The picture whose lines spread

A table in a Byzantine icon is drawn with its far edge wider than its near one. The sides of a book, a footstool, a throne, a step spread as they recede rather than converging. The convention has a name in the literature — inverse perspective, reverse perspective — and a standard explanation, which is that the vanishing point has moved to behind the viewer.

That explanation is checkable, and it is wrong.

Where the point actually goes #

Sweep the splay — the ratio of the far edge to the near — from below 1 to above it and watch the sides’ intersection.

Below 1 the sides converge upward and meet above the far edge. This is ordinary one-point perspective and the meeting point is the vanishing point of the receding direction.

At exactly 1 the sides are parallel and never meet. There is no vanishing point, and this is not a degenerate case of the other two — it is a parallel projection, with all the properties the parallel field measures.

Above 1 the sides diverge upward, which means they converge downward, and they meet below the near edge at a perfectly ordinary point on the page.

That point is in front of the eye. It is at a finite place on the picture plane. It is the vanishing point of a direction — specifically, of a direction running down and away from the viewer. There is nothing behind anybody.

Why the standard account is tempting #

The “behind the viewer” description is not arbitrary; it comes from a real intuition about what would have to be true for lines to spread.

If the object is a horizontal table top receding away, and its sides really are parallel in space, then their images must converge to a point on the horizon. They diverge instead. So — the reasoning goes — the centre of projection must be on the wrong side, and the picture must be a projection from a point behind the viewer’s head, or equivalently from a negative focal length.

Each step of that is sound and the conclusion follows only from the premise in the first clause: that the object is a horizontal table top. Drop that assumption and the divergence has an ordinary explanation, which is the subject of the next essay, and the vanishing point stays exactly where the construction puts it.

The lesson is a familiar one on this site. A construction is asked what it depicts and the answer is not the thing that was assumed; the taught two-point cube is the same finding on a different construction. In both cases the drawing is a correct picture of something, and the interesting question is what.

What a vanishing point below the picture means #

A point below the near edge is unfamiliar enough that it is worth saying what direction it is the image of, because the answer is ordinary.

A vanishing point is the image of a direction: the place points along that direction go as they recede. A point on the horizon is the image of a horizontal direction. A point above the horizon is the image of a direction that rises as it recedes — a hillside going up, a staircase climbing away. A point below the near edge is the image of a direction that descends steeply as it recedes.

Every picture of a staircase seen from above has one. Every picture of a road running down a hill has one. Nothing about the geometry is unusual and nothing about the position is exotic; it is a perfectly ordinary vanishing point of a perfectly ordinary direction, and it happens to be below the drawn object rather than above it.

The reason it feels strange in an icon is that the object is assumed to be a horizontal table, and a horizontal table’s receding direction cannot have a vanishing point down there. Which is the same assumption the whole essay keeps arriving at, and which the next one drops.

The three cases, checked rather than described #

assertDivergenceMovesTheVanishingPoint builds all three and locates the intersection in each, and the middle one is the control.

Convergent, splay 0.62: the sides meet at $y = -170$, above the far edge at $y = 140$. Off the top of the canvas, and the figure says so rather than drawing a mark where there is none.

Parallel, splay 1.00: intersect returns null. Not a very large number, not a point at the edge of the canvas — nothing. That is the control, and it matters because a construction that returned some enormous coordinate here would make the sweep continuous and the classification meaningless.

Divergent, splay 1.32: the sides meet at $y = 924$, below the near edge at $y = 330$.

The classification is therefore not a matter of degree. There are two regimes separated by a case in which the object does not exist, which is exactly the structure of the vanishing point of a direction parallel to the picture plane: as the direction turns, its vanishing point runs off to one side, ceases to exist at the parallel case, and returns from the other side. A divergent construction is on the far side of that transition.

The point’s position, as a function of the splay #

The three cases above are three samples of one expression, and writing it down turns the classification into arithmetic and supplies a consequence the samples hide.

The construction has a near edge at page height $y_{n}$, a far edge at $y_{f}$, and a splay $s$ which is the ratio of the far half-width to the near one. A side runs from the near edge’s corner to the far edge’s, and the axis of symmetry is reached at the parameter $t = 1/(1-s)$. So the meeting point sits at

$$y(s) = y_{n} - \frac{h}{1-s}, \qquad h = y_{n} - y_{f},$$

with $h$ the drawn depth on the page. On the figure’s construction $y_{n} = 330$ and $h = 190$. At a splay of 0.62 that gives $330 - 190/0.38 = -170$; at 1.32 it gives $330 + 190/0.32 = 924$; at exactly 1 it is a division by zero. Those are the three numbers the sweep reports, from one expression, with the null in the middle arriving as a pole rather than as a special case.

Three things follow that the three samples do not show.

Convergence and divergence are reflections of each other about the near edge. Splays of $1-\varepsilon$ and $1+\varepsilon$ put the meeting point the same distance $h/\varepsilon$ above and below $y_{n}$. Crossing 1 is not entering a new regime; it is passing through infinity and coming back on the other side, which is exactly the behaviour a direction turning parallel to the picture plane shows.

The meeting point is never inside the drawn quadrilateral. Landing between $y_{f}$ and $y_{n}$ would need $0 < h/(1-s) < h$, that is $s < 0$, which is a far edge on the wrong side of the axis. Every construction with a positive splay, however extreme, meets outside itself.

And the position is a badly conditioned readout of the splay. Differentiating gives $dy/ds = h/(1-s)^{2}$, so near the parallel case a small change in the splay moves the point enormously. At the modest splay of 1.08 where most of the icons’ effect lives, the point is 2,705 px below the near edge — seven times the drawn depth — and a one per cent error in reading the splay moves it by 297 px, a twelfth of its distance. Reading the far and near widths off a painted table to a per cent is optimistic; locating a vanishing point two thousand pixels outside the panel from those readings is not a measurement.

That is the same conditioning argument the height recovery makes about cross-ratios, and it has a practical form here. The splay is the quantity to measure and the vanishing point is not. The splay is a ratio of two lengths that are both drawn, both short, and both readable; the point is a construction that amplifies the error in that ratio by $h/(1-s)^{2}$ and puts the answer off the panel. An account of these pictures that argues from where the vanishing point is has chosen the derived quantity over the measured one, which is a second reason — independent of everything above — to distrust the answers it reaches.

It also explains why the divergence is easy to see and hard to quantify. The eye reads the splay directly, because the two edges are side by side on the panel; it has no access at all to the point, which is why the standard account could be repeated for a century without anybody noticing that its point was in the wrong place. The next essay takes the splay itself as the measurement and asks what object it is a picture of, which is the question the machinery can actually answer.

What “behind the viewer” would actually mean #

It is worth taking the standard account seriously enough to say what it would require, because the requirement is instructive.

A projection whose centre is behind the picture plane rather than in front of it is a real object: it is the pinhole camera’s other image, the one on the far side of the aperture, inverted. Points project through the centre and land on the opposite side, so the picture is rotated by half a turn. Every point of it, not just the edges of the table.

An icon is not rotated by half a turn. The figures are upright, the architecture stands the right way up, and only certain objects — tables, footstools, books — show the divergence. A global change of the projection would affect everything globally, and this is local.

So the standard account is not merely imprecise about where the point is; it names a global operation to explain a local effect, and a local effect needs a local explanation. The next essay supplies one, and the one it supplies is that the object is not a horizontal table top.

The other thing a splay is not #

One more account to dispose of, and this one is more nearly right.

It is sometimes said that a divergent construction is what a picture would look like if it were projected onto the viewer rather than from the viewer — a projection whose centre is at the object and whose picture plane is at the eye. That is a real construction and it does produce divergence.

It also produces a specific amount of divergence, fixed by the geometry, and the amount does not match. In icons the splay is modest — a few per cent to perhaps thirty — and varies from object to object within one picture. A construction with the centre at the object would give a splay determined by the ratio of two distances that are the same for every object at the same depth, so all the tables in one icon would splay by the same factor. They do not.

This is the kind of test worth running on any account of a convention: does it predict the variation, or only the effect? An explanation that gets the sign right and the spread wrong is describing something else.

How much splay is actually in these pictures #

The measurements below are on constructions swept through a range, and it is worth saying what range the objects occupy, because the answer bears on which explanations survive.

Icons show splays that are modest and variable. A table edge might spread by five per cent, a footstool by twenty, a book by thirty; different objects within one picture spread by different amounts; and many objects in the same pictures show ordinary convergence or no splay at all.

That variability is the strongest constraint on any account of the convention, and it eliminates the two global explanations immediately. A projection from behind the viewer is a property of the whole picture and would apply to everything. A projection onto the viewer is likewise global, and would give every object at the same depth the same splay.

Any account that survives has to be per-object, which means it has to be about the objects. That is where the next essay starts and it is why the sweep in the figure below runs a range rather than settling on a number: the interesting thing about the splay is that it varies.

What would settle it from a picture #

A useful discipline when disposing of explanations is to say what evidence would have chosen between them, and here it is specific.

If the effect were global — a projection from behind the viewer, or onto the viewer — every receding edge in the picture would splay, including the architecture and the ground. Checking that requires only looking at the picture and needs no computation.

If the effect were about the objects, only the objects with the relevant property would splay, and the amount would vary with that property from object to object within one picture.

The second is what icons show. That does not by itself identify which property, and the next essay identifies a candidate that is exactly computable — but the observation above is already enough to eliminate every global account, using nothing but the distribution of the effect across one picture.

What the construction is, stripped of interpretation #

Set aside every account and state what is on the page.

A one-point construction has a near edge, a far edge parallel to it, and two sides joining them. Three numbers describe it: the near width, the far width, and the depth on the page. Their ratios are two free parameters, and the splay is one of them.

Every value of the splay is a legitimate drawing. Every value except exactly 1 has a vanishing point somewhere on the extended picture plane. Nothing about any of them is impossible, ill-formed or self-contradictory, and the recovery machinery this site already has will answer the question what solid does this depict for every one of them — which is what the next essay does.

That is worth stating flatly because the literature on inverse perspective is largely about whether the convention is a mistake, a spiritual statement, a perceptual accommodation or a failure of skill, and the geometric answer to all of those is prior: it is a picture of something, the something is computable, and the computation takes a line of code that this site has had since its foundation.

A note on the word “inverse” #

One last thing to set aside, because the name of the convention encodes the account this essay is refusing.

Inverse perspective names the effect as a reversal of something, which presupposes that the something is the default and this is its negation. Every consequence of the name follows from that presupposition: the vanishing point is behind rather than in front, the projection is from the object rather than to it, the picture inverts the viewer’s relationship to the scene.

The measurement supports none of that. A divergent construction has an ordinary vanishing point in an ordinary place, is a projection in the ordinary direction, and puts the viewer nowhere unusual. What it has is a splay above 1 rather than below it, on a continuum that passes smoothly through the parallel case, and nothing about crossing that value is a reversal of anything.

The neutral name would be something like divergent construction, which is what it is called throughout this field. That is not a proposal to rename anything; it is an observation that the name is carrying an argument, and that the argument does not survive being checked.

What the next essay needs from this one #

Two things carry forward and both are negative results, which is the useful kind to hand on.

The vanishing point is somewhere ordinary, so any account that starts from an extraordinary projection is answering a question that does not arise. And the effect varies from object to object within one picture, so any account that is a property of the whole picture is the wrong shape.

Together those force the explanation to be about the objects, and the next essay asks the machinery what object a given splay is a picture of. The answer is not the one that was expected, which is the best thing that can happen to a question with a control attached.