Focusing a projected image

[Loren Booda]

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?

[russ_watters]

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.