One camera means one horizon, not one point

Two readings of divergent pictures now rest on one test. Four surfaces, and no one camera that draws them measured how far a picture is from being one camera’s by moving its far corners until every surface’s sides met at one place, and a camera count needs a tolerance counted cameras by asking how few such places the picture needs. Both take one camera photographing parallel surfaces to mean the surfaces’ sides meet somewhere in common.

That is true of the pictures they were run on and it is not what one camera imposes. It is stronger, and a picture exists that a real camera really took and that the stronger test rejects.

What one camera actually fixes #

A drawn quadrilateral has two families of edges. Each family, extended, meets at a vanishing point — the drawn image of the direction those edges run in the world — and the line joining a surface’s two vanishing points is the vanishing line of the plane it depicts.

The fact that matters is short. Parallel planes have the same vanishing line, because the vanishing line depends only on the plane’s direction and not on where in space the plane sits. So one camera photographing several parallel surfaces gives them all one vanishing line, whatever else it does.

It is worth seeing why that is a fact about direction alone rather than a coincidence. A vanishing point is the image of the point at infinity of a direction, and a direction has no position; a plane’s points at infinity are exactly the points at infinity of the directions lying in it, and two parallel planes contain the same directions. So the set of infinite points is the same set for both planes, and its image under one camera is the same line on the page. Moving a plane forward, back or sideways changes which of its points fall inside the frame and cannot change which directions it contains. That is the whole derivation, and it is why the test costs nothing to run: no distances are involved, so no scale, no depth and no focal length enter.

It does not give them one vanishing point. A vanishing point is the image of a direction within the plane, and two parallel table tops turned differently about their own uprights contain different directions. Both directions vanish somewhere on the one shared line; neither has any reason to vanish at the same place as the other.

So the shared-point test requires two things at once — that the planes be parallel, and that the surfaces be turned the same way in them — and only the first of those is what “one camera” means. The second is a claim about the furniture.

A picture that fails the test and passes the fact #

The cheapest way to see the gap is to make one. Four rectangles lie on parallel planes leaning 35° toward a pinhole; each is turned in its own plane by its own angle, by 0°, 28°, −17° and 41°, and the pinhole photographs them.

Their sides meet at columns 345, −137, 622 and −443: a spread of 1,065 pixels, wider than the picture. And they meet at one height, 957.637 px down the page, to within 3 × 10⁻¹² px, which is arithmetic and not agreement within a tolerance.

Two of those columns are negative, which is to say the meeting points lie off the left edge of the picture entirely. That is ordinary rather than exotic — a surface turned toward the viewer’s left recedes to the left, and its sides converge somewhere out beyond the frame — and it is a reminder that a meeting point is not a mark on the page but a construction off it. The heights, meanwhile, agree to the twelfth decimal place across surfaces whose meeting points are more than a picture-width apart.

That is exactly what the fact predicts. Every surface’s near and far edges are level on the page here, so each surface’s second vanishing point is the horizontal point at infinity, and each surface’s vanishing line is therefore the horizontal line through its own meeting point. Four surfaces on one plane direction, four horizontal lines through four meeting points, one shared line: the meeting points have to be at one height and are free in column.

A test that accepts everything is not a test #

Every control so far is a picture the vanishing-line reading accepts. A reading that has never refused anything is not evidence that a picture passed it, so the reading must be handed a picture it refuses, and one that a careless version of it would not.

The picture to build is one camera photographing four surfaces whose planes are not parallel: leaning 35°, 20°, 48° and 62° toward the eye. They are turned in their planes as well, by the same 0°, 28°, −17° and 41°, so that a refusal cannot be credited to the turning — the turning is held fixed and only the parallelism is broken.

The four lines come apart by 1,152 px, further than the constructed stack’s 588. So the reading refuses a picture one camera really took — correctly, because what it tests is not whether one camera was present but whether the surfaces are parallel, and here they are not.

That is worth dwelling on, because it is the shape of everything this test can do. A refusal says these surfaces are not parallel in the world, and the two ways of getting one are a second camera and a second plane direction. Nothing in the marks distinguishes them. A reader who knows the surfaces are parallel learns about cameras; a reader who knows there was one camera learns about the furniture; a reader who knows neither learns that one of the two assumptions is wrong and not which. It is the same shape of conditional that runs through every recovery in this collection — what a single view cannot give is not a defect of any particular method but a count of what the marks contain — and the right response is to say which assumption is being spent, not to look for a reading that spends none.

What the stricter test charges #

Run the shared-point redraw on the turned picture and it returns 28.7 px. A reader applying the earlier measurement to it would conclude that no single camera drew it, and would be able to quote a number.

Sweeping the turn from nothing to forty-five degrees separates the two readings cleanly. At no turn they agree and both charge zero, which is why the gap has been invisible. As the turn grows the shared-line reading stays at zero to twelve decimal places and the shared-point reading climbs without limit — the charge is not bounded by anything, because two meeting points can be pushed arbitrarily far apart by turning two surfaces toward each other’s sides.

The charge is also not an error that gets smaller with more evidence. Adding a fifth turned surface adds a fifth meeting point somewhere else along the same line, which makes the shared-point redraw worse and leaves the shared-line reading at zero. That is the signature of a mis-specified test rather than a noisy one: more data makes it more confident and more wrong, where noise would make it less confident and eventually right.

It is also why the error could not have surfaced as a poor fit. The shared-point redraw handed a turned picture does not fail to converge or return something ragged; it converges on a definite common point — 422 px across, in this case — and reports a clean root-mean-square distance to it. A reader watching only the residual would see a number of a size that occurs all the time, on a picture where the truth is exactly zero.

How large the mistake would have been #

The charge is easy to dismiss as small until it is set against the numbers this field has been quoting.

The shared-point redraw’s charge rises at about 1.05 px for every degree of turn, nearly linearly over the range a room supplies. So a photograph of four parallel surfaces whose furniture is turned by 16.9° is charged 17.63 px — which is, to three decimal places, exactly what the constructed divergent stack is charged. A photograph of an ordinary room with a stool pulled round by seventeen degrees would be reported as being precisely as far from one camera’s picture as the icon construction is, and the icon construction was built to be as unlike one camera’s picture as four splayed surfaces can conveniently be.

The mixed stack’s 23.76 px is matched at a turn of 23.2°, which is a chair pushed back from a table.

Those are not marginal confusions to be resolved by quoting a tolerance. They are the measurement’s headline numbers reproduced exactly by a picture containing no divergence at all, and a reader who had both numbers and no knowledge of which picture produced which would have nothing to go on. It is the clearest statement of why the test had to be corrected rather than merely noted.

Why nothing earlier moves #

The natural next worry is that two readings have been published on the wrong test. They have not, and the reason is worth stating precisely rather than announced.

Every surface in every constructed stack this field has measured is drawn symmetric about one column of the page — the near edge and the far edge share a centre, and every surface shares that centre with every other. For surfaces of that shape a meeting point is forced onto the shared centre column, so same height and same point are the same condition, and a redraw that enforces either enforces both.

The arithmetic bears that out where it can be checked against itself. On the constructed stack the shared-line redraw and the shared-point redraw both return 17.632 px, to every digit; on the stack whose surfaces disagree in sign both return 23.758 px. The two tests are not merely close on these pictures, they are the same test.

So the earlier numbers stand, and what changes is the size of the claim they support. The 17.6 px is the distance from the nearest one-camera drawing of surfaces drawn square to one another, and the constructed stack qualifies because it was built that way. It was never the distance from the nearest one-camera drawing of anything at all, and reading it as the latter would be reading a measurement of a narrow family as a measurement of a wide one.

The same correction applies one level up. The camera count is a count of the groups a picture needs given that its surfaces are drawn square to one another, and a picture of a turned room would be counted too high — not because more cameras drew it, but because a surface turned by twenty degrees looks, to a test that asks for a shared point, exactly like a surface photographed from somewhere else.

Which test to run, and on what #

There is no reason to keep the weaker-hypothesis test out of the way; the shared-line reading is available on any picture on which the shared-point reading is, and it is the one to run.

The practical form is simple for the pictures this field is about. For a surface drawn with its near and far edges level on the page, the vanishing line is the horizontal through its meeting point, so the test is: are the meeting heights equal? — and the columns are to be ignored. For a surface whose edges are not level, both vanishing points are finite and the line is the one through them, which is the same line that every pair of parallel lines in that plane runs to.

Two cautions about running it. The vanishing line must be computed homogeneously, because for the level-edged surfaces that make up most of these pictures one of the two vanishing points is at infinity — that is not an awkward case to be handled separately, it is the common case, and a construction that dehomogenises first has to special-case the very surfaces it will spend most of its time on.

And the lines must be compared as lines on this page rather than by the angle between them, for the same reason the redraw was quoted in pixels rather than in degrees of tilt. Two vanishing lines a tenth of a degree apart are the same line for a picture the width of a hand and different lines for a wall painting, so the comparison worth making is how far apart the two lines run at the picture’s own left and right edges. A vanishing line is, after all, the picture’s horizon for that family of planes, and a horizon is something a reader locates on the page rather than in the world. That is a distance in pixels, which is the currency the redraw already spends and the currency a brush line can be quoted in.

The third hypothesis, and why there is no test below it #

Two tests have now been separated, and they sit in an ordered list of hypotheses with a definite bottom.

Same plane direction and same in-plane orientation: the surfaces share a vanishing point. The strongest hypothesis, and the one the earlier readings assumed.

Same plane direction only: the surfaces share a vanishing line. This is what “one camera photographed parallel surfaces” means, and it is the right place to stand.

No shared direction at all: one camera imposes nothing whatever. This is the bottom, and it is worth seeing why it is a genuine bottom rather than a weaker test. A single drawn quadrilateral is the picture of a rectangle on some plane at some tilt under some lens — that is the whole content of the single-surface reading — so any collection of quadrilaterals whatever is one camera’s picture of a collection of rectangles on unrelated planes. There is no picture a camera could not have taken, and so there is nothing to test.

That is why the assumption of parallelism was never optional and never a technicality. It is the entire source of the constraint. A fact about the depicted world is what makes a picture testable at all here, and weakening it to the point of honesty leaves nothing behind. The correction this essay makes is to weaken it by exactly the right amount: parallel planes, yes; furniture squared up to the same wall, no.

What this establishes and what it does not #

It establishes that the shared-point test is stronger than the fact it stands for. One camera and parallel planes require a shared vanishing line, and a picture satisfying that can have its meeting points arbitrarily far apart in column.

It establishes that the stronger test rejects real photographs. A pinhole’s picture of four parallel surfaces turned by 0°, 28°, −17° and 41° is charged 28.7 px by the shared-point redraw; the charge rises at about 1.05 px a degree, and at 16.9° of turn it equals the figure the constructed divergent stack was reported at.

It establishes that the earlier readings are unaffected. On both constructed stacks the two tests return the same number to every digit, because those pictures’ surfaces are drawn symmetric about one column. The verdicts stand; what narrows is the family they are verdicts about.

It does not establish anything about paintings. Every picture here is constructed, and no tradition’s pictures have been measured for whether their furniture is squared up. Whether a divergent picture’s surfaces are turned or square is a question about that picture, and a reader who ran the shared-point test on one containing a turned stool would get a number and no warning.

And it does not make the correct test strong. A shared vanishing line is one linear condition on each surface, so a picture of two surfaces passes it for free — any two lines meet somewhere and any two points lie on a line. The test only starts to say anything at three surfaces, and a divergent picture with two objects in it is untestable for the same reason a picture with one object is.

That last point cuts against the earlier readings in a direction the correction does not. Weakening the test from a point to a line does not merely rescue turned pictures; it weakens what a passing picture proves. Three surfaces sharing a point is a coincidence of two numbers twice over; three surfaces sharing a line is a coincidence of one number twice over. So the same picture that would have been strong evidence of one camera under the old reading is half as much evidence under the right one — and the honest version of the earlier verdicts keeps their refusals intact while making their acceptances cheaper.

Still open: what a picture with two vanishing lines in it would look like #

The count and the test now agree about what a group is, and the pictures they have been run on all have their surfaces’ edges level on the page, which is what makes each vanishing line a horizontal and each surface’s contribution a single number.

A picture whose surfaces are drawn with neither family of edges level is a different object. Each surface then has two finite vanishing points and its vanishing line is genuinely a line, with a slope as well as a height, so a group must agree about two numbers instead of one and a picture can fail the test by tilt as well as by height. The question that leaves is whether the count changes character there: whether the extra number makes groups easier to tell apart, because two lines must now agree about more, or harder, because a redraw has a second free parameter to spend absorbing the hand’s scatter — and whether the floor that a hand’s own jitter puts under the count rises or falls when the thing being fitted is a line rather than a level.