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What does focusing mean? What does it have to do with the formula

1 / u + 1 / v = 1 / f

u is the object distance,

v is the image distance,

f is the focal length.

Does focusing mean that both sides of the formula are equal?

enbin
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  • When I look through the viewfinder of my camera I do not do mathematical equations, I turn the focusing ring until what I have chosen or what I feel is the subject is in focus. For me it has nothing to do with that formula. Perhaps there’s a mathematical equation to explain what I feel but I do not care to know what that is. – Alaska Man Mar 30 '20 at 08:31

2 Answers2

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Both sides of the formula are always equal (for theoretical, thin lenses; for real lenses, the formula is only an approximation). What it says is where you should place the image plane (the film or the sensor) to have a sharp image.

In practice

  • focal length f is fixed (because you have a prime lens, or because you have choosen the desired focal length of your zoom lens)
  • the object distance u is also fixed (almost: it changes a bit when moving the lens for changing focus)
  • then it follows there is one specific image distance v

Focusing means changing the distance between lens and film/sensor so that it matches v.

Again, only approximately because real lenses are not theoretical thin lenses; also focusing generally moves the lens so that object distance u changes a bit.

Roel Schroeven
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  • If u + v is fixed and f is also fixed, then when v is what, the depth of field is infinite. – enbin Mar 30 '20 at 11:45
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    If u + v is fixed, you can still move the lens back and forth: that increases u and decreases v by the same amount, or vice versa. What do you mean by 'the depth of field is infinite'? It absolutely isn't. – Roel Schroeven Mar 30 '20 at 12:41
  • When v is equal to what, the depth of field is ♾. – enbin Mar 30 '20 at 12:47
  • @enbinzheng, to have the DOF extend to infinity you have to change u (the focus distance) so that it is ≥ the hyperfocal distance for the lens and aperture setting. – Steven Kersting Mar 30 '20 at 13:08
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    This formula calculates what u or v need to be for a perfectly sharp image. DOF calculations show how far u and v can be off their ideal position while still having an acceptably sharp image. – Orbit Mar 30 '20 at 13:12
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    For focus at infinity, 1/u = 0, so 1/v = 1/f and v = f. In other words, the sensor/film should be exactly at the focal point of the lens. Note that focus at infinity is not the same as infinite depth of field. Is focus at infinity what you're after? Or rather focusing at the hyperfocal distance, which comes closest to infinite depth of field? – Roel Schroeven Mar 30 '20 at 14:04
  • @RoelSchroeven When thinking about infinite depth of field, the relationship of u, v, f. – enbin Mar 30 '20 at 16:02
  • @RoelSchroeven What is the relationship at the hyperfocal distance, u, v, f? – enbin Mar 30 '20 at 16:22
  • For u=v=1, both sides are not equal, because 1 does not equal two. Just because you were able to guess the context, does not mean that it does not need to be established. – Carsten S Mar 30 '20 at 16:51
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    @enbinzheng: other factors come into play (circle of confusion) when talking about depth of field and hyperfocal, thigs I'm not very knowledgeable about and certainly don't fit in a comment. Maybe best put it into a new question. – Roel Schroeven Mar 30 '20 at 18:24
  • @CarstenS: u=v=1 => 1/1 + 1/1 = 1/f => 2 = 1/f => f = 1/2. The equation is still satisfied. But I am confused about what your comment is a reply to, or what point you are trying to make? – Roel Schroeven Mar 30 '20 at 18:28
  • @CarstenS u is the object distance, v is the image distance, and f is the focal length. – enbin Mar 30 '20 at 22:06
  • @RoelSchroeven https://photo.stackexchange.com/q/26841/91249Can you answer this question? – enbin Mar 31 '20 at 00:02
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    @enbinzheng: That's an old question on a different subject, and already has an accepted answer. I'm perfectly happy to answer a question now and then if I feel I have anything valuable to add, but I don't have the time to go on wild tangents, sorry. – Roel Schroeven Mar 31 '20 at 08:12
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Yes, it means changing either f or v (u+v being constant) so that a point on the subject is just a point on the sensor/film (ie, v is the distance from the lens to the sensor).

In other words, the formula is always true, but if v isn't the distance between the lens and the sensor the subject is out of focus. Focusing is moving the lens (or changing its focal length) to that v is the distance between lens and sensor.

In practice, real camera lenses are not the theoretical "thin lenses" on which your formula is valid...

xenoid
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  • Does focusing mean that both sides of the formula are equal? – enbin Mar 30 '20 at 11:09
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    The formula is always true. But if v isn't the distance between the lens and the sensor the subject is out of focus. Focusing is moving the lens (or changing its focal length) to that `v| is the distance between lens and sensor. – xenoid Mar 30 '20 at 11:43
  • If u + v is fixed and f is also fixed, then when v is what, the depth of field is infinite. – enbin Mar 30 '20 at 11:59
  • If u+v is fixed, you can still move the lens longitudinally: (u-Δ)+(v+Δ)=u+v – xenoid Mar 30 '20 at 12:05
  • So what v will have infinite depth of field? – enbin Mar 30 '20 at 12:39
  • https://photo.stackexchange.com/q/116336/91249What do you think of this problem? – enbin Apr 02 '20 at 00:15
  • Already said in the comments there. – xenoid Apr 02 '20 at 00:39
  • I have modified the problem a lot and it has been perfected. Please look again. – enbin Apr 02 '20 at 01:00
  • @enbinzheng You seem to be conflating focus at a point of infinity with infinite depth of field. The two, though related, are not remotely the same. – Michael C Apr 05 '20 at 09:06
  • @MichaelC https://youtu.be/446Bhuyx1Os Is my explanation correct? – enbin Apr 05 '20 at 09:09
  • Sorry, I'm not clicking on a link to a YouTube video. – Michael C Apr 05 '20 at 09:17
  • Still based on false assumptions... 1) when you look at the moon on the horizon the moon is sharp so we focus on it on not on trees nearby, and 2) for short focal lengths focus at at 380000km, 3km or even 30m is nearly identical (you demo uses much closer objects...) and 3) human vision isn't only a matter of optics, sharpness also comes from small eye movements. Now feel free to write a book about this or even start a sect, but stop pestering us. – xenoid Apr 05 '20 at 10:26