Focusing is a zoom

A focal length is not an angle argued that the phrase a 50 mm lens is an incomplete description of a picture, because the angle a lens covers depends on the rectangle behind it: fifty millimetres is 39.6° across full frame and 8.7° across a phone sensor. It ended with a qualification — that the focal length is not quite a distance either, since a real lens defines it against a nodal point whose position depends on the design.

There is a second way the phrase is incomplete, and it applies to one lens on one camera. The fifty millimetres engraved on a lens is its focal length, which is the distance from lens to sensor when the lens is focused at infinity. Focus it on anything nearer and it has to stand further from the sensor, and a picture is a projection from wherever the lens actually stands. So a lens focused close makes a picture with a longer principal distance than its engraving, and everything that depends on that distance — the angle covered, the camera recovered from the picture, the angles read off it, the light that reaches the sensor, the place a reader should stand — moves with the focus ring.

Where the lens stands #

The simplest way a lens focuses is to move as a whole, away from the sensor, without changing its own shape. Photographers call it unit focusing, and it is how many older and simpler lenses work. It is the case this essay measures, because it is the case where the geometry is exactly the thin-lens equation.

A thin lens of focal length $f$ brings a point at distance $u$ in front of it to focus at a distance $v$ behind it, where

$$\frac{1}{u} + \frac{1}{v} = \frac{1}{f}, \qquad v = \frac{fu}{u - f}.$$

Focused at infinity, $v = f$, and the lens stands exactly its focal length from the sensor. Focused at three metres, a 50 mm lens stands 50.85 mm away. At one metre, 52.63 mm. At half a metre, 55.56 mm. At thirty centimetres, 60.00 mm. The extension grows slowly at long distances and quickly at short ones, and at a magnification of one — the subject as large on the sensor as in life — it reaches twice the focal length.

Nothing about the picture’s geometry cares how the lens got there. The sensor records rays that pass through the lens’s centre of projection and meet the sensor plane, and that centre now stands $v$ from the sensor. The picture is a pinhole picture with principal distance $v$ — not $f$ — and every quantity computed from a focal length in a picture is really computed from $v$.

The angle narrows #

A sensor 36 mm wide behind a lens $v$ away subtends $2\arctan(18/v)$. At infinity that is 39.60°. Focused at three metres the angle is 38.99°, at one metre 37.76°, at half a metre 35.90°. Focusing from infinity to half a metre narrows the picture by nearly four degrees — the frame that held a doorway at infinity holds a tenth less of it at close focus, from the same place.

Cinematographers call this focus breathing, and treat it as a defect because a focus pull changes the framing during a shot. From the point of view of the picture it is not a defect at all. It is a change of camera: the lens has moved, the principal distance has changed, and the picture is exactly the correct pinhole picture of a camera with that principal distance. What changed is only the gap between the number engraved on the barrel and the camera actually in use.

That makes it the mirror image of stepping closer is not zooming. There, changing the focal length left every ratio between things in the picture untouched while moving the camera changed them. Here, turning the focus ring does change the principal distance — it is a zoom, a small one, with no change of position — and so it leaves every ratio between things in the picture untouched and changes only the angle covered. A reader comparing a close-focused picture with one focused at infinity from the same place sees a crop, not a change of perspective.

What the picture reports about its own camera #

This site’s standing habit is to recover a camera from its picture and compare. Recovering the camera reads the focal length out of a photograph from the vanishing points of perpendicular directions, with no knowledge of the lens at all: two horizontal directions at right angles vanish at two points, and the focal length is the square root of minus the product of their distances from the principal point along the horizon.

Run that recovery on a picture made by a 50 mm lens focused at each distance, and it returns the image distance every time: 50.847 mm at three metres, 52.632 mm at one, 55.556 mm at half a metre. It never returns fifty millimetres except at infinity.

That is exactly what the recovery should do, and it is worth saying plainly because the conclusion drawn from a mismatch is usually the wrong one. A calibration that reports 55.6 mm for a lens engraved 50 mm, from a picture taken at half a metre, is not wrong about the lens and not suffering from noise. It has measured the camera in use, which is a different camera from the one on the engraving, and the difference is the focus setting.

The corollary is the practical one. Photographs commonly carry a record of the lens’s focal length alongside them, and that record is the engraving, not the principal distance. A measurement pipeline that takes the recorded focal length as its camera, for a photograph taken at close focus, starts from a camera 11% wrong at half a metre — and every quantity built on that camera inherits the error.

A right angle, read with the wrong camera #

How large that error is depends on what is measured, and angles are where it shows most directly.

Photograph two horizontal directions at right angles, each 45° off the axis. Their vanishing points sit at plus and minus the principal distance along the horizon — at $\pm v$. Read those vanishing points back through the engraved focal length, and the two directions come back as $(\pm v, 0, f)$, which make an angle of $2\arctan(v/f)$ rather than 90°.

Focused at three metres, the right angle reads 90.96°. At one metre, 92.94°. At half a metre, 96.03°. At thirty centimetres, 100.39°. And at a magnification of one, where $v = 2f$, it would read 126.87°. An angle measured off a close-focused photograph of a room, with the engraved focal length, is wrong by several degrees at the distances a room is photographed from, and the error is systematic: every right angle in the picture is read too wide, by the same amount.

The error runs the way it does because the picture was made by a camera with a longer principal distance than the reading assumes. Reading it with a shorter one treats every off-axis mark as further off the axis in angle than it really was, so directions that were 45° apart on each side of the axis are read as more than 45° off it.

The same extension dims the picture #

The engraving that misdescribes the angle misdescribes the light as well, and for the same reason.

An f-number is the focal length divided by the diameter of the entrance pupil, and like the focal length it is engraved for a lens focused at infinity. What sets how much light reaches a point on the sensor is the cone of rays converging on it, and that cone’s steepness is the pupil’s diameter against the distance the lens actually stands from the sensor — $v$, not $f$. So a lens focused close works at an f-number of $N\,v/f$. An f/2.8 lens focused at half a metre works at f/3.11, and the light on the sensor falls by $(v/f)^{2}$, which is 0.30 of a stop.

This is the familiar correction for close-up photography, where a lens pushed out on bellows needs a longer exposure, and it has exactly the geometry of the angle narrowing. The same ratio $v/f$ that shrinks the width covered by a tenth at half a metre dims the picture by nearly a fifth. A meter reading the light through the lens sees the loss and compensates without anyone knowing. A separate meter, or an exposure calculated from the engraving, does not.

At ordinary distances the effect is small, and it is worth knowing how small. The sharp band is a decision worked its depth of field for an f/2.8 lens focused at three metres, where the lens stands 50.85 mm out and works at f/2.85 — under two per cent from its engraving, well inside every other approximation in that arithmetic. At half a metre the engraving is 11% out, and at thirty centimetres, where the lens stands at 60 mm, it is 20% out.

Where a reader should stand #

This site’s central claim about any picture is that it is correct from exactly one distance, and a focal length is not an angle wrote that distance in the sensor’s terms: a print $W_{d}$ wide made from a sensor $w$ wide is correct from $d = f\,W_{d}/w$. Written with the principal distance the picture actually has, that is $d = v\,W_{d}/w$.

For a 160 mm print from full frame, the picture made at infinity is correct from 222.2 mm. The same lens focused at half a metre makes a picture correct from 246.9 mm — eleven per cent further back, a difference of two and a half centimetres at reading distance. The point to stand at is set by the picture, and the picture is set by where the lens stood, so a close-focused photograph asks to be held further from the eye than a distant one taken with the same lens.

The same factor runs through every link of the chain. The screen sets the distance put a 50 mm frame at 9 cm from a phone, 83 cm from a monitor and 16.7 m from a cinema seat; every one of those distances is proportional to the principal distance, so every one of them is 11% further for a picture taken at half a metre. In practice nobody holds a print at either distance, and standing in the wrong place is the ordinary state of every reader. What the number establishes is that the correct distance is not a property of the lens; it moves with a ring the photographer turns for a different reason entirely.

A longer lens breathes more at the same distance #

The size of the effect depends on how far the subject is in units of the focal length, which means a longer lens at a given subject distance changes its principal distance by more.

A 100 mm lens focused at half a metre stands 125 mm from the sensor: the picture’s principal distance is a quarter longer than the engraving, where the 50 mm lens at the same distance was 11% longer. The extension is $v - f = f^{2}/(u - f)$, so doubling the focal length at a fixed subject distance roughly quadruples it. Portrait lenses, which are long and are used close, are the lenses whose engraved focal length is furthest from the camera actually in use.

The converse is why a wide lens is the one whose engraving can be trusted. A 24 mm lens focused at half a metre stands 25.21 mm from the sensor, five per cent beyond its engraving where the 50 mm lens was eleven; focused at three metres it stands 24.19 mm out, under one per cent. As a fraction of the engraving the extension is $f/(u - f)$, which is close to the focal length divided by the subject distance, so what matters is not how far away the subject is but how many focal lengths away. A wide lens photographing a room is within a few per cent of its engraved camera. A portrait lens photographing a face from the same spot is not — and the two are routinely used at the same distance, sometimes on the same afternoon, with the same measurement pipeline reading both pictures.

When the extension is too small to see #

The recovery returns the principal distance exactly on a perfect picture. On a real one it returns it with an error, and the question of whether focus shows in a recovered camera is a question of which is larger.

The extension falls off quickly with distance. For the 50 mm lens it is 11.1% at half a metre, 5.26% at one metre, 1.69% at three, 1.01% at five and 0.50% at ten. A floor with a referent measured how precisely a vanishing-point recovery can return a focal length: through a pinhole, with enough points, to a few hundredths of a per cent; through a lens with a barrel coefficient of −0.05, no better than a floor of 0.72% of the focal length, because the distortion the recovery does not model sets a limit no number of points removes.

Put those side by side. On a picture from a perfect pinhole, the recovered camera would show the focus setting at every distance measured here. On a picture from the mildly distorted lens that essay measured, focus at one metre is seven times the floor and unmistakable; focus at five metres is barely distinguishable from it; focus at ten metres is inside it, and the recovered camera says nothing about where the lens was focused. Past a few metres, focus is real and invisible in the recovery, and the engraving is as good a camera as the picture can give.

What this is and is not a claim about #

It is exact for a lens that focuses by moving as a unit. That model is a thin lens that moves; its principal distance is $v = fu/(u - f)$ with no approximation. Many real lenses focus differently — internal focusing moves groups inside the barrel and changes the lens’s own focal length as it focuses, which can shorten the principal distance at close focus rather than lengthening it, or hold it nearly constant by design. The direction and size of the effect in any particular lens is a question about that lens’s design, and nothing here measures one.

It does not depend on the sensor. Every number above is for full frame, but the principal distance is a property of the lens and the focus setting. On a smaller sensor the angles are smaller and the ratios between them — and the right angle’s misreading, and the viewing distance’s percentage change — are the same.

It moves the centre of projection too. A lens that focuses by moving as a whole carries its entrance pupil with it, 5.56 mm further from the sensor at half a metre, and the eye is a place, not a point noted that this is why panorama heads are set up at the focus distance the panorama will be shot at. On a wide lens the pupil also walks with the angle, which is a separate effect of comparable size.

And the effect is not an aberration. A lens focused close makes a sharp, geometrically exact pinhole picture. The only thing wrong is the description of the camera, and the only fix is to use the principal distance rather than the engraving, which a recovery from the picture does automatically — where the extension is large enough for the recovery to see.

Still open: whether a picture carries its magnification #

The extension that lengthens the principal distance is the magnification in another form: $v/u$ is the ratio of an object’s image size to its real size, 0.11 at half a metre for a 50 mm lens and 1 at twice its focal length. A picture carries its principal distance, which a recovery reads; the open question is whether it therefore carries its magnification, and with it the subject’s distance — which a single picture has so far never been able to give, because the one thing a single view cannot give is a size. A direct test recovers the principal distance from a close-focused picture, combines it with the engraved focal length, solves the thin-lens equation for the distance the lens was focused at, and measures how precisely that returns the subject’s distance — and therefore a scale — as the focus distance grows, finding where the extension sinks below the recovery’s own error and the old ambiguity returns.