The centre of the picture is not the centre of the paper

Almost every perspective construction ever taught puts one number on a sheet of paper before it computes anything: the middle. The horizon goes halfway up it if the picture is level. The viewing distance is stepped out from it along a normal. The centre of vision is marked at it, and the whole cone of vision is drawn around that mark. All of that is a statement about a piece of paper, and it is smuggled in as though it were a statement about a projection.

The picture’s optical centre — the principal point — is where the camera’s principal ray, the one it looks straight along, meets the image plane. A lens decides it, and nothing on the page does. The middle of the sheet is decided by whoever last trimmed it. On the rectangle a camera exposes the two happen to coincide, which is why the confusion is invisible and why every account can afford to be silent about it. A crop moves one of them and cannot move the other, and from that moment the horizon is not the half height, the normal through the middle is not the ray the picture was made along, and nothing on the sheet says so.

What follows measures the gap on one photograph, at one trim, and then asks what it costs a reader who does not know the trim happened — which is the ordinary condition of anyone holding a printed picture.

A crop is a translation, and a translation has no focal length in it #

The arithmetic is short enough to be worth doing rather than asserting.

A pinhole camera’s intrinsics are a focal length $f$ and a principal point $(c_x, c_y)$: a world direction reaches the image at $f$ times its slope, offset by the principal point. Removing $a$ pixels from the left of the image and $b$ from the top renumbers every surviving pixel, so the principal point becomes $(c_x - a,\, c_y - b)$ in the new coordinates and $f$ is written down again unchanged. Removing pixels from the right and the bottom does not renumber anything at all; it moves the rectangle, and with it the point a reader will call the middle.

Either way the transformation on the image plane is a translation, and a translation carries no scale. The focal length is a scale, so it survives. The principal point is a position, so it does not.

The figure at the head of this essay is that statement drawn. Before the trim the two marks are one mark. After a trim of a fifth off the right and a fifth off the bottom of a picture 690 across they are 81.3 pixels apart — 69 across and 43 down — and the focal length is the number it always was, to every digit a double carries.

That asymmetry is the whole subject. A quantity a construction cares about a great deal has moved, and a quantity it also cares about has not, and there is no way to tell which is which by looking. Both are invisible on a print. The difference is that one of them can be recovered from the picture’s own content, as the last section but two shows, and the other never needed recovering.

It is worth noticing what this rules out immediately. The crop factor a photographer uses to compare formats is a ratio of sensor sizes, and it converts a focal length into an equivalent angle of view. It says nothing about where the principal point is, because a sensor smaller than the image circle is not necessarily concentric with it. A cropped picture and a picture from a smaller sensor share the arithmetic of angle and differ entirely in whether the optical axis still hits the middle.

The horizon is not the half height, and the taught rule is not wrong #

The rule that a level picture puts its horizon halfway up the paper is exact, and it is exact about the wrong rectangle.

For a camera whose optical axis is horizontal, the horizon is the image of every horizontal direction, and it passes through the principal point. On the exposed rectangle the principal point is the middle, so the horizon is the half height, and the rule holds to the last bit of the arithmetic. Trim the rectangle asymmetrically and the horizon has not moved on the picture at all. What has moved is the denominator.

Two things follow from the shape of that pair of lines, and the second is the more useful.

The first is the size. A fifth is a modest trim, the kind a printer applies to fit a plate to a page, and it moves the horizon by an eighth of the sheet’s height. Anyone reading the picture backwards — taking the horizon’s height on the paper as the eye height in the scene, which is the standard inference and the one the essay on the horizon and the fraction sets out — is now reading an eye height that is out by a quarter.

The second is that the control is flat. Removing the same amount of picture symmetrically changes nothing, so the damage is not done by losing picture. It is done by losing more of one side than the other, which is a fact about the scissors and not about the camera. That is what makes this a property of the board rather than of the projection, and it is why no amount of care with the lens prevents it.

The rule survives, restated: the horizon passes through the principal point. It is only the identification of the principal point with the middle of the sheet that fails, and that identification was never part of the geometry.

What the trim costs a reader, in centimetres #

The site’s signature measurement is the distance a picture is correct from: scale the focal length to the width the picture is displayed at, and the answer is a station point in centimetres that a reader could actually occupy. The essay that established it computes the distance and stops there, because on an untrimmed picture the direction is not in question — the correct eye is on the normal to the sheet through the middle.

On a trimmed one the distance and the direction come apart, and only the distance is what a reader guesses.

The direction is the part worth dwelling on, because it is the part that has no analogue in the untrimmed case. A picture’s correct station is not merely at a distance; it is on the principal ray produced backwards out of the sheet, and on an untrimmed print that ray leaves the paper through the middle, so nobody has ever had to think about which normal it is. On a trimmed print there are two candidate normals a body’s length apart at ordinary reading distance, and only one of them is the principal ray.

Enlarging the trimmed picture back to a full page multiplies its focal length in units of print width, so the reading distance grows from 22 cm to 27.5 — that is the honest half of what a crop does, and it is the half every account already describes as an increase in effective focal length. The other half is the offset, and the two are the same translation seen twice.

Four point nine degrees is a small angle and it is not a negligible one. It is roughly the angle a reader’s head turns in shifting attention from one side of a print to the other, so it cannot be felt; and it is well inside the range over which standing in the wrong place measures a real change in the reconstructed scene. The consequence is not that the picture looks wrong. It is that a reader who works from the picture — measuring an angle, judging a proportion, deciding where a wall runs — is doing so from a station the picture did not draw.

And the offset does not shrink with the print. Doubling the size of the print doubles both the correct distance and the sideways error, so the angle is fixed. It is a property of the trim, not of the presentation, which is exactly the property a reader has no access to.

The control, and it is the same amount of picture #

An essay whose every measurement confirms its thesis has not been tested. The test here is the neighbouring trim in which the effect is absent, and it is the one that removes exactly as much picture.

That the symmetric case reads exactly zero rather than merely small matters, because it says the effect has a mechanism rather than a magnitude. If a symmetric trim had come back at a fifth of the asymmetric one, the right conclusion would have been that cropping degrades a picture and that doing it evenly degrades it less. The zero says something stronger: the quantity being measured is the displacement of the rectangle’s midpoint, and a symmetric trim does not displace it.

It also explains why the fault is so rarely met by anybody who thinks about it deliberately. A photographer trimming for composition crops off one side, because composition is asymmetric. A conservator trimming a damaged edge crops off one side. A printer fitting a plate to a page crops off whichever side does not fit. The symmetric case is the one nobody has occasion to perform.

The same distinction settles a question the trim raises about the cone of vision. The sixty-degree cone is a rule about the angle subtended at the eye by the picture’s extremes, and a trim changes which extremes those are. A symmetric trim narrows the cone about its own axis; an asymmetric one narrows it and swings it, so the picture that survives is off-axis rather than merely tighter.

Two readers, one photograph, one focal length #

The strongest version of the claim is not that the trim damages a reading. It is that one way of reading is immune to it and the taught way is not, and the two differ by which point they take the principal point to be.

A picture containing anything rectangular carries its own vanishing points, and recovering the camera from the picture it drew turns three of them into a focal length and a principal point together, with nothing about the sheet entering the computation. The elementary two-vanishing-point account does something cheaper: it takes two horizontal vanishing points, supplies the principal point as the middle of the frame, and reads the focal length off the relation $f^{2} = -(\mathbf{v}_1 - \mathbf{p})\cdot(\mathbf{v}_2 - \mathbf{p})$.

There is a way to see why the assumption is expensive without computing anything. The relation is a product of two vectors measured from the assumed principal point, so a displacement of that point perturbs both factors at once, and the vanishing points of a shallow pavement are far outside the picture — hundreds or thousands of pixels away. A small absolute error in a point, multiplied by two long lever arms, is not a small error in their product. The same arrangement that makes the two-vanishing-point method convenient, which is that its inputs are far apart and therefore well determined by short bundles of edges, is what makes it sensitive to the one input it does not measure.

That is the mechanism the essay on the principal point and the shifted frame prices at one and a half per cent for a shift of a fifth of a frame; here a comparable displacement costs rather more, because a trim that removes a fifth also enlarges what is left.

The useful part is the gap between the two lines, not either line alone. The information needed to get the focal length right is present in the photograph — it is in the pavement’s own directions — and the taught method throws it away in exchange for one fewer bundle of edges to identify. That is a legitimate trade on an untrimmed frame, where the assumption is free. On a trimmed one it is a trade of accuracy for convenience, made silently, by a method that never mentions the assumption it is making.

A shift lens is a crop chosen before the exposure #

Nothing above depends on the picture having been cut. The principal point can be off the middle of the rectangle at the moment of exposure, and the equipment that does it deliberately is ordinary professional kit.

The two operations are the same translation of the image rectangle relative to the projection, performed at different moments. A shift lens does it with glass before the exposure; a pair of scissors does it with the print afterwards. Everything in this essay applies identically to both, which is a small piece of evidence that the quantity being measured is real: it does not care how the rectangle came to be where it is.

There is one practical difference and it runs the other way from intuition. A shifted picture is usually documented as shifted, because the photographer chose the shift and the equipment records it. A trimmed print carries no record at all. So the deliberate off-centre picture is the safe one and the accidental one is not, which inverts the usual relation between doing something on purpose and doing it by accident.

The same holds of the digital operation. Rendering a tile of a larger frame — the arrangement a tiled renderer or a multi-projector wall uses — is an off-centre frustum, and its principal point is off the tile’s middle by construction. That is understood by the people who build such systems and is written into the projection matrix. The failure this essay is about is not the off-centre picture; it is the off-centre picture read as though it were centred.

What this does not settle #

The measurement is a statement about geometry and it is worth being exact about how far it reaches.

It does not say a trimmed picture looks wrong. Nothing here is a claim about perception, and the site does not own one. A reader standing 2.36 cm from the correct station sees a scene reconstructed slightly differently from the one photographed, and whether any part of that difference is noticeable is a question for a different discipline entirely. Every essay here that leans on a viewing distance is making a geometric statement about which scene the picture is a projection of, not a psychological one about what a viewer experiences.

It does not say the trim can be undone. The principal point can be recovered from the picture’s own content when the picture contains enough structure — three bundles of parallel edges, or a horizon plus two perpendicular pairs. A photograph of a beach contains none of that, and for such a picture the trim is unrecoverable in principle rather than merely difficult. The recovery in the section above works because the subject is a colonnade.

And it does not say the taught constructions are wrong. They are exact, and their exactness is the reason this is worth writing down: an exact construction fed a false premise returns a false answer with no residual to show for it. That is a different and worse failure than an approximate construction, which at least announces its tolerance. The constructions assume the principal point is the middle of the sheet; when it is, they are correct to the last digit, and when it is not, they are wrong by an amount nothing in the drawing reports.

The board is part of the geometry #

The general shape of this is worth naming, because the collection keeps meeting it.

A perspective construction is carried out on a physical object with a physical instrument, and a handful of the things it assumes are facts about that object rather than about projective geometry. That the sheet’s middle is the picture’s centre is one. That a straightedge is long enough to reach the vanishing point is another, and dividing to a point off the board is what a draughtsman does when it is not. That two recipes for the same pavement start from the same free parameter is a third, and the two rules that agree to a hundredth of a pixel and part by twenty-four turn on the numbers their wordings invite rather than on the geometry.

In each case the geometry is exact, the instrument is fine, and the error is in an unstated premise about the board. None of the three fails a test the construction itself performs, because a construction tests its own steps and cannot test its own assumptions.

The consequence for reading pictures is simple enough to state as a habit. The principal ray is a thing to be found in a picture, not a thing to be measured off its edges with a ruler, and finding it needs content — a horizon, two perpendicular directions, a bundle of verticals. A picture that supplies that content supplies its own centre and cannot be lied to about it. A picture that supplies none is a picture whose station point is unknown, and the middle of its sheet is a guess in the shape of a measurement.

There is a version of this that is entirely practical and worth stating in one line, because it is the only part of the essay a reader can act on. Given a printed picture and a reason to measure something in it, the first question is not what lens took it but whether anything in the frame fixes the centre — a run of parallel edges, a horizon with two perpendicular directions crossing it, a set of verticals. If something does, the measurement is available and the sheet is irrelevant. If nothing does, the honest answer is that the picture’s station point is undetermined, and the number that comes out of assuming the middle should be quoted with the assumption attached rather than as a measurement.

That is the reason Brunelleschi drilled a hole through his panel rather than writing an instruction on the back of it. A hole is a mark on the board that cannot come adrift from the projection, because the projection was made through it, and the first demonstration in the European record is the only picture discussed in this collection whose principal point is not in doubt.