The hole a scene actually sees

The eye is a place, not a point established that a camera has a definite centre of projection, that rotating about anywhere else costs parallax falling as one over the distance, and that the centre is called the entrance pupil. It did not say where the entrance pupil is, and the omission matters because the obvious answer is wrong.

The obvious answer is the aperture — the hole visible from in front of the lens, whose blades can be counted. It is not.

The smallest model that shows it #

A thin lens of focal length f, with an aperture stop a distance s behind it. That is the whole model, and it is enough because the entrance pupil is the image of the stop through whatever glass is in front of it, and here the glass in front of it is the lens.

Imaging the stop backwards through the lens: an object at distance a in front images at s behind when 1/a + 1/s = 1/f, so a = fs/(s − f) and the magnification is f/(s − f).

At f = 50 mm and s = 18 mm those come out negative: a = −28.125 mm, which means the pupil is a virtual image sitting 28.1 millimetres behind the lens, and the magnification is −1.5625, so it is 1.56 times the size of the stop.

So the hole a scene sees is in a different place from the stop and is bigger than it. Neither of those is a defect; both are what imaging a stop through glass does.

The chief rays, traced #

The claim is not a definition. It is that the chief rays are concurrent at the computed point — for every object distance — which is the statement that the picture is a projection from there.

A chief ray is the one from a world point that passes through the centre of the stop. Traced through the lens, four object distances from 0.9 metres to 60 and three lateral positions apiece, the crossings of their object-side extensions agree to 3.6 × 10⁻¹⁵ millimetres and sit at the entrance pupil’s own computed position to the same order.

That is the round trip this collection runs on everything: compute the object one way, recover it another, compare. The pupil is computed from an imaging formula and recovered from a bundle of rays, and the two agree at the arithmetic floor.

Two pupils, and which one the geometry uses #

A lens has two, and they do different jobs.

The entrance pupil is the stop imaged by the glass in front of it, seen from the scene. It is the centre of projection: every chief ray passes through it, so the geometry of the picture — which is the whole subject of this collection — is a pinhole projection from that point.

The exit pupil is the stop imaged by the glass behind it, seen from the sensor. It sets the cone of light arriving at each sensor point, so it decides the blur disc’s size, the angle of incidence on the microlenses, and how much light reaches a corner.

So the two rungs of this field’s aperture work belong to different pupils, which is a division a pupil sees around an edge leans on without stating. The centre has an area measures a patch whose size comes from the exit pupil and whose centre comes from the entrance pupil, and in the thin-lens model those are the same object, which is why the model can get away with one.

Why this is not pedantry #

Three places where the difference has consequences a reader can measure.

Panoramic rotation. Rotating a camera about the entrance pupil is what makes a stitched panorama free of parallax; rotating about the stop, the tripod screw or the front element is what makes it ghost. The pivot that is not the eye and the parallax a stitcher cannot remove measure exactly that, and the offset in this model — ten millimetres — is the order of magnitude that matters at close range.

Depth-of-field arithmetic. The formulas quoted in the sharp band is a decision are written for a symmetric lens, where the two pupils are the same size. For an asymmetric one — a retrofocus wide angle, a telephoto — the pupil magnification enters, and the near and far limits shift. At the pupil magnification here, 1.56, the correction is a few per cent, which is inside most people’s criterion and outside a careful macro calculation.

And calibration. The recovered centre of projection in a photogrammetric calibration is the entrance pupil, not any physical part of the lens. Trying to measure the offset with a ruler against the barrel is measuring the wrong thing, and the discrepancy is exactly the ten millimetres above — a systematic of the same size as the shift the principal point is not the centre measures at the other end of the same instrument.

How fast the pupil moves, and why it runs away #

The figures walk the stop from 8 mm behind the glass to 36, and the captions say the pupil moves with it and by more. Both quantities have one-line forms, and writing them down turns “by more” into a number and explains why one end of the sweep is quiet and the other is not.

Put the lens at the origin and the stop $s$ behind it. The entrance pupil sits $fs/(f-s)$ behind the glass, so its offset from the stop is

$$\frac{fs}{f-s} - s = \frac{s^{2}}{f-s},$$

and the pupil magnification is $f/(f-s)$. The two are the same factor applied to different things: the pupil’s distance behind the glass is the stop’s setback times $f/(f-s)$, and its diameter is the stop’s diameter times the same. Position and size are one fact.

At $f = 50$ mm those give, across the sweep the figures draw:

The offset is quadratic in the setback at the near end and hyperbolic at the far end, and the derivative of the pupil’s own position is $f^{2}/(f-s)^{2}$ — 2.44 mm of pupil movement per millimetre of stop movement at the setting the numbers are quoted from. That is the “by more”, and it is worth having as a number because it says the pupil’s position is a more delicate quantity than the stop’s, not a less delicate one: whatever moves the stop by a little moves the centre of projection by more than twice as much.

The runaway at $s \to f$ is the telecentric case arriving. As the stop approaches the front focal plane the magnification and the offset both go to infinity, the chief rays leave the lens parallel, and there is no finite point the picture is a projection from. That limit is not a numerical accident to be guarded against; it is a lens design, and the model refuses it rather than returning a large number.

Two practical readings follow. The offset is a design consequence rather than a nuisance: a lens with its stop well back has a distant pupil by arithmetic, which is why a telephoto’s rotation point is nowhere near its barrel’s middle. And the offset is what parallax is proportional to, so its consequences scale as offset over subject distance: 10.1 mm against a subject at one metre is one part in a hundred, which is a visible seam in a stitched panorama and nothing at all at fifty metres. That is the same one-over-the-distance law the previous rung measured, with the numerator now computed from the lens rather than assumed.

Where the pupil actually sits, in real lenses #

Worth some concrete cases, because the model above is deliberately minimal and its answer — behind the lens — is one of several possibilities.

A normal lens has its entrance pupil somewhere inside the barrel, usually a little in front of the stop. A retrofocus wide angle has it well forward, sometimes ahead of the front element, which is why such lenses need to be rotated about a point in mid-air. A telephoto has it well back. And a lens whose stop sits exactly at the front focal plane is telecentric: the stop images to infinity, there is no finite entrance pupil, and the projection is orthographic rather than perspective.

That last case is the one this collection has a whole field about. A telecentric lens is a picture with no eye built in glass — its rays are parallel, no distance in a room makes the picture correct, and the model here refuses it explicitly rather than returning a large number.

How to find it on a real lens #

Two methods, and the second is the one worth using.

By parallax. Put two objects at different distances in line, rotate the camera about a trial point, and look for the setting at which they stay in line. That is the standard nodal-slide procedure and it works, and it is a nulling measurement, so it is as precise as the null is sharp.

By looking. Point the lens at a ruler, look into the front of it from the scene’s side, and note where the image of the stop appears to sit against the barrel. That image is the entrance pupil, at its actual size and its actual place, and this is the measurement the model above computes. It also makes the magnification visible: the hole looks bigger than the stop is, by the factor the arithmetic gives.

The second is worth preferring because it measures the object directly rather than a consequence of it, which is this collection’s usual preference — and because it makes it obvious that the thing being located is an image rather than a piece of the lens, which is the whole point.

The name that is wrong #

The point is widely called the nodal point, and it is not one.

Nodal points are a real and different pair: the two points at which a ray entering aimed at one leaves aimed at the other with the same angle. They are properties of the refracting surfaces and they coincide with the principal points when the medium is the same on both sides. They have nothing to do with the aperture stop and are not where a camera should be rotated.

The eye is a place, not a point records the confusion; what this rung adds is why the right object is easy to miss. The nodal points are properties of the glass and can be looked up; the entrance pupil is an image of the stop and moves when the aperture ring does not — but it does move when the lens is focused, because focusing moves elements and therefore moves what is in front of the stop.

So the correct rotation point is a function of the focus setting, which is a fact that a nodal slide’s single detent quietly denies.

What moves when the aperture ring turns #

A question the model answers cleanly and that is easy to get backwards.

Turning the aperture ring changes the stop’s diameter and not its position, so the entrance pupil stays where it is and changes size by the same magnification. The centre of projection does not move with the f-number.

That is why a panoramic head’s setting does not need adjusting between apertures, and it is also why the parallax nulling measurement is best done wide open — a large pupil makes the depth of field short and the null sharp, and it locates the same point a small pupil would.

What does move it is focusing, because focusing moves elements and therefore changes the glass in front of the stop. On an internally focusing lens the movement can be several millimetres across the focus range, which is why a nodal slide calibrated at infinity is wrong at close range — and close range is exactly where parallax is largest.

So the practical rule has two halves that pull in opposite directions: the offset matters most at close subject distances, and that is where the calibration is least likely to hold.

What the model does not have #

Stated plainly, because a two-parameter model of a lens is a small object.

It has one refracting element, so it cannot show the entrance and exit pupils being different sizes, which is the pupil magnification that a real asymmetric lens has and that enters the depth-of-field correction above. It has no aberrations, so its pupil is a perfect image of the stop rather than one that shifts and distorts with field angle. And it has no thickness, so the distinction between principal planes does not arise.

What it does have is the one thing this essay is about: a stop that is not at the centre of projection, and a centre of projection that is an image rather than an object. Those two facts are what a thin lens with a displaced stop is the minimal carrier of, and everything else on the list is a refinement of a picture that is already qualitatively right.

The pupil and the picture’s own distance #

One connection back to this collection’s premise, because it is the reason the entrance pupil is worth this much attention.

Every figure here prints the distance a reader must be at for the picture to be a correct projection, computed from the focal length scaled to the width the figure is displayed at. That number is a property of the picture, and it is a distance from the reader’s eye to the page.

Its counterpart on the camera’s side is a distance from the entrance pupil to the scene. The two are the two ends of one projection: the picture is correct when the reader’s eye stands to the page as the entrance pupil stood to the world. So a wrong idea about where the pupil is does not affect the reader’s number at all — that comes out of the focal length — and it affects everything about reproducing the camera’s position.

Which is a clean division of labour worth naming. The reader’s distance is a fact about the picture and needs no knowledge of the lens; the camera’s position is a fact about the instrument and needs exactly this.

What the sign of the pupil’s position means #

The model returns a distance that comes out negative on this arrangement, and reading the sign correctly is the difference between a pupil in front of the lens and one behind it.

a = fs/(s − f) is positive — a real point in front of the glass — when the stop sits further behind the lens than its focal length. It is negative when the stop is nearer than that, which is the ordinary camera arrangement, and the pupil is then a virtual image: the rays only appear to come from it, and nothing is there.

That is not a defect and not an approximation. A virtual centre of projection is as good as a real one for every geometric purpose, because the chief rays’ extensions are concurrent there and a projection is a statement about lines rather than about light arriving somewhere. It does mean the point can be inside the barrel, behind the sensor, or in mid-air in front of the lens depending on the design, and a photographer looking for it on a nodal slide finds it wherever the parallax nulls.

The short version #

A picture is a projection from the entrance pupil, which is the aperture stop imaged by the glass in front of it — not the stop itself, and not any physical part of the lens.

On a 50 mm thin lens with its stop 18 millimetres behind, the entrance pupil sits 28.1 millimetres behind the glass, 10.1 millimetres from the stop, at 1.56 times its size. The chief rays from four object distances cross the axis there to 3.6 × 10⁻¹⁵ millimetres, which is what says the picture is a projection from that point and not from the stop.

The exit pupil is the other one, and it is what decides the blur disc’s size rather than the picture’s geometry. Rotating a camera about the stop, the screw or the front element leaves parallax; rotating it about the entrance pupil does not.