A vanishing line with a slope in it

One camera means one horizon, not one point narrowed the test a picture of parallel surfaces has to pass. Requiring every surface’s sides to meet at one point charges 28.7 pixels to a picture one camera really took, because one camera gives parallel planes a shared vanishing line and gives them a shared point only when the furniture is all square to the same wall. Turn a footstool a few degrees and the picture is still one camera’s.

Every picture that verdict was reached on has something else in common, and it is not a decision anyone made. The plane is tilted toward the eye about the camera’s own horizontal axis, so one of the plane’s two directions runs parallel to the image and its vanishing point sits at horizontal infinity. Each surface’s vanishing line is therefore a horizontal, a horizontal is fixed by one number, and a surface’s whole contribution to the test is a height.

Turn the plane about the view direction as well and none of that survives. Both vanishing points become finite, the vanishing line acquires a slope as well as a height, and a group of surfaces must now agree about two numbers. The question is whether the count changes character there — whether the extra number makes groups easier to tell apart, because more has to agree, or harder, because the redraw has a second free number to spend absorbing a hand’s scatter.

The answer is neither, to within a few per cent. What the slope does do is expose a defect in the redraw the count has been priced in since it was first measured.

A plane tilted two ways #

The picture is the same furniture as before — a floor, a footstool, a table and a book, each turned by its own angle in its own plane, all four on parallel planes, photographed by one pinhole. The only change is that the shared plane is turned about the view direction before anything is laid on it.

One camera and parallel planes still give one vanishing line exactly — that is the control the whole test rests on, and it survives the turn to the arithmetic floor. What changes is what the line is. At no turn it is a horizontal and a surface reports a height; at twenty-two degrees it runs at 22.000 degrees across the page and a surface reports a height and a slope.

That is not a cosmetic difference. A picture can now fail the test by tilt as well as by height: two surfaces whose lines cross in the middle of the page agree there and disagree at both edges, and a test that read the lines at one place would call them the same line. The spread between two lines is therefore read at both page edges and charged at the worse of them, which is how one camera means one horizon, not one point already compared them and why nothing in that comparison had to change.

The redraw that has nothing left to spend #

A test that says yes or no is not a count. What makes a count possible is a distance — how far a picture is from passing — and a camera count needs a tolerance established that the honest distance is a redraw: how many pixels the far corners must move before the surfaces do share a line.

The redraw those readings use keeps each surface’s near edge and the height of its far edge, and slides the two far corners sideways. Write the far edge’s demand as a single number and the arithmetic is short: only the difference of the two moves is constrained, so a surface has one of its two moves left over, and splitting the demand evenly spends it. One constraint, two moves, one to spare.

Put a slope in the picture and the spare one is gone. Sliding both far corners along the near edge’s direction no longer leaves the far edge parallel to anything in particular, so the edges’ vanishing point must be steered onto the target line as well as the sides’. Two conditions, two moves, nothing left.

A determined move is not a small one. At the line the picture actually has, sliding the far edge costs $3\times10^{-12}$ pixels; a quarter of a degree away it costs 75,444. The cost rises without bound as the target approaches the right line from either side, because steering a vanishing point onto a nearly-right line means pushing it out toward infinity along that line, and the corner that pushes it there has to travel.

A quantity that is not continuous in its target is not a distance. The count is built on minimising over target lines, and a minimisation over a razor valley is a search for a needle rather than a measurement. So the instrument has to change, and the change is not a patch.

The redraw that rebuilds the surface #

The way out is to stop nudging the drawing and start rebuilding it. Let both far corners move anywhere on the page: four free numbers against the same two conditions, with two left over — the same arithmetic the level picture enjoyed, now arrived at honestly rather than by the target line happening to be horizontal.

Every quadrilateral with a given near edge and a given vanishing line is reached exactly once by choosing where its sides meet on that line and which of the lines through a fixed point its far edge is. So the nearest such drawing is found rather than approximated, the objective is smooth, and the same figure shows it: 0.0747 pixels a quarter degree off, 0.299 at one degree, 0.602 at two, and nothing at all at the line.

The two numbers a group must now agree about are not worth the same, and the rebuild prices them against each other directly. Moving the target line one degree off the picture’s own costs 0.299 pixels a corner; moving it ten pixels up the page, with no change of angle, costs 0.189. A degree of slope is worth about sixteen pixels of height, on a picture 690 pixels wide, and the exchange rate is the page’s width rather than anything about the furniture — a line turned a degree has moved six pixels at each edge and none in the middle, and a surface is charged for where its own corners are.

That matters for reading a real picture, because the two disagreements arrive from different places. A group of surfaces drawn too high or too low is a group whose sides meet in the wrong place, which is the error a painter makes stepping a construction. A group drawn at the wrong angle is a group whose plane is turned, which is a different mistake and a rarer one. The rebuild charges both in one currency and does not say which it found; separating them is the sweep the last section makes.

This is the second time this field has found a test measuring its own convenience. The shared-point test charged a real camera’s picture 28.7 pixels because it demanded a shared receding direction nobody had asked for. The sliding redraw charges one because it holds each far edge at the height it happens to be drawn at, and a height on the page is not a property of the drawing when the drawing has been turned.

The control is the useful half. On the stack whose surfaces are all symmetric about one column — the picture every earlier reading was run on — the two redraws agree to $3\times10^{-14}$ pixels, so nothing measured there is disturbed. The charge appears exactly when a surface is turned in its own plane, which is what one camera means one horizon, not one point introduced and what a room full of furniture looks like. Just under a pixel is not large beside the tolerances that essay swept, and it is charged to a picture that is innocent, so it belongs in the count as an error and not as evidence.

What the slope costs the count: almost nothing #

With a redraw that survives the slope, the question one camera means one horizon, not one point left can be put directly. Take a picture one camera drew, let a hand displace every far corner by a stated amount, and measure the redraw the resulting drawing needs — the tolerance at which a one-camera picture stops reading as one camera. Then do it again at every turn of the plane from none to thirty-eight degrees.

The median moves by nine per cent across the sweep, against a quartile band twenty-four per cent wide — which is to say it does not move. A group must agree about two numbers instead of one, and the line it must agree on has two numbers instead of one, and the two cancel.

That cancellation is worth stating as arithmetic rather than as a coincidence. Each surface contributes two conditions, whether the line is level or sloped, because a vanishing line is two numbers either way; what the level picture hides is that one of those conditions is satisfied for free, by the far edge staying level when the corners slide sideways. The freedom the redraw gains is exactly the freedom it needs to satisfy the condition it also gains. Neither the count nor its floor changes character, and the expectation that more constraints must mean sharper separation was wrong in the direction that matters.

Where the slope does bite #

The count does not change and the floor does not change, but one thing about the picture genuinely does, and it is the thing a reader with a ruler would be caught by.

Two groups in a picture with no slope have vanishing lines that are both horizontals, so they are parallel: their separation is one number, the same everywhere across the page, and it does not matter where a reader measures it. Two groups in a turned picture can have lines at different angles, and lines at different angles cross.

At ten degrees apart the two lines run at 22.00 and 32.00 degrees and cross at x = 7, which is on the page. Their separation is 1.4 pixels at the left edge and 150.9 at the right. A reader comparing the two lines at one place — the middle of the picture, or wherever the furniture happens to be — could read anything between those and report it as the disagreement.

That is why the spread between two lines is charged at the worse of the two page edges rather than measured once, and it is the one place where a level picture’s habits are actively misleading rather than merely incomplete. The redraw does not have the problem, because it never compares two lines at all: it asks what it costs to put every surface on one line, and the crossing case costs 1.315 pixels a corner, which a hand good to two pixels would hide.

The count, in degrees of the depicted world #

A count in pixels of redrawing is honest and hard to use. What a reader of a picture wants to know is how far two pieces of furniture have to be from parallel before the picture says so, and the rebuild answers that directly, because its charge is very nearly proportional to the disagreement.

A degree of disagreement between two groups’ planes costs 0.247 pixels a corner, and it costs the same whether the picture is level or turned twenty-two degrees. A hand that places each far corner to within a pixel costs 0.76 pixels, which is 3.1 degrees; to within two pixels, 6.2 degrees; to within four, 12.3.

So a divergent picture whose table and footstool are drawn on planes ten degrees apart is, in the hands of an ordinary painter, a one-camera picture. A camera count needs a tolerance put a floor under the count in pixels and found it running from 0.17 to 1.34 times a hand’s own scatter across a hundred and twenty hands; this converts that floor into the quantity a reader cares about, and the conversion is a single constant. That is a strong statement and it is the right shape for the argument this field is making: the reason a divergent picture does not convict its maker of using several viewpoints is not that the test is weak but that the hand is wide, and the width is measurable.

What this does not settle #

The rebuild is powerful, and it can fit a picture nobody photographed. Handed four surfaces on planes that are genuinely not parallel — the stack this field keeps for exactly this purpose — it charges 3.69 pixels rather than refusing outright. That is $10^{13}$ times what it charges a real camera’s picture and it is still only as much as a four-pixel hand. The instrument has no absolute verdict in it, only a distance, and the paragraph above is what the distance buys.

A crossing pair is harder to find than to describe. The section above builds two groups whose planes differ about the view direction, which makes their lines cross near the page. Two groups differing about the tilt axis instead — a table steeper than the floor under it — keep parallel lines however far the picture is turned, and every measurement in this essay’s sweep is of that kind. Which disagreements a real divergent picture actually holds is not something this settles.

The plane’s turn is not the painter’s. Turning the plane about the view direction is the cleanest way to give every vanishing line a slope while keeping one camera’s guarantee exact, and it is not a claim that icon painters turned their tables that way. A picture drawn freehand has slopes in it for reasons that are not a rigid turn of anything, and those are not modelled here.

The hand’s slip was modelled as independent at every corner. A painter who is consistently out in one direction across a whole picture puts a systematic error into the very quantity the line is fitted to, and the sweep above contains none of that. It is the same limit the rows count hands, not cameras records for its own reading, and it is the error both are least protected against.

And the rows are untouched. The rows count hands, not cameras found that the lines inside a strip charge a camera nothing at all and fix a hand’s habit instead. Nothing in this essay changes that: a turn of the plane moves where the rows fall on the page and leaves the habit they report exactly where it was, since a habit is read from a strip’s own sides and rows and a turn takes both with it.

And nothing here touches the lens. The count is a statement about drawn quadrilaterals and stays one; turning it into degrees of depicted tilt still needs a focal length nobody can read off the page, which four surfaces, and no one camera that draws them measured at a factor of seven between a short lens and a long one. The degrees in the section above are degrees between two planes, which is a difference and needs no lens.

The slope, the redraw and the count #

A plane turned about the view direction gives every surface two finite vanishing points and every vanishing line a slope, and one camera still puts four such surfaces on one line to $9\times10^{-14}$ pixels.

The redraw the count was priced in does not survive that. Sliding the far corners along the near edge is two moves against two conditions, so the move is determined and the cost explodes — 75,444 pixels a quarter of a degree from the right line, against $3\times10^{-12}$ at it. Rebuilding the surface instead leaves two numbers to spend, costs 0.0747 pixels at the same place, and repairs something the level picture was already hiding: the sliding rule charges 0.9547 pixels to a picture one camera really took, as soon as its surfaces are turned in their own planes.

With that redraw the slope changes nothing. Four pixels of a hand cost the test 3.000 pixels at no turn and 3.272 at thirty-eight degrees of it, inside a quartile band three times as wide. And the charge is linear in what a reader actually wants to know: 0.247 pixels for every degree two groups’ planes are apart, so a hand good to two pixels hides six degrees of disagreement and a hand good to four hides twelve.

Still open: whether a slope lets the far edges be trusted #

Both redraws here move only the far corners, and that is a choice nothing has justified. It is inherited from the constructions the earlier readings measured, where a painter draws the near edge of a strip first and the far edge last, so the far edge carries the accumulated error and the near edge is the datum.

A turned plane makes that assumption checkable for the first time. With no slope, moving a near corner sideways and moving the opposite far corner sideways do very nearly the same thing to a surface’s vanishing line, so there is nothing in the drawing to say which was misplaced. With a slope, the two act differently — a near corner moves the line’s height and its slope in one proportion and a far corner in another — and a redraw allowed to move all four corners, at a stated cost each, would distribute the blame rather than assume it.

The measurement that follows gives every corner a move and a weight, sweeps the weight from “the near edge is exact” to “all four are equally suspect”, and asks two things: how much the count changes across that sweep, and whether the redraw’s own distribution of blame agrees with where the error was actually put when the picture was built by displacing near corners instead of far ones. If it does, a picture can be asked which of its edges its painter drew first.