Character arches are among the least stable architectural structures.
An arch is a physical structure in the shape of half of a McDonald's sign. An arc is a path or a curve in the same shape.
If a fictional character follows some kind of, you know, path, throughout the course of a narrative, it would be a "character arc." Now people who keep typing "character arch" might get defensive and claim that "arch" works too because it's the same shape and, um, you know, living language, blah blah. Well, make them pronounce it. I bet you they're going to say "ark" with the hard K sound, which is never how you pronounce "arch" as a separate word, and at that point, you go, "Come on, buddy, we all know you meant 'arc,' and slipped up. Just fess up and laugh it off and we'll have a drink."
My Writing Mentor
The disturbing thing is that most people who are talking about character arches in the first place are people who are, or want to be writers. The quote above is from a writing mentor and the one below is from an aspiring screenwriter. Eeesh. Not a good start, guy.
Done Deal Message Board
"Come on, honey. I heard all main characters have to go through one of these."