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For the formula 1 / u + 1 / v = 1 / f, what is infinite depth of field or hyperfocal distance? In other words, when u, v, and f, we have infinite depth of field or hyperfocal distance?

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Hyperfocal distance itself doesn't really have anything to do with the lens formula you quote.

The formula for hyperfocal distance is (approximate, but good enough in practice):

H = f² / (Nc)

where:

  • H: hyperfocal distance
  • f: focal length
  • N: f-number (aperture)
  • c: circle of confusion diameter limit

(see https://en.wikipedia.org/wiki/Hyperfocal_distance)

To apply that to the lens formula: the hyperfocal distance is the object distance u. Fill that in, and the focal length, and then you can calculate image distance.

I'm not sure what the practical use for that is though.

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    And, further, in this case, c is not just a simple number - it depends on many things, including the size of sensor elements on your particular camera, the final size of the print or image you're producing, the distance from which that image will be viewed, and to some extent, the resolution of your lens. And maybe some other stuff... – twalberg Mar 31 '20 at 11:48
  • photo.stackexchange.com/q/116336/91249What do you think of this problem? – enbin Apr 2 '20 at 0:06
  • @enbinzheng: There are already lots of comments on your question there. Adding mine would not contribute anything useful. – Roel Schroeven Apr 2 '20 at 7:50
  • @RoelSchroeven Do you think I'm right? – enbin Apr 2 '20 at 12:58
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    @enbinzheng Depth of Field is an illusion. Only one distance is in best focus. All others are blurrier to one degree or another. Since hyperfocal distance is a special case of DoF, it is also an illusion. Only the hyperfocal distance itself is in focus. The rest is blurry, just not blurry enough for our eyes to notice it's blurry. – Michael C Apr 5 '20 at 8:55

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