How can an intervening lens correct an out-of-focus projected image?

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Loren Booda
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Why can an out-of-focus picture on a screen from a projector be corrected by an
intervening lens, while a lens cannot correct the out-of-focus image once
impinged on the screen?
 
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When the light hits the screen, the direction the rays travel is not preserved due to the specular reflection. Replace the screen with a mirror and you can re-focus the light.