The one third into the scene rule

I am regularly surprised when I find people saying that you should focus one third of the way in the scene between the nearest and furthest things you want in focus. That rule presumes that the far DOF is normally about twice the near DOF. Some simple calculations show that this is true just exactly when the plane of exact focus is at distance

hyperfocal_distance/3 + focal length

Since the far DOF approaches infinity at the subject distance approaches the hyperfocal distance, it is clear that the one third rule can't be generally true. So why do so many people swear by it?

I think I've found a couple of reasons.

If your criterion to is to balance cocs in the film plane for points coming to focus on opposite sides of that plane, the usual analysis tells you that the correct place on the rail is the harmonic mean of the bellows extensions for the near and far points. (The harmonic mean is obtained by averaging the reciprocals of the two other distances and taking the reciprocal of that.) If the two distances are close together, which is always the case for bellows extensions outside the close-up range, the harmonic mean is very close to the actual mean, and that justifies the usual advice to set the position on the rail halfway between the near and far points.

What about subject distances? It turns out there is a basic symmetry, and the rule behind the lens translates into the same rule for subject distances: you should focus at the harmonic mean of the near and far subject distances (at least if the subject distance is less than the hyper-focal distance, in which case the far point is at infinity). But how about the ratio of far DOF to near DOF in that case. It turns out that this ratio is the same as the ratio of the far distance to the near distance. If that ratio is close to 1/3, then the 'one third into the subject' rule will be pretty close. A typical example might be a stone wall at about 5 meters and a wall of trees at 15 to 20 meters. On the other hand it is clear there are many circumstances where the rule won't work very well. It won't work if the far point is very far away compared to the near point, and it also won't work in the close-up range, where the DOF (usually) is roughly equal on both sides of the exact focus.

However, there is one thing that will save you if you use the one third rule outside the close-up range, at least if the far point isn't effectively at infinity, in which case, the rule doesn't make any sense. For distant points, large changes in subject distance translate into small changes in position on the rail. That means the difference on the rail between the proper position and the incorrect position won't be very large. Stopping down may easily compensate for any error.

It seems to me much simpler to use the mid-point along the rail instead of trying to estimate subject distances into the scene. But when will that be accurate? The same rule applies. The ratio of the bellows extensions for near and far points gives you the ratio of the distances of those points. So under what circumstances is that ratio close to one? Calculations show that this will be true outside the close-up range. Within the close-up range, further calculation shows that the ratio starts to depart significantly from one as you move to short focal length lenses with very small apertures. But, for close-up subjects, DOF at constant magnification is roughly independent of focal length, so there is no need to use short focal length lenses unless you want the resulting perspective. In such a circumstance, you should use the harmonic mean to find the position to focus at, whichever side of the lens you concentrate on. A further consequence of these calculations is that, for close-ups, the equality rule for DOF on both sides of the focus distance tends to break down when you use short focal length lenses with small apertures.

Sorry, for some errors in my article.

"If that ratio is close to 1/3, then the 'one third into the subject' rule will be pretty close. A typical example might be a stone wall at about 5 meters and a wall of trees at 15 to 20 meters."

should read

" If that ratio is close to 2:1, then the 'one third into the subject' will be pretty close. A typical example might be a stone wall at about 5 meters and a wall of trees at 10 to 15 meters."

This is a rule-of-thumb. It's off for close-ups and it is way-off for landscapes, which approach 1/inf. But it is a decent rule-of-thumb.

Focus_Distance = (2 x Near_Limit x Far_limit) / (Near_Limit + Far_Limit)

Ralph,

Thanks for clarifying the issue. My aim, which I didn't make clear, was to explain why the rule of thumb often works. Note in particular that even for landscapes, it won't be too far off, at least if the far point is not essentially at infinity, for the reason I gave. If the far point is at infinity, 'one-third into the scene' doesn't make sense, unless you focus at infinity. If you believe Merklinger's argument for focusing at infinity, which is not based on the usual coc considerations and is controversial, you could say the one third rule even works in that case.

Thank you Michael for another way to find the harmonic mean. The same formula works for the position on the rail:

v = 2 x v_near x v_far/(v_near + v_far)

where v is the bellows extension for the focus position and v_near and v_far are the bellows extensions for the near and far points. Using this formula, it is easy to see that

v - v_far = v_far x (v_near- v_far)/(v_near + v_far)

and

v_near - v = v_near x (v_near - v_far)/(v_near + v_far).

It then follows that the ratio

(v_near - v)/(v - v_far) = v_near/v_far

which explains why if the two distances v_near and v_far are close, as is typically true for distant points, then the ratio is close to one and the focus position v is close to the midpoint.

For the subject distances, the same algebra works, but the ratio is

far_limit/near_limit.

This ratio is not usually close to one for distant points. It is close to one in the close-up range for normal and long focal lengths, but may not be for short focal lengths. The failure for short focal lengths was new to me since the usual rule of thumb is that DOF is roughly equal on both sides of the point of focus in the close-up range.

For those who do not work with rangefinders and mathematical formulae, there's got to be a starting point that can be easily visualized/imagined. The reasonable options are 1/2 or 1/3 of the distance, and of the two 1/3 is a better approximation for first order iteration.