by ʇolɐǝz ǝɥʇ qoq

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I have been studying lenses for two days now and the aperture is very confusing. Based on the question here it appears that the aperture should not crop the image at all but when I look through a lens directly it does crop the image. When I look through the viewfinder it does not crop the image but only makes it darker as expected. Why is the image cropped when looking through a lens directly?

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When you're looking at the lens, this is what you see - the lens. Your eye will focus on the lens and see its parts.

When you're looking through a viewfinder, you're not actually looking at the lens; you're looking at the image that is projected by the lens on a ground glass. The glass is matte, so you can't actually see things behind it, only light rays that have been focused on that piece of glass. On the path of those rays, the aperture mechanism is so out of focus that it will only affect brightness and depth of field of the image.

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