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1. To cast a light or illuminated image of someone or something, as from a film or slide, onto some surface. In this usage, a noun or pronoun is used between "project" and "on." We darkened the room and projected the film on the wall. Would you mind erasing the writing on the white board so I can project the notes for today on it? He projected a beam of light on the car so I could see what I was doing.
2. To attribute some subconscious element, such as grief, insecurity, pain, etc., on another person or thing as a way of externalizing and avoiding it. In this usage, a noun or pronoun can be used between "project" and "on" to specify what the person is projecting. It sounds like she might be projecting on your relationship a bit—I know that she's got a lot of baggage from her childhood that she's still trying to cope with. I wish you would stop projecting your insecurities on me—it's ruining our friendship!
project something on (to) someone or somethingand project something upon someone or something
to show a picture, such as from a film, transparency, etc., onto something, such as a screen, wall, etc., or even onto a person. Henry projected the pictures onto the screen as he discussed each one. The teacher had to project the slides upon the wall.