but ultimately, he does not completely relinquish his morality in battling the Joker. This Batman occupies a no man's land outside law enforcement and the civil justice system; he uses torture on the mobster Maroni to elicit information he needs, but he is not a murderer and chooses not to kill the Joker when he has the chance.  Batman carries the deaths of Dent and Rachel as the consequences of his own failings.  He faces questions of guilt and responsibility when he chooses to save Gordon's son though it means that he can't save Dent.  It should be noted that Batman offers to sacrifice his hidden identity and his heroic reputation by casting himself as an outlaw vigilante so that the city of Gotham can continue to believe in the justice that the law provides.
The Joker (played by Heath Ledger to great acclaim) is emblematic of the forces of chaos.  He is not driven by greed, a desire for power, nor any form of ideology, and thus he cannot be thwarted using conventionally moral or orthodox methods.  In a confrontation with Batman, he tells him, "The only sensible way to live in this world is without rules."  The Joker is a freelance agent of destruction who has long since abandoned his humanity.  In Batman, he sees a doppelganger, and he tells him, "This is what happens when an unstoppable force meets an immovable object" and "I think you and I are destined to do this forever."  Indeed, because Batman will not cross the line and kill the Joker, and nothing short of death will stop the Joker, their animus has no foreseeable ending.
https://en.wikiquote.org/wiki/The_Dark_Knight_(film)

https://www.newyorker.com/magazine/2008/07/21/past-shock

https://www.nytimes.com/2008/07/18/movies/18knig.html