This is the best I could come up with. Hope it helps at least a bit.
I have a printshop that wants me to send them my artwork in a .curve What in the world is that?
Nobody knows and because of that its stands to reason that this is a proprietary format of their routing/ laser cutting software. Essentially it means nothing. Could be anything from a simple ASCII list format (formatted text file/ CSV) to a super-secret binary format. Anyway, if your printshop expects you to deliver your stuff in such an exotic format and can't deal with a simple AI file (AI 8 at least), then it's time to find another service partner. Seriously.
While looking for explanations about the .curve format myself, google lead me here - hi.
While I'm uncertain about the standard format for a printshop, I just wanted to add that the .curve format is used by the data visualization tool 'VisIt' (https://wci.llnl.gov/codes/visit/). I found a brief description of the format on pg.12 of https://wci.llnl.gov/codes/visit/1.5.4/GettingDataIntoVisIt1.5.4.pdf.
Hope that helps,
Europe, Middle East and Africa