A focal length is not an angle

Every camera on this site has had a focal length in pixels. That is deliberate, and its own docstring says why: everything is a ratio until a physical size is named, and naming one prematurely disguises which claims are about geometry and which are about a particular piece of equipment.

This field names them. And the first thing that follows is that the phrase everybody uses — a 50 mm lens — is an incomplete specification of a picture, in a way that is not a quibble.

$$\theta = 2\arctan\!\left(\frac{w}{2f}\right)$$

One line, and the $w$ in it is the width of the sensor. A 50 mm lens is 39.6° across a full-frame sensor, 26.6° across APS-C, 19.6° across Micro Four Thirds, and 8.7° across the sensor in a phone. Those are four different pictures of the same scene from the same place.

Where the sensor came in #

It is worth being clear about what has changed, because the site’s machinery has not.

This site’s camera takes a focal length in pixels or a field of view in degrees and derives the other, and refuses to be given both, on the stated ground that a figure whose stated field of view does not match its focal length “is exactly the kind of thing this site is about”. The pixels there are the picture’s pixels, and the picture is whatever the figure is drawn at.

A real camera has one more rectangle in the chain: the sensor. The focal length is a physical distance from the lens’s rear nodal point to the sensor, and the sensor has a physical width, and the ratio of the two is the only thing that decides the angle. Which is exactly the same statement the site’s own camera makes with $f/W$, with a millimetre in place of a pixel.

So nothing has been added. What has happened is that a ratio the site has always used has been given two named ends, and giving them names makes visible something that was invisible while both ends were the same abstract picture.

The format cancels — and this is the site’s own claim #

Here is the result worth having, and it is a statement about where the reader has to stand rather than about photography.

This site’s central claim is that a picture is correct from exactly one distance, and that the distance is computable: scale the picture to a display width $W_d$ and the focal length scales with it, and the scaled focal length is the viewing distance. In the sensor’s terms,

$$d = f \cdot \frac{W_d}{w}$$

and every symbol on the right is a length in millimetres.

Now take two cameras matched on angle rather than on focal length — a 50 mm on full frame and the 32.8 mm that gives the same 39.6° on APS-C. Their prints, at the same size, are correct from the same distance: 222.2 mm for a 160 mm print, both ways, to nine digits.

They must be, because $d$ depends on $f/w$ and the two setups were chosen to have the same $f/w$. But it is worth checking rather than deducing, because it is the kind of statement that is easy to write down and easy to get backwards, and because the check discriminates: the same focal length on the smaller sensor gives 339 mm rather than 222, so the assertion is not satisfied by a function that ignores its arguments.

That is a more interesting result than it first looks. A photograph does not carry its format. Nothing about a print says what size the sensor behind it was, and nothing needs to: the only quantity that survives into the picture is the angle, and the angle is what the station point is computed from. The whole vocabulary of formats and crop factors is about how to achieve an angle with the equipment in hand, and it disappears the moment the picture exists.

Crop factor is a diagonal, and it disagrees with itself #

The trade’s way of handling this is the crop factor: the ratio of the 35 mm frame’s diagonal to the sensor’s, used to convert focal lengths into “equivalents”.

It is a genuinely useful number and it is exactly right only on the diagonal. Micro Four Thirds has a crop factor of 2.00 on the diagonal, and:

The spread is because Micro Four Thirds is 4:3 and the 35 mm frame is 3:2. A “50 mm equivalent” lens on that format gives the same diagonal angle and a narrower horizontal one and a wider vertical one, and which of those a photographer cares about depends entirely on what is being photographed.

This is not a criticism of the convention — one number cannot reconcile two aspect ratios and the diagonal is the sensible compromise. It is a reason to be careful with the word equivalent, which is doing more work than it can carry. The gate asserts the disagreement rather than the agreement: the crop factor is computed on all three axes and the check requires them to differ substantially, because a crop factor that agreed on all three axes would mean the two formats had the same shape and the whole subject would be trivial.

The spread depends only on the two shapes #

The three crop factors for Micro Four Thirds — 1.999, 2.081 and 1.846 — are quoted as a property of that format, and they are not. Divide each by the diagonal figure and what is left depends on nothing but the two aspect ratios.

For a format of aspect ratio $a = w/h$, the width is $a/\sqrt{1+a^{2}}$ of the diagonal and the height is $1/\sqrt{1+a^{2}}$. So comparing a format of ratio $a_{2}$ against the 35 mm frame’s $a_{1} = 1.5$:

$$\frac{\text{width factor}}{\text{diagonal factor}} = \frac{a_{1}/\sqrt{1+a_{1}^{2}}}{a_{2}/\sqrt{1+a_{2}^{2}}}, \qquad \frac{\text{height factor}}{\text{diagonal factor}} = \frac{\sqrt{1+a_{2}^{2}}}{\sqrt{1+a_{1}^{2}}}.$$

For 4:3 against 3:2 those come to 1.041 and 0.923 — which reproduce 2.081 and 1.846 from the diagonal’s 1.999 exactly, and would do so for a sensor of any size with that shape. The sensor’s dimensions cancel out of the disagreement entirely. A crop factor’s spread is a fact about rectangles, not about cameras.

That makes the table extendable without measuring anything, and the extension is where the convention hurts most:

The widescreen case is worse than the 4:3 case, which is the opposite of the usual intuition, because 16:9 is further from 3:2 in shape than 4:3 is. A “50 mm equivalent” on a 16:9 sensor is nineteen per cent narrower horizontally than it is vertically, relative to what the equivalence promises, and video is where the phrase is used most casually.

Two practical readings follow, and the second is the one that decides whether the convention can be used at all.

The equivalence is exact for a subject that fills the diagonal and for nothing else. A landscape framed edge to edge on 4:3 gets four per cent less angle than the “equivalent” says; a portrait framed top to bottom gets eight per cent more. Neither is large and both are systematic, and a photographer choosing a lens to match a remembered framing is matching the wrong axis about half the time.

There is a third reading, and it is the one a reader of this collection will want. The spread is a pure function of two shapes, so it is computable before any equipment is bought and it is the same for every lens — which puts it in the same class as the ratio a dome port returns: a quantity whose dimensioned parts have cancelled, leaving something that transfers between instruments unchanged. A number of that kind is worth learning once, and a number that has to be re-measured per camera is worth looking up each time.

And the spread vanishes exactly when the shapes agree. Two formats of the same aspect ratio have one crop factor, correct on every axis, and the word equivalent means what it says. Full frame against APS-C is nearly that case — 3:2 against 3:2 — which is why the convention feels reliable to photographers who have only ever moved between those two, and why it starts misbehaving the moment a phone, a cinema camera or a square crop enters.

None of this touches the essay’s main result, which is that the picture does not carry its format at all. Once the photograph exists, its own angle fixes where to stand and every rectangle that produced it is forgotten. The crop factor is a tool for reaching an angle with equipment in hand, and the spread above is the price of expressing a two-dimensional fact as one number — the same price a crop factor’s cousin pays whenever a quantity that varies across a frame is quoted as a single figure.

What the sensor does not change #

Three things, each of which is regularly attributed to sensor size and none of which is a consequence of it.

Perspective. The relationship between near and far things in a picture is set by where the camera is, and by nothing else. Two cameras at the same place with different sensors produce two pictures that are crops of each other, so every ratio between things in the scene is identical in both. That is the next essay’s whole subject, and the invariance is asserted to machine precision.

The viewing distance for a given angle. Established above: it depends on $f/w$, so any two setups with the same angle agree.

What a straight line does. A rectilinear lens draws straight lines straight, and a crop of a rectilinear picture is a rectilinear picture. Sensor size cannot bend anything.

What sensor size does change is the angle for a given lens, the depth of field for a given angle and aperture, and how much light each photosite collects. The first is this essay. The second and third are optics and photometry, which this site does not own and does not take a position on — the honest limit stated wherever it matters is that this site computes the geometry of pictures.

The phrase “wide angle” belongs to the angle #

There is a small vocabulary problem worth clearing up, because it makes an argument the site has already made harder to state.

Wide-angle is not distortion argues that the stretch at the edge of a wide photograph is not a defect of the lens but a correct consequence of projecting a large angle onto a flat plane — and that it disappears entirely when the picture is viewed from the point it is correct from, which for a wide picture is very close.

That argument is about the angle, and the word “wide-angle” says so. But the trade’s usage attaches it to the focal length — a “wide-angle lens” is one with a short focal length — which is a property of the glass and not of the picture. A 24 mm lens on a phone sensor is a substantial telephoto, subtending about 18°.

So the rule for reading any claim in this area: if it is about the picture, it is about the angle; if it is about the equipment, it is about the focal length. The two are related by one line of arithmetic and the sensor’s width, and confusing them produces claims that are true of one camera and false of the next.

Which dimension the angle is measured across #

A sensor is a rectangle and has three obvious widths — horizontal, vertical and diagonal — so “the field of view” is three numbers, and the trade quotes whichever suits.

For full frame at 50 mm: 39.6° horizontal, 27.0° vertical, 46.8° diagonal. Cinema quotes horizontal, still photography usually quotes diagonal, and the specification of a rendered picture is almost always horizontal — with a vertical variant that appears whenever somebody wants the picture to behave sensibly when the window is resized.

That last case is worth a sentence because it is a genuine design decision hiding as a convention. A picture whose horizontal angle is fixed shows the same width whatever the window’s shape, so making a window taller adds scene at the top and bottom. A picture whose vertical angle is fixed does the reverse. Neither is more correct; what is wrong is not saying which, because a viewer who resizes a window and finds the framing changed has no way to tell whether the picture is a wider view or a closer one.

For this site the relevant one is horizontal, because the site’s viewing-distance claim is stated against the width a figure is displayed at, and the arithmetic has to use the same dimension at both ends. Mixing them is the same category of error as the crop factor’s — a ratio taken on one axis applied to a length on another — and it is worth naming for the same reason.

The chain, and where it goes next #

Everything so far concerns one end of the chain. The full chain has three links and each contributes one ratio:

$$\text{scene} \xrightarrow{\ f/w\ } \text{picture} \xrightarrow{\ W_d/w\ } \text{print or screen} \xrightarrow{\ d\ } \text{reader}$$

The first ratio decides what is in the picture. The second decides how large it is shown. The product decides where the reader has to be, and the screen field is about the second and third links, where the answer turns out to be that nobody is standing there.

What makes it worth doing as a chain rather than as a single number is that the two ends are decided by different people, at different times, with no communication between them. The photographer picks $f$ and $w$; the reader picks $W_d$ and $d$, usually without knowing that either is a choice. Nothing in between records what was intended.

The focal length is not quite a distance either #

One qualification, made here rather than left implicit, because the whole essay treats $f$ as a length and a real lens complicates that.

The formula $\theta = 2\arctan(w/2f)$ is a statement about a pinhole: rays converge on a point $f$ in front of the sensor. A lens is not a point, and its focal length is defined against a nodal point whose position depends on the design and is regularly nowhere near the middle of the glass. A retrofocus wide-angle lens has its nodal point in front of the front element; a telephoto has it behind the rear one.

None of that changes the arithmetic, because the nodal point is exactly the point the geometry needs and its position inside the barrel is irrelevant to the picture. What it changes is the intuition that a “50 mm lens” is 50 mm long, which almost none of them are.

And it matters in one place this site has already measured. The eye is a place, not a point shows that the centre a panorama must be rotated about is the entrance pupil — a specific point which is neither the front element nor the sensor nor the middle of the barrel — and that rotating about the wrong one turns a rotating camera into a translating one, with parallax that no alignment removes. So the question “where exactly is the camera” has an answer, it is not where anybody assumes, and it is the same question the focal length’s definition depends on.

Two conveniences that are worth keeping #

Having spent the essay separating the focal length from the angle, it is only fair to say why the trade quotes the focal length and not the angle. There are two reasons and both are good.

A focal length is a property of one object. A lens has one, permanently, whoever puts it on whatever camera. An angle is a property of a pairing, and quoting it requires naming both halves every time.

And focal length composes. Teleconverters, extension tubes and focal reducers multiply it. Two lenses of the same focal length behave identically as regards magnification whatever else differs. Angles do none of that cleanly.

So the convention is right for the person holding the equipment and wrong for the person looking at the picture, and this site is written for the second one. Which is the same split the whole site runs on — a construction is described from the draughtsman’s side and measured from the viewer’s — and it is worth naming rather than treating the trade’s usage as sloppiness. It is precise about a different thing.