The sharp band is a decision

The centre has an area established that a world point images as a disc of diameter

$$c = \frac{f^2}{N}\cdot\frac{|Z - Z_f|}{Z\,(Z_f - f)}$$

and that its centre is exactly where a pinhole would have put it. Depth of field is one question about that formula: where is c under some number? A frame is an interval asks the same shape of question about time, and gets the same shape of answer.

The number is not supplied by the optics. It is supplied by a reader.

The band, five times over #

A 50 mm lens at f/2.8, focused at three metres, with the criterion swept over a factor of sixteen:

• half a pixel: the band runs 2.76 to 3.28 metres — half a metre deep;
• one pixel: 2.56 to 3.63;
• two pixels: 2.23 to 4.58;
• four pixels: 1.78 to 9.66;
• eight pixels: 1.26 metres to infinity.

Same lens, same aperture, same focus setting, same discs. The band goes from half a metre to unbounded, and the only thing that moved is what a reader will accept — which makes it a member of the same family as the sixty-degree cone of vision, a rule about a reader dressed as a rule about a picture.

Why the criterion is quoted in pixels here #

The usual unit is a “circle of confusion” in millimetres of sensor — 0.03 mm for full frame, 0.02 for APS-C — and those numbers come from a chain of assumptions that is almost never stated: a print of a given size, viewed from a given distance, by an eye of a given acuity.

This collection computes viewing distances for a living. The point to stand at is the whole premise: a picture is correct from one place, and the place is arithmetic. So the criterion is quoted where a reader can check it — as a diameter in the pixels of the figure in front of them — and the millimetres of sensor follow from the format rather than preceding it.

The traditional 0.03 mm on full frame corresponds to about 0.6 pixels here. That is a strict criterion by the standards of this figure, and the bands it produces are correspondingly narrow.

The hyperfocal distance is one number’s arithmetic #

Set the far limit to infinity and solve for the focus distance. What comes out is

$$H = \frac{f^2}{N c} + f$$

and two exact consequences fall out of it that are usually quoted as rules.

Focus at H and everything from H/2 to infinity is inside the band. Exactly half, checked here against the closed form to a part in a million rather than asserted.

And H is the whole of the depth-of-field story. Near and far limits at any focus distance are Z_f(H−f)/(H−f ± (Z_f−f)), so a photographer who knows H for their lens, aperture and criterion knows the band at every focus setting without further arithmetic.

On the 50 mm at f/2.8 with a two-pixel criterion, H is 8.61 metres. At f/1.4 it is 17.2, at f/5.6 it is 4.33, at f/11 it is 2.23 — halving with each stop, because H goes as 1/N.

The one-third rule, measured #

“A third of the depth of field lies in front of the subject and two thirds behind” is taught, printed on lens barrels, and true at exactly one distance.

Swept across focus distances on the same lens with a two-pixel criterion, the share in front is 0.468 at 0.6 metres, 0.433 at 1.2, 0.328 at 3, 0.035 at 8, and 0.000 at 20 — where the far limit has gone to infinity and the share is meaningless.

So the curve passes through a third at about three metres for this lens and this aperture, and it is a different distance for every lens and every aperture. Close up the share approaches a half, because the band is short and nearly symmetric; approaching the hyperfocal distance it approaches zero, because the far limit runs away.

The rule is a reading of a curve at one point, presented as a property of lenses.

Why the band is asymmetric at all #

The formula makes it look like arithmetic; the geometry makes it obvious.

The disc’s diameter depends on 1/Z rather than on Z — it is the reciprocal of the distance that enters linearly, which is the same reciprocal depth is a reciprocal is about in the stereo field. So the band is symmetric in 1/Z, and a symmetric interval in the reciprocal is an asymmetric one in the distance, stretched at the far end.

That single observation explains the whole family of quoted rules: the near limit approaches the focal length and never reaches it, the far limit runs to infinity at a finite focus setting, and the share in front falls monotonically. All four are the same statement about a reciprocal.

It also connects two fields that look unrelated. A stereo pair’s depth uncertainty is symmetric in disparity and asymmetric in depth for exactly the same reason, and the two asymmetries have the same shape — which is why the range a pair cannot see past and the hyperfocal distance are the same calculation.

What the criterion actually is #

Worth pinning down, because “how large a blur a reader will accept” sounds unmeasurable and is not.

It is the angle the blur disc subtends at the reader’s eye, compared with the angle at which they can resolve detail — about one arcminute for a person with good sight. Everything else is arithmetic: the disc’s size on the sensor, times the enlargement from sensor to print, divided by the viewing distance.

Which means the criterion depends on how the picture will be shown, not on how it was taken. The same negative has a different depth of field on a contact print and on a billboard, and this is not a figure of speech — the interval in metres genuinely changes, because the number in the formula does.

This collection has the machinery to make that concrete. The point to stand at computes the distance a picture is correct from; a reader at that distance sees the picture at the camera’s own field of view, and the criterion in pixels is then fixed by their eye alone. That is the one viewing arrangement in which depth of field is a property of the photograph.

The other cosine: what the corner receives #

A note toward the next rung, because the band is computed on the axis and the frame has corners.

The pupil seen from off the axis is foreshortened, so the patch a corner receives is an ellipse rather than a disc — axis ratio 0.920 at the corner of a full-frame picture with a 50 mm lens — and its area is smaller by the same factor. A criterion stated as a diameter is therefore satisfied slightly earlier in the corner along one axis and at the same place along the other.

The effect is small compared with everything else going on in a corner, and it is worth naming only because it is geometry rather than aberration: it happens to a perfect lens, and no design removes it.

The closed form, checked against the discs #

Every number above comes from a closed form, and this collection does not quote closed forms unchecked.

So the near and far limits are computed from the algebra and then handed back to the disc calculation: at the near limit the diameter comes out 2.000000 pixels against a two-pixel criterion, and at the far limit the same. Six figures of agreement between an algebraic solution and the geometry it was solved from.

That is a small check and it has caught things elsewhere on this site — the four-centre ellipse’s own assertion is written the same way, against an exact value rather than a tolerance, precisely because a bound on the size of an error holds where the effect is small, which is where the control lives.

The hyperfocal claim gets the same treatment. Focused at H, the far limit is not merely large: the arithmetic returns infinity, because the denominator is exactly zero. And the near limit is asserted at exactly half of H to a part in a million rather than “about half”.

Why the band moves with the aperture and not with the focal length #

A source of confusion worth clearing, because the formula has both in it.

H = f²/(N·c) grows as the square of the focal length, which suggests a long lens has much less depth of field than a short one. It does — at the same subject distance. At the same framing, the subject distance grows with the focal length, and the two effects nearly cancel: a 100 mm lens at six metres and a 50 mm at three, both at f/2.8, give bands whose depths differ by a few per cent rather than by a factor of four.

What does not cancel is the aperture. Halving N halves H and halves the band at any framing, which is why the f-number is the control a photographer actually reaches for.

The residual difference between focal lengths at fixed framing shows up in the background, not in the band: a long lens magnifies the out-of-focus background more, so the same disc diameter covers more of the picture. That is a statement about what the blur looks like rather than about how deep the sharp region is, and conflating the two is the usual mistake.

The hyperfocal distance is the pupil’s diameter in disguise #

The claim below that two formats with the same field of view and the same entrance pupil diameter have identical depth of field is worth deriving rather than asserting, because the derivation collapses the whole subject to one physical length.

Write the criterion as a fraction $\varepsilon$ of the frame’s width — which is what quoting it in pixels does — so $c = \varepsilon w$. Fix the field of view, which fixes $f/w = 1/(2\tan(\theta/2))$. And write the entrance pupil’s diameter as $D = f/N$. Then

$$H \approx \frac{f^{2}}{Nc} = \frac{fD}{\varepsilon w} = \frac{D}{2\,\varepsilon\,\tan(\theta/2)}.$$

Every trace of the format and of the focal length has gone. The hyperfocal distance depends on the pupil’s diameter in millimetres, on how much blur the reader accepts as a fraction of the frame, and on the field of view. Nothing else.

Checked against this essay’s own numbers: the 50 mm at f/2.8 has a pupil 17.86 mm across, the two-pixel criterion on a 690 px frame is $\varepsilon = 0.0029$, and the field is 39.6°, giving $H = 8.56$ m against the 8.61 quoted. One expression, three inputs, none of them a focal length.

Three readings follow, and the third is the one that settles a long argument.

The pupil is the only thing about the equipment that enters. A photographer who wants a shallower band is choosing a larger hole, measured in millimetres, and there is no other way to get one. That is why fast long lenses are large physical objects — an 85 mm at f/1.4 has a 61 mm pupil and the glass in front of it has to be at least that wide — and why the size is not a manufacturing inconvenience but the specification itself.

“Equivalent aperture” becomes exact rather than approximate. Matching a full-frame 50 mm f/2.8 on a format half as wide at the same field of view needs a 25 mm lens at f/1.4, because $D = f/N$ must be held and $f$ has halved. The crop factor applies to the $f$-number for depth of field for exactly the reason it applies to the focal length for field of view, and both are the same statement about $f/w$.

And it prices the small sensor honestly. A phone’s 4 mm lens at f/1.8 has a 2.2 mm pupil, so at the same field and the same acceptance its hyperfocal distance is 1.06 m against the 50 mm’s 8.6 — eight times nearer, which is exactly the ratio of the two pupils. Everything from that focus setting outward is inside the band, which is why a phone photograph is sharp from the near foreground to the horizon and why producing background blur on one requires either a different lens or a computation.

What this is not #

Three things, because depth of field attracts more folklore than any other quantity in photography.

It is not a property of a format. Two formats with the same field of view and the same entrance pupil diameter have the same depth of field, exactly, because the formula depends on f²/N and on the enlargement, and those two changes cancel. Quoting a format as “having more depth of field” is quoting a habit about which apertures people use on it — the same conflation a focal length is not an angle unpicks for field of view.

It is not diffraction. Stopping down narrows the disc and widens the band without limit in this model; in a real lens diffraction eventually widens the spot again. That is optics rather than geometry and this collection does not compute it, which is a limit worth stating rather than a gap to be embarrassed by — and it sits beside the honest limit wide-angle is not distortion draws between what a projection does and what a viewer perceives.

And it is not focus. There is exactly one distance at which the disc is a point. Everything else is a disc, and the band is the interval where a reader has agreed not to mind.

The criterion in a viewfinder, and why it is never the one used #

A practical aside that also explains why so much of this is folklore.

A photographer judging sharpness through a viewfinder is looking at a small, dim, magnified image at a fixed apparent size, and the criterion their eye applies there has nothing to do with the criterion the final print will demand. So the band they see is not the band they get, and the discrepancy is in the pessimistic direction for a big print and the optimistic direction for a small one.

Depth-of-field scales on a lens barrel are engraved for one criterion, usually a strict one, chosen when the expected output was a modest print. Digital sensors out-resolve those scales, so the engraved marks are now generous by a stop or two — the scale has not changed and the acceptable disc has.

None of that is a defect in anything. It is what happens when a quantity that depends on the viewer is engraved into the instrument, and it is the same category of error as the sixty-degree cone: a decision about a reader, hardened into a property of a device, and then inherited by people who never saw the decision made.

One number, four ways of saying it #

The hyperfocal distance is f²/(Nc) + f, and every quantity a photographer quotes about depth of field is a rearrangement of it. Worth listing them together once, because seeing that they are one expression is what makes the topic small.

The band at any focus setting is Z(H−f)/(H−f ± (Z−f)). The near limit at the hyperfocal distance is H/2. The share in front is (Z−near)/(far−near), which is a function of Z/H alone. And the depth of field at close range, where Z ≪ H, collapses to 2Z²c N/f² — symmetric, and proportional to the square of the distance.

Four rules, one number, and the number contains a choice.

The short version #

Depth of field is the interval in which the blur disc stays under a diameter somebody chose. On one 50 mm lens at f/2.8 focused at three metres, that interval is 2.76–3.28 metres at half a pixel and 1.26 metres to infinity at eight, with nothing about the optics changing between them.

The hyperfocal distance is the focus setting whose far limit is infinity, its near limit is exactly half of it, and it is the single number the whole band arithmetic follows from — 8.61 metres for this lens and criterion. The one-third rule is a reading of one curve at one point: the share in front runs from 0.468 close up to 0.000 at the hyperfocal distance, and passes a third at about three metres here.