A cut-in is a technique used in film editing, involving two shots. The first shot shows a distant view of a scene, and the second shot shows a closer view of the same scene. There is an instant change between the distant view and the close view in a cut-in, otherwise known as a cut.
A cut-in is distinguished from a zoom-in by, naturally enough, the cutting. A zoom-in involves only one shot, and gradually enlarges some part of the scene. Furthermore, a cut-in is accomplished through editing, whereas a zoom-in is a purely cinematographic effect.
A diagram might be helpful (I apologize in advance for the crude ASCII art).
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Shot 1: This shot shows a long shot of a person, which makes him seem quite small.
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Shot 2: This shot shows a close-up of the same person, which shows his neck and head. A cut-in will instantly transition between shot 1 and shot 2.
Cut-ins are very common in films, used most often to transition from a distant shot to a close-up of a character. Cut-ins are also used when zoom-ins would be inappropriate. Cut-ins can telescope space much faster than zoom-ins, and might be a better choice when there is a great distance between the distant framing and the closer framing. Cut-ins also allow filmmakers to cheat a little bit with the camera angles between the two shots, allowing them to more effectively frame the action.
There is a nice example of a cut-in in Fight Club, Everything2's favorite film. Warning: Very mild spoiler ahead.
Around 23 minutes into the film, the Narrator (Edward Norton) has a conversation with an airline employee. After the conversation, he spots someone who looks like Tyler Durden (Brad Pitt) jumping into a convertible. The shot frames the Narrator on the right, foreground side of the screen, and Tyler and the car are in the distance on the left. The next shot cuts-in to show a much closer view of Tyler and the car, and they are both centered in the shot. Note that the angle of the car in the first shot is slightly different than the angle in the second shot. By using a cut-in instead of a zoom-in, the director was able to better frame the second shot.
Notes
David Bordwell and Kristin Thompson. Film Art: An Introduction, Sixth Edition. (New York: McGraw-Hill, 2001).
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