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Consider for example the starship enterprise (registry NCC1701) from the star trek tv series. If someone takes a picture of the ship from the tv show and sticks it on a poster, it is clearly a derivative work. But what if they only show part of the ship? What if they only show a small enough part that it is not recognizably from the enterprise? Or what if they just produce a gray (the starship enterprise is gray) image, with "NCC1701" written on it?

Is that a derivative work? Would it matter if it used a different font, or a different color background? Would it make a difference if it was less iconic, e.g. a reproduction of a different ship that only appeared in one episode?

My question is not specific to start trek, but regarding any works which have distinctive subcomponents (e.g. the monolith from 2001, the rebel alliance symbol of from star wars, the one ring from the lord of the rings etc.) - how much of an original work needs to be reproduced for it to infringe the original copyright?

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Any amount that exceeds fair use/dealing in the relevant jurisdiction. This is a combination of quantity and intent.

For the examples you give they are all probably not fair use. As an added complication, the USS Enterprise is almost certainly trade mark which is a different type of IP with different rules.

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