Your eyes focus themselves to get a clear image of the object they are focussing on. Any object we can only see because light bounces off of the object and into our eyes. An image is just a bunch of light rays in a certain order bounced off of an object and into our eyes (or the light is sent out of the screen directly). Its that object that your eyes then focus on, not what is being projected on it. So that can be a picture of something far away, but because the camera that took that picture already did the focussing for you, now you can see the same as that camera did when it took the picture, but projected on an object that is closer than the image subject was to the camera.
It’s because the screen doesn’t preserve the angle of the light coming from the original image. Focusing is required because the pupil of your eye is a few mm across, so that light entering on one side of the pupil enters at a slightly different angle than from the other side. The angle difference will be tiny for distant objects but much greater for nearby objects.
Screens generate new light that fans out in all directions from up close so you have to focus close. Mirrors preserve the relative angles as they reflect light so they don’t reset the focus distance. Projection screens reflect light from a projector but they’re diffuse reflectors that scatter light in all directions so, again, you need to focus on the screen.
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