Pinhole Geometry: Blur, Focal Distance, Field of View and Falloff

A pinhole camera has exactly four numbers in it: the diameter of the hole, the distance from the hole to the emulsion, the size of the sheet, and the thickness of the metal the hole is pierced in. Every other property of the camera — how sharp it is, how wide, how dim, how dark the corners go, how close you can work — follows from those four by straight lines and similar triangles. This page derives the lot. By the end of it you should be able to take a format you have chosen and a focal distance you fancy, and say in advance what the negative will look like.

Nothing here needs waves. The single place where the straight-line account fails is the reason the next lesson exists.

Focal distance, not focal length

A lens has a focal length: the distance behind it at which parallel light converges to a point. It is a property of the glass, fixed by its curvature and its refractive index, and it is what makes focusing possible — move the film to the plane where the converging cone is narrowest and the image is sharp; move it away and the image degrades.

A pinhole converges nothing. Light that enters the hole travels on in exactly the direction it was going. There is no plane of best focus behind a pinhole, because there is no convergence to be best at. There is only the plane you decided to put the paper on.

So the course calls that distance the focal distance, or the pinhole-to-film distance, and reserves focal length for glass. This is not pedantry, and three consequences follow immediately:

• There is no focusing. Nothing you can do to a pinhole camera makes one distance sharper at the expense of another.
• There is no depth of field in the usual sense. Depth of field is the range of subject distances over which blur stays below a stated tolerance, and it exists because blur has a minimum at the plane of focus. Pinhole blur has no minimum; it has a floor.
• “A 50 mm pinhole camera” is a construction dimension, not an optical property. If your back sits 3 mm further from the plate than the drawing said, you have a 53 mm camera, and every number on this page changes accordingly. This is why the build page ends by measuring the focal distance rather than trusting the cut list.

The blur circle

Take a single point in the world, at distance u in front of the hole. It radiates in all directions. Some of that light falls on the hole and gets through; the rest hits the plate and stops. What gets through is a cone with the point at its apex and the hole as its cross-section, and behind the plate that cone keeps expanding at the same rate.

The apex is at u from the hole. At the hole, the cone’s diameter is d. A further f on, at the film, the cone has travelled a total of u + f from the apex, so its diameter has grown in proportion:

b = d (u + f) / u = d (1 + f / u)

The geometric blur circle

b is the diameter of the patch of light one object point makes on the film, d the pinhole diameter, f the focal distance and u the subject distance, all measured from the plane of the hole. Every point in the scene makes such a patch, and the picture is the sum of them all: a photograph in which every point has been replaced by a disc. This is the geometric blur, and it is the first of the two that decide how sharp a pinhole can be.

One point, one cone, one disc on the film

The rays here are computed from the element positions, so the inversion and the image height are the geometry's, not the draughtsman's. The hole is drawn as a gap because that is all it is.

What the formula says at its limits

A distant subject blurs to the size of the hole. As u grows, f/u goes to nothing and b → d. A landscape photographed through a 0.25 mm hole is built out of 0.25 mm discs, no matter how far away the hills are. This is the floor: no subject is ever rendered with a blur smaller than the hole itself.

A subject at the focal distance blurs to twice the hole. Put u = f and b = 2d. That is also the life-size case, since the magnification is one.

In between, the penalty is modest and predictable. At f = 50 mm, a subject 2 m away gives b = 0.25 × 1.025 = 0.256 mm, which is 2 per cent worse than infinity and invisible. The same subject at 300 mm gives b = 0.25 × 1.167 = 0.292 mm, 17 per cent worse.

Angular blur, and the nearest useful subject

For a distant subject the blur on the film is d, and it subtends an angle d/f at the hole. That angle is the resolution limit of pure geometry: two distant points closer together than d/f in angle put their discs on top of each other and merge. At 50 mm and 0.25 mm that is 0.005 radians, or about 17 minutes of arc — some seventeen times coarser than the eye. Notice what the expression does not contain: the subject distance. Geometry alone says a pinhole’s angular resolution depends only on the ratio of hole to focal distance.

If you will tolerate a blur no worse than k times the distant-subject value, then

1 + f / u ≤ k, so u ≥ f / (k − 1)

Nearest useful subject distance

Accept 10 per cent extra blur and the nearest subject is ten focal distances away — 500 mm at f = 50 mm. Accept 25 per cent and it is four focal distances, 200 mm. Both are much closer than a lens of the same angle of view would manage without extension, which is where the pinhole’s reputation for infinite depth comes from.

The whole table, for the 50 mm camera with a 0.25 mm hole, is short enough to memorise:

Blur you will accept	k	Nearest subject u	Blur there
2 per cent worse than infinity	1.02	2.5 m	0.255 mm
5 per cent	1.05	1.0 m	0.263 mm
10 per cent	1.10	500 mm	0.275 mm
25 per cent	1.25	200 mm	0.312 mm
50 per cent	1.50	100 mm	0.375 mm
Twice	2.00	50 mm	0.500 mm

Read the last row again. At the focal distance itself — 50 mm in front of the hole — the blur has only doubled. There is no cliff anywhere in this table, which is the real difference from a lens: a lens has a distance at which the image collapses, and a pinhole has a gentle slope you can walk down as far as you like as long as you know what you are paying.

✕It is not depth of field, and calling it that will mislead you

Matt Young’s 1971 summary of pinhole optics advertises “virtually infinite depth of field”, and the phrase has been repeated ever since. It is true in the sense that everything from a few hundred millimetres to the horizon is rendered equally. It is false in the sense a lens photographer will hear it. A lens has a plane where blur genuinely goes to nothing, and depth of field is the region either side where it stays acceptably small. A pinhole has no such plane. Everything is unsharp by at least d, and the honest statement is not that everything is in focus but that nothing is, to the same degree, everywhere.

Image size, magnification and the bellows question

The two triangles either side of the hole are similar, so the image height is the object height scaled by the ratio of the distances:

m = h′ / h = f / u

Magnification

A doorway 2.0 m high at 10 m, on a camera with f = 50 mm, images 2000 × 50 / 10000 = 10 mm high. On a 127 mm sheet that is a twelfth of the frame, which is why pinhole pictures of buildings need either a very close viewpoint or a long focal distance.

Now the exposure question. Take a small patch of an evenly lit subject, of area A at distance u. The hole, of area a, subtends a solid angle a/u² at the patch, so the light that gets through is proportional to A a/u². That light lands on the image of the patch, whose area is A m², or A f²/u². Dividing, the illuminance at the film is proportional to a/f² — and u has cancelled out. Write the hole’s area as proportional to d² and the result is the familiar one: illuminance goes as (d/f)².

🧮Why a pinhole camera has no separate bellows factor

For a lens, the bellows factor exists because the f-number is defined from the focal length, while the light actually travels the image distance v = f(1 + m), which is longer whenever the subject is close. Correcting for that gives the familiar N_eff = N(1 + m).

A pinhole has no focal length to be caught out by. The distance in N = f/d is the image distance, because there is nowhere else for it to be measured from. So the correction is already inside the number, provided f is the real pinhole-to-emulsion distance and not the one on the drawing. Set N_eff = v/D for the lens and the two statements become the same statement.

The trap is practical rather than algebraic. Stack an extra frame to photograph something close and f changes, so N changes, and if you keep metering at the old N you will be short. Going from 50 mm to 100 mm at a fixed 0.25 mm hole takes you from f/200 to f/400: two stops, which is the bellows factor arriving through the front door. And a film holder that sets the emulsion 3 mm behind where you assumed costs (53/50)² = 1.12, about a sixth of a stop, which is not worth worrying about but is worth knowing the size of.

This course has not been able to consult Ray’s Applied Photographic Optics or Kingslake, so the derivation above is its own, from the geometry stated. If you have either book, check it.

Angle of view

The hole sees the whole of the sheet behind it, and the sheet’s edges define the field. For a format dimension w — the width, the height, or the diagonal, taken at the film —

angle = 2 arctan( w / 2f )

Angle of view

For a 4 × 5 inch paper negative, 101.6 × 127 mm with a 162.6 mm diagonal, at f = 50 mm: 90.9° across the short side, 103.6° across the long side, 116.8° corner to corner. Those are extraordinarily wide numbers by lens standards, and Young’s abstract notes exactly this, that a pinhole’s angular field “can be made to exceed 90°”.

Angle of view across the diagonal, against focal distance, for three sheet formats

• 4 × 5 in (163 mm diagonal)
• 5 × 7 in (218 mm diagonal)
• 8 × 10 in (325 mm diagonal)

Show the numbers behind this plot

Three curves of diagonal angle of view against focal distance, all falling steeply at first and then flattening. The 4 by 5 inch format, whose diagonal is 163 mm, starts at 152 degrees at a focal distance of 20 mm, passes 117 degrees at 50 mm, 78 degrees at 100 mm, 57 degrees at 150 mm and 30 degrees at 300 mm. The 5 by 7 inch format, diagonal 218 mm, is above it throughout: 159 degrees at 20 mm, 131 degrees at 50 mm, 95 degrees at 100 mm, 72 degrees at 150 mm and 40 degrees at 300 mm. The 8 by 10 inch format, diagonal 325 mm, is higher again: 166 degrees at 20 mm, 146 degrees at 50 mm, 117 degrees at 100 mm, 95 degrees at 150 mm and 57 degrees at 300 mm. The three curves are the same curve slid sideways, because the angle depends only on the ratio of the diagonal to the focal distance: an 8 by 10 sheet at 100 mm sees the same angle as a 4 by 5 sheet at 50 mm, since both have a diagonal twice the focal distance.

Series	Focal distance, mm	Diagonal angle of view, degrees
4 × 5 in (163 mm diagonal)	20.00	152.40
4 × 5 in (163 mm diagonal)	30.00	139.50
4 × 5 in (163 mm diagonal)	40.00	127.60
4 × 5 in (163 mm diagonal)	50.00	116.80
4 × 5 in (163 mm diagonal)	60.00	107.20
4 × 5 in (163 mm diagonal)	80.00	90.90
4 × 5 in (163 mm diagonal)	100.00	78.20
4 × 5 in (163 mm diagonal)	125.00	66.10
4 × 5 in (163 mm diagonal)	150.00	56.90
4 × 5 in (163 mm diagonal)	200.00	44.30
4 × 5 in (163 mm diagonal)	250.00	36.00
4 × 5 in (163 mm diagonal)	300.00	30.30
5 × 7 in (218 mm diagonal)	20.00	159.30
5 × 7 in (218 mm diagonal)	30.00	149.30
5 × 7 in (218 mm diagonal)	40.00	139.80
5 × 7 in (218 mm diagonal)	50.00	130.80
5 × 7 in (218 mm diagonal)	60.00	122.40
5 × 7 in (218 mm diagonal)	80.00	107.60
5 × 7 in (218 mm diagonal)	100.00	95.10
5 × 7 in (218 mm diagonal)	125.00	82.30
5 × 7 in (218 mm diagonal)	150.00	72.10
5 × 7 in (218 mm diagonal)	200.00	57.30
5 × 7 in (218 mm diagonal)	250.00	47.20
5 × 7 in (218 mm diagonal)	300.00	40.00
8 × 10 in (325 mm diagonal)	20.00	166.00
8 × 10 in (325 mm diagonal)	30.00	159.10
8 × 10 in (325 mm diagonal)	40.00	152.40
8 × 10 in (325 mm diagonal)	50.00	145.80
8 × 10 in (325 mm diagonal)	60.00	139.50
8 × 10 in (325 mm diagonal)	80.00	127.60
8 × 10 in (325 mm diagonal)	100.00	116.80
8 × 10 in (325 mm diagonal)	125.00	104.90
8 × 10 in (325 mm diagonal)	150.00	94.60
8 × 10 in (325 mm diagonal)	200.00	78.20
8 × 10 in (325 mm diagonal)	250.00	66.10
8 × 10 in (325 mm diagonal)	300.00	56.90

Computed from 2 arctan(w/2f) with the nominal sheet diagonals; nothing here is measured. Sheet film in a holder is masked by the rebates and its true diagonal is smaller than the nominal one, so measure your own back before using these figures for film. The curve is drawn to show the shape, not measured from a real material. Your own materials will differ, and measuring them is what the sensitometry part of the course is for.

Wide, normal and long on one format

Photographic convention calls a focal length roughly equal to the format diagonal “normal”, because it gives an angle near 53°, close to the angle the eye takes in comfortably. For a 4 × 5 sheet that would be a 163 mm pinhole. Almost nobody builds one, and the reason is in the next two sections: long pinholes are dim and their corners are dark. In practice pinhole cameras live between about a quarter and three-quarters of the diagonal, which is to say wide to very wide, and that is the pinhole’s natural territory rather than a limitation to be apologised for.

The effective f-number

Define the effective f-number exactly as for a lens: the focal distance divided by the aperture diameter.

N = f / d

Effective f-number

A 0.25 mm hole at 50 mm is N = 200, written f/200. That is why pinhole cameras sit in a range no lens reaches. The next lesson shows that the best hole diameter grows only as the square root of the focal distance, d ∝ √f; put that into N = f/d and

N ∝ f / √f = √f, so exposure ∝ N² ∝ f

Why long pinholes are slow

Doubling the focal distance therefore doubles the exposure — not quadruples it, because the hole gets bigger too. Working the actual numbers for blue light at the optimum gives f/151 at 25 mm, f/213 at 50 mm, f/302 at 100 mm and f/604 at 400 mm. That is where the f/100 to f/500 range of practical pinhole cameras comes from: it is not a convention, it is what the square-root law does over the range of focal distances anyone builds. Rayleigh made the same point in 1891 in the other direction, observing that the brightness of a properly proportioned pinhole camera falls as 1/f.

The square law then does the rest. From a meter reading of f/16 to f/200 is 2 log₂(200/16) = 7.3 stops, a factor of 156. A scene that a hand meter calls 1/125 s at f/16 needs about 1.25 s at f/200 before any correction for reciprocity failure — and that correction is usually the larger of the two. The exposure lesson (pinhole-exposure-and-reciprocity-correction) does this properly.

The image circle and the cos-fourth law

The image circle is what the hole throws at the film plane, and the centre of a pinhole negative is always the brightest part of it, and the corners are always darker. This is not a defect of the hole; it is three separate pieces of geometry multiplying together, and it happens behind a lens as well. Rodenstock’s enlarging-lens data sheets plot exactly this quantity and label it 1 − cos⁴.

Take a point on the film at field angle θ, measured at the hole from the axis. Three things are different about it compared with the point on the axis.

Where the four cosines come from

1. 1Inverse square: the slant distance is f / cos θ — contributes cos² θ
2. 2Oblique aperture: the hole looks like an ellipse of area a cos θ from P — contributes cos θ
3. 3Oblique film: the beam is spread over 1 / cos θ more area — contributes cos θ

Multiply the three and the illuminance at field angle θ is cos⁴ θ times the illuminance on the axis. Nothing in the argument mentions the aperture's contents, which is why a lens obeys the same law.

In words, the three factors are these. The slant distance is longer: the corner is f/cos θ from the hole rather than f, and by the inverse-square law that costs cos² θ. The aperture is foreshortened: seen from the corner, the round hole is an ellipse of area a cos θ, costing another cos θ. The film is oblique: the beam meets it at θ to its normal, so the same light covers 1/cos θ more area, costing a third cos θ. Multiply, and the product is the cosine-fourth law.

E(θ) = E(0) · cos⁴ θ

The cos-fourth illumination law

E is illuminance at the film and θ is the angle, at the hole, between the axis and the point in question. Because the three factors are purely geometric, the law is the same for a pinhole and for a lens; what differs is that a lens designer can partly fight it with the shape of the entrance pupil, and a hole in a plate cannot.

Falloff in stops

Express the loss as stops, which is what the exposure will be judged in: stops = −log₂(cos⁴ θ), or equivalently −4 log₂(cos θ). Two of the values are worth memorising because they are exact.

Field angle θ	cos⁴ θ	Loss, stops
15°	0.87	0.2
30°	0.5625	0.83
45°	0.25	exactly 2
50°	0.171	2.6
55°	0.108	3.2
60°	0.0625	exactly 4
65°	0.032	5.0
70°	0.0137	6.2

At 45° the corner is two stops down; at 60° it is four. Those follow from cos 45° = 1/√2 and cos 60° = 1/2, so they are not approximations.

The practical image circle is wherever you decide the falloff has stopped being a picture and started being a fault. Paper negatives have a long enough scale to carry two stops without the corners going empty, so a reasonable working rule is to keep the corner angle at or under 45°, which means a half-diagonal no greater than the focal distance. On a 4 × 5 inch sheet, whose half-diagonal is 81 mm, that would need f ≥ 81 mm. At 50 mm the corners sit at 58.4° and are 3.7 stops down — which is the dark-cornered look pinhole photographs are known for. It is a legitimate choice. It is not, however, an accident, and once you have this table you cannot claim you did not know.

Set out over the whole range of focal distances anyone would build on a 4 × 5 sheet, the trade is stark and monotonic:

f	Corner angle	Corner loss, cos⁴ only	Diagonal angle of view
25 mm	72.9°	7.1 stops	145.8°
40 mm	63.8°	4.7 stops	127.6°
50 mm	58.4°	3.7 stops	116.8°
65 mm	51.4°	2.7 stops	102.7°
81 mm	45.1°	2.0 stops	90.2°
100 mm	39.1°	1.5 stops	78.2°
163 mm	26.5°	0.6 stop	53.0°

Every extra degree of angle of view is bought with corner darkness, and the currency never varies, because both quantities are functions of the same ratio of half-diagonal to focal distance. A 25 mm pinhole on 4 × 5 inch paper is not “a very wide camera with a vignetting problem”; it is a camera whose corners are seven stops down, which on any material is black. If you want that angle, use a smaller sheet.

🧪Measure your own falloff, on one sheet of paper

You can test cos⁴ before the camera exists. Tape a sheet of paper to a wall in even, indirect daylight and photograph nothing but the wall: point the camera at a large blank area, fill the frame with it, and expose normally. Any negative made this way is a map of the camera’s illumination, because the subject has none of its own.

Process it, then read it in three places — centre, halfway to a corner, and the corner itself. If you have a densitometer or a phone app that will read reflected values, use it; if not, print the negative and compare the three patches against a stepped print made from a grey card. Convert the differences to stops and plot them against the field angle you compute for each point from arctan(r/f).

Two questions to answer in the notebook. Does the centre-to-45° difference come out near two stops? And is the corner worse than cos⁴ predicts — which is what plate thickness should be doing to it? The commissioning experiment later in this part runs the same measurement properly, but doing it once now, roughly, is the difference between believing a table and having tested one.

📓A number this course could not find

Real pinhole falloff is generally worse than cos⁴, for the reason the next section gives and because a hand-pierced hole is not a perfect circle in an infinitely thin screen. This course could not find a published measurement of pinhole falloff to put a figure on the excess, and it does not invent one. Your own camera can supply it: the commissioning experiment (commissioning-the-pinhole-camera) photographs an evenly lit blank surface and reads centre, mid-field and corner densities, which is the measurement. Record the excess over cos⁴ in stops, with your plate thickness beside it.

Plate thickness and the tunnel effect

The blur derivation quietly assumed the hole was a hole in a plane of no thickness. A real hole is a short cylinder of length t, the thickness of the metal, and a cylinder is a tunnel. Look down a tunnel from the side and you see less of the far end.

This is the tunnel effect, and it is true vignetting — a physical obstruction of the light path — rather than the geometric falloff of the previous section, which is why the two must be kept apart and why the total is worse than either. On the axis you see the full circle, of area proportional to d². At angle θ, the near opening and the far opening are displaced relative to your line of sight by t tan θ, and what gets through is the overlap of two circles of diameter d offset by that amount. When the offset reaches d the circles no longer overlap at all and the hole has closed:

θ_c = arctan( d / t )

Tunnel cut-off angle

The same hole in three plates, seen in section

1. 1Thick plate, t = d — cut off at 45°; corners of a wide picture receive nothing
2. 2Thin plate, t = d/5 — cut off at 79°; about 0.4 stop of extra loss at 45°
3. 3Dished plate — stiff where you hold it, thin where the light goes

Drawn to show the geometry, not to scale: a real 0.05 mm plate beside a 0.25 mm hole is far thinner in proportion than anything legible on a page.

The numbers are unforgiving. For a 0.25 mm hole:

Plate thickness t	t / d	Cut-off angle	Extra loss at 45°	Extra loss at the 58° corner
0.05 mm	1/5	78.7°	0.4 stop	0.75 stop
0.10 mm	2/5	68.2°	1.0 stop	2.1 stops
0.20 mm	4/5	51.3°	3.3 stops	dark
0.25 mm	1	45.0°	dark	dark

Illumination falloff against field angle: cos-fourth alone, and with a thick plate

• cos⁴ only (plate of no thickness)
• plus tunnel effect, t = d/5
• plus tunnel effect, t = 2d/5

Show the numbers behind this plot

Three curves of light loss in stops against field angle from zero to seventy degrees, all rising from zero at the axis and steepening. The lower curve is the cos-fourth law alone, for a plate of no thickness: 0.36 stop at 20 degrees, 0.83 at 30, exactly 2.00 at 45, exactly 4.00 at 60 and 6.19 at 70. The middle curve adds the tunnel effect of a plate one fifth as thick as the hole is wide: 0.50 stop at 20 degrees, 1.06 at 30, 2.42 at 45, 4.82 at 60 and 7.76 at 70, with complete cut-off at 78.7 degrees. The upper curve is a plate two fifths as thick as the hole: 0.65 stop at 20 degrees, 1.33 at 30, 2.99 at 45, 6.36 at 60, and complete cut-off at 68.2 degrees, so the curve runs off the top of the plot before 70 degrees. The gap between the curves widens with angle, which is the point: plate thickness costs almost nothing in the middle of the picture and a great deal at the corners.

Series	Field angle, degrees	Loss, stops
cos⁴ only (plate of no thickness)	0.00	0.00
cos⁴ only (plate of no thickness)	10.00	0.09
cos⁴ only (plate of no thickness)	20.00	0.36
cos⁴ only (plate of no thickness)	30.00	0.83
cos⁴ only (plate of no thickness)	40.00	1.54
cos⁴ only (plate of no thickness)	45.00	2.00
cos⁴ only (plate of no thickness)	50.00	2.55
cos⁴ only (plate of no thickness)	55.00	3.21
cos⁴ only (plate of no thickness)	60.00	4.00
cos⁴ only (plate of no thickness)	65.00	4.97
cos⁴ only (plate of no thickness)	70.00	6.19
plus tunnel effect, t = d/5	0.00	0.00
plus tunnel effect, t = d/5	10.00	0.15
plus tunnel effect, t = d/5	20.00	0.50
plus tunnel effect, t = d/5	30.00	1.06
plus tunnel effect, t = d/5	40.00	1.88
plus tunnel effect, t = d/5	45.00	2.42
plus tunnel effect, t = d/5	50.00	3.07
plus tunnel effect, t = d/5	55.00	3.85
plus tunnel effect, t = d/5	60.00	4.82
plus tunnel effect, t = d/5	65.00	6.06
plus tunnel effect, t = d/5	70.00	7.76
plus tunnel effect, t = 2d/5	0.00	0.00
plus tunnel effect, t = 2d/5	10.00	0.22
plus tunnel effect, t = 2d/5	20.00	0.65
plus tunnel effect, t = 2d/5	30.00	1.33
plus tunnel effect, t = 2d/5	40.00	2.32
plus tunnel effect, t = 2d/5	45.00	2.99
plus tunnel effect, t = 2d/5	50.00	3.81
plus tunnel effect, t = 2d/5	55.00	4.88
plus tunnel effect, t = 2d/5	60.00	6.36
plus tunnel effect, t = 2d/5	65.00	8.96

Computed, not measured: cos⁴ θ multiplied by the overlap area of two circles of diameter d offset by t tan θ. The overlap model treats the bore as a clean cylinder, which a hand-pierced hole is not, so treat these as the best case for a given thickness. The curve is drawn to show the shape, not measured from a real material. Your own materials will differ, and measuring them is what the sensitometry part of the course is for.

This is the whole argument for thin metal, and for the sanding cycle in the pinhole lab: dimple, pierce, sand the burr flat, re-pierce. Sanding does two jobs at once. It removes the burr, which would otherwise be a ragged extension of the tunnel, and it thins the plate locally, leaving the rest of it stiff enough to handle. A dished plate is the best of both.

Measuring the focal distance you actually built

Every number on this page is a function of f, and f is the one quantity a builder is most likely to get wrong, because it is not a dimension of any single part. It runs from the plane of the pinhole to the emulsion, and both ends move.

At the front, the pinhole plate sits in a carrier, the carrier sits in a rebate, and the plate has a thickness of its own. Measure to the middle of the bore, not to the front face of the body.

At the back, the trap is worse. A sheet of paper taped to a flat back sits where the back is. A sheet of film in a holder does not: the holder’s own body stands proud of the camera’s back, and the film sits some distance inside it behind the dark-slide rails. The design lesson (designing-the-modular-camera) owns the holder geometry, and this course has not verified a manufacturer’s drawing for the standard 4 × 5 inch pattern, so the instruction here is to measure the holder you own rather than to trust a figure. The method is simple: close an empty holder into the back, push a straight rod or a strip of card through the pinhole until it touches the film plane through the open dark slide, mark the rod at the outer face of the plate, and subtract the plate thickness.

Then recompute. A 3 mm error at 50 mm is 6 per cent in f, which is 6 per cent in N, 12 per cent in exposure and a sixth of a stop — negligible. The same 3 mm at 25 mm is 12 per cent, a third of a stop, and it also moves the angle of view by four degrees. Record the measured value in the register beside the design value, and use the measured one everywhere afterwards.

Perspective

Pinhole perspective is ordinary perspective. The relative sizes and positions of things in a picture are fixed entirely by where the aperture is — the viewpoint — and not at all by what is in it. A pinhole standing where a lens stood gives exactly the perspective the lens gave, over whatever part of the field they share. Change the focal distance and you change the angle of view and the image scale; you do not change the perspective, and the only way to change perspective is to move the camera.

On a flat film plane the pinhole is also exactly rectilinear: straight lines in the world render as straight lines in the picture, with no barrel or pincushion at all. The proof is one sentence. A straight line in the world, together with the pinhole, defines a plane; the image of the line is where that plane cuts the film; and a plane cuts a flat surface in a straight line. Young’s 1971 abstract lists “freedom from distortion” as one of the pinhole’s two chief virtues, and this is why it is not merely small but exactly zero. No lens can claim that.

Curve the back and the same proof gives a different answer: a plane cuts a cylinder in an ellipse. Lines parallel to the cylinder’s axis still render straight, because their planes contain the axis direction; everything else bows. That is the whole subject of the curved-plane experiment (curved-plane-and-multiple-pinhole-experiments), and it is a choice about mapping, not a defect.

Offset and multiple pinholes

Slide the hole sideways by s, parallel to the film, and the entire image — and the circle of illumination with it — slides by s in the same direction. Nothing rotates, so verticals stay vertical. This is precisely the shift movement of a view camera, and it is what lets a pinhole photograph a tall building from ground level without tipping the camera up and making the verticals converge. Part VII uses it. What limits the shift is everything in the two sections above: shifting towards one edge raises the field angle on that side, so the falloff and the tunnel effect are what run out first, not the shift mechanism.

Two holes at once, separated by s, give two complete images displaced on the film by

Δ = s (1 + f / u)

Displacement of two images from two holes

which for a distant subject is simply s. Look at that bracket. It is the same (1 + f/u) that governs the blur circle, and for a very good reason: a pinhole of diameter d is a continuous set of pinholes spanning d, and the blur circle is the displacement between the two extreme ones. The blur formula and the twin-image formula are the same formula. Making holes deliberately far apart is only making that displacement large enough to see as two pictures rather than as softness.

One hole, offset; and two holes at once

1. 1One hole, offset by s — the image circle moves by s; verticals stay vertical
2. 2Two holes, s apart — two whole images, displaced by s for a distant subject
3. 3The doubled band — width of sheet minus s; the ends carry a single image only

Both panels are the same geometry. The circle of illumination belongs to the hole, and it goes where the hole goes.

🧮DO THE MATHS: a complete design for a 4 × 5 inch paper negative at 50 mm

The format. A sheet of grade-neutral resin-coated paper cut to 4 × 5 inches is 101.6 × 127 mm. Its diagonal is √(101.6² + 127²) = 162.6 mm, and the half-diagonal, which is what the corner falloff needs, is 81.3 mm.

Choosing the hole. The next lesson derives the optimum family; taken at 450 nm, the wavelength blue-sensitive paper actually works at, it brackets the answer between 0.21 mm and 0.29 mm. Take d = 0.25 mm, which sits in the middle and is a hole you can make and measure.

The f-number. N = f/d = 50 / 0.25 = f/200. Relative to a meter reading at f/16 that is 2 log₂(200/16) = 7.3 stops, a factor of 156. A reading of 1/125 s at f/16 becomes 156 / 125 = 1.25 s at f/200, before reciprocity correction.

The blur. A distant subject: b = d = 0.25 mm. A subject at 2 m: 0.25 × (1 + 50/2000) = 0.256 mm. A subject at 300 mm: 0.25 × (1 + 50/300) = 0.29 mm. Diffraction adds about 0.22 mm on top of these, which the next lesson works out; the total is a little under half a millimetre. Taking a line pair as needing two blur widths, the resolving power is about 1 line pair per millimetre on the negative — against roughly 47 lp/mm for a decent lens at f/8, computed the same way.

The angle of view. Across the short side, 2 arctan(50.8/50) = 90.9°. Across the long side, 2 arctan(63.5/50) = 103.6°. Corner to corner, 2 arctan(81.3/50) = 116.8°.

The corner falloff. The corner sits at θ = arctan(81.3/50) = 58.4°. cos 58.4° = 0.523, and cos⁴ = 0.075, so the corner receives 7.5 per cent of the centre: 3.7 stops down by cos⁴ alone. Add a plate 0.05 mm thick — one fifth of the hole — and the tunnel effect takes it to about 4.5 stops. With a 0.1 mm plate it is 5.8 stops, at which point the corners of the negative have nothing in them.

The verdict. This is a fast, extremely wide camera with soft, very dark corners, and every one of those four adjectives has a number attached. If you want the corners inside two stops you need the half-diagonal no larger than the focal distance, so f ≥ 81 mm on this format — which costs you angle of view and, from the square-root law, about 0.70 stop more exposure. That is the whole design trade, and now you can make it deliberately.

Recap

• A pinhole has a focal distance, not a focal length, because it converges nothing. The distance is a construction dimension and has to be measured on the finished camera.
• Every object point becomes a disc of diameter b = d(1 + f/u). Distant points blur to exactly the hole diameter; that is a floor, not a minimum, so nothing is ever sharp.
• m = f/u, and the illuminance goes as (d/f)² whatever the subject distance, so a pinhole’s f-number already contains what a lens needs a bellows factor for.
• Angle of view = 2 arctan(w/2f), and N = f/d. Because the best hole grows as √f, N grows as √f and exposure grows in proportion to f.
• Illumination falls as cos⁴ θ: two stops at 45°, four stops at 60°, both exactly.
• A thick plate makes it worse, cutting off entirely at arctan(d/t). Keep t ≤ d/5.
• Perspective is the viewpoint’s, and on a flat plane a pinhole is exactly rectilinear. Offsetting the hole shifts the image; two holes give two images displaced by s(1 + f/u), the same bracket as the blur.

Check your understanding

Question 1. A camera is built with a 0.4 mm pinhole at a focal distance of 75 mm, on 4 × 5 inch paper. What is its effective f-number, its diagonal angle of view, and the corner falloff in stops from the cos-fourth law alone?

• f/188, 95°, and about 2.2 stops
• f/188, 47°, and about 2.2 stops
• f/300, 117°, and about 3.7 stops
• f/188, 117°, and about 3.7 stops

Show the answer and why

Answer: f/188, 95°, and about 2.2 stops

Three separate calculations. N = f/d = 75/0.4 = 187.5, so f/188. The diagonal of a 4 × 5 inch sheet is 162.6 mm, so the diagonal angle is 2 arctan(162.6/150) = 2 × 47.3° = 94.6°. The corner field angle is half of that, 47.3°: cos 47.3° = 0.678, and 0.678⁴ = 0.211, which is −log₂(0.211) = 2.24 stops. Compare that with the 50 mm design in the worked example, where the corner sat at 58.4° and lost 3.7 stops: moving to 75 mm on the same format has bought 1.5 stops of corner brightness at the cost of 22° of angle, and about 0.58 stop of overall exposure. Option 2 forgets the factor of two and quotes the half-angle; options 3 and 4 reuse the 50 mm figures.

Question 2. A subject stands 300 mm in front of a pinhole camera whose focal distance is 50 mm and whose hole is 0.25 mm. Compared with a distant subject, how much larger is the blur circle, and does the exposure need to change?

• 17 per cent larger, and the exposure must be increased by about half a stop
• 17 per cent larger, and the exposure does not change
• Six times larger, and the exposure must be increased by two stops
• The same size, and the exposure does not change

Show the answer and why

Answer: 17 per cent larger, and the exposure does not change

b = d(1 + f/u) = 0.25 × (1 + 50/300) = 0.25 × 1.167 = 0.292 mm against 0.25 mm for a distant subject: 17 per cent larger. The exposure is a separate question, and the answer is no. The illuminance at the film works out proportional to (d/f)² with the subject distance cancelling, so as long as f is the true pinhole-to-emulsion distance, f/200 is f/200 whatever you point it at. A lens would need a bellows factor here because its f-number is defined from focal length rather than image distance; a pinhole has no focal length to be caught out by. What would change the exposure is stacking an extension frame to make the subject bigger, because that changes f and therefore N.

Question 3. Why does a thick pinhole plate narrow the usable image circle even though the hole diameter is unchanged?

• Because the thicker metal absorbs some of the light passing through it
• Because a hole in a thick plate is a short tunnel, so an oblique ray sees only the overlap of the entry and exit openings, and that overlap shrinks to nothing at arctan(d/t)
• Because a thick plate cannot be pierced accurately enough to be circular
• Because diffraction is stronger in a thicker plate

Show the answer and why

Answer: Because a hole in a thick plate is a short tunnel, so an oblique ray sees only the overlap of the entry and exit openings, and that overlap shrinks to nothing at arctan(d/t)

The aperture is not a hole in a plane, it is a cylindrical bore of length t. Seen along the axis it presents a full circle. Seen at angle θ, the far opening is displaced relative to the near one by t tan θ, and only the overlap of the two circles passes light; when t tan θ reaches d there is no overlap and the corner of the picture receives nothing at all. For a 0.25 mm hole in 0.2 mm metal that cut-off is at 51°, which is inside the corner angle of a 4 × 5 sheet at 50 mm. Absorption is irrelevant — the metal is opaque either way — and the diffraction of the next lesson depends on the hole diameter, not on the plate.

Question 4. A student proposes a 300 mm focal distance on 4 × 5 inch paper, calling it a "telephoto pinhole". Which two consequences make this hard to use?

• The diagonal angle of view falls to about 30°, so the camera sees very little
• The optimum hole grows to about 0.6 mm and the f-number to about f/520, so the exposure is roughly six times that of the same camera at 50 mm
• The corner falloff becomes worse than at 50 mm
• Perspective becomes distorted at long focal distances

Show the answer and why

Answer: The diagonal angle of view falls to about 30°, so the camera sees very little, The optimum hole grows to about 0.6 mm and the f-number to about f/520, so the exposure is roughly six times that of the same camera at 50 mm

Angle of view is 2 arctan(162.6/600) = 30.3°, which on a format this size is a long lens and rules out most of what a pinhole is good at. Exposure is the second problem: at the optimum the hole grows only as √f, so N grows as √f and exposure grows as f — a factor of six from 50 mm to 300 mm, on top of an already long time, and every extra second of it runs further into reciprocity failure. The other two are wrong in instructive ways. Corner falloff gets better, not worse, because the half-diagonal of 81 mm now sits at arctan(81/300) = 15°, only 0.2 stop down. And perspective does not depend on focal distance at all; it depends only on where the camera stands.

Question 5. Two pinholes are drilled 20 mm apart in one plate, at a focal distance of 60 mm, and a 100 mm wide sheet is exposed to a distant landscape. What appears on the negative?

• One image, blurred by 20 mm
• Two complete images displaced by 20 mm, with an 80 mm band across the middle that has received both and 10 mm at each end that has received one
• Two complete images displaced by 20 mm, with an 80 mm band that has received both and a 20 mm strip at each end that has received one
• Two images displaced by 40 mm, because the displacement doubles at the film

Show the answer and why

Answer: Two complete images displaced by 20 mm, with an 80 mm band that has received both and a 20 mm strip at each end that has received one

The displacement is Δ = s(1 + f/u), and for a distant subject f/u vanishes, leaving Δ = s = 20 mm. Each hole casts a complete image; slide one 20 mm relative to the other on a 100 mm sheet and they overlap over 100 − 20 = 80 mm, leaving a 20 mm strip at each end covered by one image only. Those single strips will print about a stop lighter than the doubled band, which is the visible signature of a two-hole exposure. Option 1 confuses displacement with blur — though the two are the same formula, and if you made s equal to the hole diameter you would get exactly the blur circle.

Question 6. Which statement about the cos-fourth law is correct?

• It applies to pinholes but not to lenses, because a lens focuses the light
• It is an empirical rule fitted to measurements of real cameras
• It follows from three geometric factors — inverse-square slant distance, foreshortening of the aperture, and obliquity of the film — and therefore applies to any small aperture, lens or hole
• It describes the diffraction spreading at the edge of the aperture

Show the answer and why

Answer: It follows from three geometric factors — inverse-square slant distance, foreshortening of the aperture, and obliquity of the film — and therefore applies to any small aperture, lens or hole

The derivation never asks what is in the aperture. The slant distance to a point at field angle θ is f/cos θ, and inverse-square gives cos²θ; the aperture seen from that point is foreshortened to a cos θ, giving another cos θ; and the beam arrives at θ to the film normal so it is spread over 1/cos θ more area, giving a third. Three factors, cos⁴ θ. Lens manufacturers plot it — Rodenstock label the fall-off axis of their enlarging-lens data sheets "1 − cos⁴" — and lens designers can only partly fight it by shaping the entrance pupil, which a hole in a plate cannot do. Diffraction is a different effect entirely and is the subject of the next lesson.

Check my answersClear