Visual Plane
every picture states the point it is correct from
Where to stand

An anamorph has one eye

From the design point exactly — a camera's single eye — the floor marks give the design back to sixteen decimal places. A head carries two eyes 63 mm apart, and neither of them is the design point. The difference between the disparity the floor gives and the disparity an upright board would give runs to 47 arcminutes, against a stereoacuity of a few tens of arcseconds. This is why pavement paintings are photographed.
19 min read 13 figures The viewer is in the geometryNot a projection at allOne machine used twice

Worth reading first: A set cut for one eye · A floor anamorph is three numbers · The point you have to stand at.

The design point of an anamorph is a point in the air. Standing there, the floor marks read back as the intended picture — exactly, to sixteen decimal places, which is what this collection means when it says a construction is a construction rather than an approximation.

Nobody has an eye at a point in the air. A person has two eyes, about 63 millimetres apart, and neither of them is the design point. Both are 31 millimetres off it, in opposite directions, across the line of sight — which is the direction the anamorph forgives least, where the whole tolerance at a ten-millimetre error budget is 14 millimetres.

So a head standing at the marked spot does not fit inside the room the design allows.

Two eyes in a room built for oneThe same section with the two eyes of one head drawn in it, 63 mm apart across the line of sight. The room is 29 mm across at its narrowest, so a head does not fit inside it in the direction that matters.toward the designup63 mm apart10 mm14 mm across36 mm along the sight line
Fig. 1 The two eyes of one head drawn inside the region a ten-millimetre tolerance permits, at true separation. The room is 14 millimetres across at its narrowest and the head is 63 millimetres wide, so the two eyes are outside it in the direction that matters, on opposite sides.
What the second eye is toldThe difference between the disparity the floor marks give and the disparity the same design on an upright board would give, at 55 mm of eye separation. The worst is 40.7 arcminutes and the middle of the design is 19.3; human stereoacuity is a few tens of arcseconds.0.06 m up the design1.9′0.32 m up the design10.6′0.58 m up the design19.9′0.84 m up the design29.9′1.10 m up the design40.7′floor against board, per point of the designone eye reads it exactlyand nobody has one eye
Fig. 2 The same measurement at a child’s eye separation. The signal is smaller in proportion and still two orders of magnitude above the threshold at which a person can detect it.

What each eye gets

Each eye is a wrong eye, and a wrong eye is a known object: it composes the intended picture with a central collineation whose axis is the ground line, so the error is zero at the bottom of the design and grows with height. That is where the anamorph still works.

The two eyes are displaced in opposite directions, so they get opposite errors — the left eye sees the design sheared one way, the right eye sheared the other. Neither sees the design.

But shear is not the interesting part, because a person cannot compare their two eyes’ pictures point by point. What the visual system does compare is the directions of the same feature seen from the two eyes, and the difference between them is a disparity, which is what depth perception is made of.

How the disparity is computed

The measurement wants stating, because the naive version of it gives a number that is large and means nothing.

Each floor mark has a direction from the left eye and a direction from the right eye, and the angle between those two directions is the mark’s disparity. Every point at a finite distance has one; a mark two metres away has a much larger disparity than a mark six metres away, and the near marks of a pavement anamorph are much nearer than the far ones. So the raw disparities of the floor marks run over a wide range — about 37 arcminutes from end to end of this design — and quoting that range would be quoting the fact that the floor recedes.

What matters is the difference between the disparity the floor gives and the disparity the intended picture would have given. The intended picture is an upright board standing at the ground line: it also has a range of distances, since its top is further from the eye than its bottom, and it also produces disparities. If the two sets of disparities agreed, the two eyes would have no way to tell the floor from the board.

They do not agree, and the disagreement is the signal. Point by point up the design, the floor’s disparity and the board’s differ by up to 46.6 arcminutes, with a median of 22.1.

Quoting the raw floor disparity instead would have been the same kind of error this collection recorded when a search returned the same wrong answer at three iteration counts: a self-consistent quantity, correctly computed, that is not the quantity the argument needs.

Disparity to depth, and the wall the reader's own head puts inz = b·D/(b − d). Zero disparity puts the point on the screen at 3.2 m; crossed disparity brings it forward; and at d = 63 mm — the separation of the eyes — the point reaches infinity. Past that the display is asking the eyes to diverge, which they cannot do, so the depth budget is set by the width of the reader's head and by nothing about the scene.010203040-2502550disparity on the screen — millimetreswhere the point is depicted — metres from the eyesd = 63 mm — the eyes' separationon the glassscreen at 3.2 m · eyes 63 mm apartthe ceiling is the head, and it does not move when the screen does
Fig. 3 The relation the visual system is exploiting. Disparity and depth are tied together by the eye separation, so a set of disparities is a depth map, and the depth map two eyes build of a pavement painting is a map of a floor.

The comparison that matters

An anamorph’s marks are on the floor, at a range of distances from the viewer. The picture they stand for is an upright board at the ground line.

Both give the two eyes a disparity — anything at a finite distance does. The question is whether the disparity the floor gives is the disparity the board would give, because that difference is the signal that says these marks are not where the picture says they are.

What the second eye is toldThe difference between the disparity the floor marks give and the disparity the same design on an upright board would give, at 63 mm of eye separation. The worst is 46.6 arcminutes and the middle of the design is 22.1; human stereoacuity is a few tens of arcseconds.0.06 m up the design2.2′0.32 m up the design12.2′0.58 m up the design22.8′0.84 m up the design34.3′1.10 m up the design46.6′floor against board, per point of the designone eye reads it exactlyand nobody has one eye
Fig. 4 The difference between the two, point by point up the design, at 63 millimetres of eye separation. The worst is about 47 arcminutes and the middle of the design is about 22. Human stereoacuity is a few tens of arcseconds — that is, a few hundredths of this.

The worst disagreement is 46.6 arcminutes and the median is 22.1. Stereoacuity — the smallest disparity a person can detect — is a few tens of arcseconds at these distances, which is a few hundredths of a single arcminute.

The signal is therefore three orders of magnitude above threshold. A viewer standing at the mark is not being asked to notice something subtle; they are being told, loudly and continuously, that the marks are on the ground.

Why it fails in person and works in the photograph

This is the point at which a measurement can be misread, so it is worth being careful.

A pavement painting standing up as a hole in the road, looked at with two eyes from the marked spot, does not usually work. Anyone who has stood on one knows it: the illusion is weak in person and overwhelming in the photograph on the poster next to it. That has always been explained by saying photographs have one eye, which is true and is usually left at that.

The measurement above puts a number on it, and the number says the explanation is not merely qualitatively right — the disparity signal is enormous, far above threshold, and it is the strongest cue in the scene. Nothing else about the picture is competing on that scale.

One boundary before that. Everything measured here is a disparity — an angle between two directions, computed from two centres of projection — and a disparity is geometry. Whether a person notices one, how their visual system weighs it against the other cues, and what they end up believing about the floor are facts about seeing, and this collection has no standing on any of them. The stereoacuity figure quoted above is borrowed from that literature to say the signal is not marginal; nothing else here depends on it, and the geometry would be the same for an observer who could not fuse two images at all.

What is competing is everything monocular: perspective, texture gradient, occlusion, shading. Those are strong cues too, and in a well-made anamorph they all point the other way. The result is a conflict, and conflicts get resolved differently by different people, at different distances, with different amounts of squinting — which is exactly the reported experience of standing on one.

Closing one eye resolves it immediately and completely, and that is the test the measurement predicts.

What the conflict actually looks like

The measurement says the stereo signal is three orders of magnitude above threshold, and experience says that people standing on a good pavement painting sometimes do see the illusion. Both are true, and the reconciliation is worth having because it is the only part of this subject that is not geometry.

Depth cues are combined rather than adjudicated. A person looking at a painted hole in the road receives:

  • disparity, saying flat floor, at enormous strength;
  • perspective and texture gradient, saying deep hole, also strong;
  • occlusion, saying nothing, because a painting occludes nothing;
  • accommodation and vergence, saying flat floor, weakly at these distances;
  • motion parallax, saying flat floor the moment the viewer’s head moves at all.

Four of those say floor and two say hole, and the four include the strongest. So the expected outcome is that the illusion fails, and it does — the reported experience of standing on one is that it looks like a painting until one eye is closed.

What survives with both eyes open is not the depth illusion but the rendering: the shading, the cast shadows and the perspective all read as a solid object, and the object reads as a solid painted flat on the ground. Which is exactly what it is, and is why these things are still worth looking at.

The photograph is a different matter entirely, and the difference is one number: the camera’s single centre of projection removes the four cues that say floor, leaving only the two that say hole.

Moved 50 mm sideways, the picture becomes an elationThe intended design and the one a displaced eye actually sees, drawn over each other. The map between them fixes the ground line pointwise, so the departure is exactly zero there and reaches 42.5 mm at 1.38 m up.ground line — right from anywherefaint: intended · solid: seendeparture (mm) against height (m)01.38the error map is an elation, with the ground line as its axisno characteristic ratio — nothing off the axis is fixed50 mm sidewayszero on the axis, 42.5 mm at the top
Fig. 5 What one eye of a pair is being given, drawn at the shortest step the slider offers — 50 mm, a little more than the 31.5 mm each eye is actually displaced by. The error is zero along the ground line and grows upward, and the other eye gets the mirror image of it.
The room the eye may stand inA section through the design eye, containing the line of sight and the vertical, at a tolerance of 10 mm on a design 1.8 m wide. 36 mm along the sight line, 14 mm across it, and 31 cm³ altogether.toward the designup10 mm14 mm across36 mm along the sight line
Fig. 6 The room a single eye may stand in, for comparison with the head drawn into it above. Its narrow axis is across the line of sight, which is exactly the axis a pair of eyes is separated along.

The built set, which has the same problem

The same measurement applies to something physical rather than painted, and this collection has already built it.

A forced-perspective set is a colonnade four metres deep cut so that its picture is the picture of one eighteen metres deep. From the design eye the match is exact to sixteen decimal places. What gives it away is the second eye — and the giveaway is not the ratio anybody would predict, because the two eyes’ disparity for a real eighteen-metre colonnade and for a four-metre one cut to look like it are different by an amount set by the depths, not by the sizes.

The film industry’s answer is the same as the pavement painter’s: use one lens. A forced-perspective set photographed by a single camera is exact; the same set shot in stereo falls apart, which is why the technique nearly vanished when stereo photography arrived and returned when it left.

A set cut for one eye, and what the second one readsFive columns in 2 m of real depth, cut so that their picture is the picture of a row in 16 m — the far one at 0.222 of full size, and the match is exact to 2.8e-17. The disparity between two eyes 63 mm apart is 56.3% of the deep row's, which is a ratio of reciprocal depths and not the 8.0× the set was cut at.eyefaint: the row it imitates · solid: the row that is builtbuilt row15.8 pxthe deep row28.0 pxdisparity across the row, two eyes 63 mm apartfar column cut to 0.222the eyes read 56.3%, not 12.5%
Fig. 7 The set, and the eye it was cut for. The taper is forced rather than chosen — height and width both scale as the real depth over the intended one — and the match from the design eye is exact to sixteen decimal places.

Two eyes are a pair of pictures

There is a cleaner way to say all of this, which connects it to a subject this collection has already built machinery for.

Two eyes are two cameras with a known baseline. A pair of pictures determines the scene’s shape — two views give shape and no size — so a pair of eyes looking at an anamorph determines the shape of what is in front of them, and what is in front of them is a floor with marks on it. The reconstruction is correct. There is no illusion for the pair to be fooled by; the pair is doing its job.

The illusion is available only to a single view, because a single view is what leaves the depth free. That is the same fact from the other side: an anamorph exploits precisely the ambiguity a monocular picture has, and adding a second view removes the ambiguity and the exploit together.

Which makes the phrase “correct from one point” more literal than it sounds. The anamorph is correct from one point, and two eyes are not a point.

A point on the left is a line on the rightFive points in the left picture. Each one fixes a line in the right picture — the image of the ray it came along — and its match lies on that line to 1.4e-11 px. The five lines meet at the epipole, 3.9e-10 px from concurrent. Knowing where a point is in one picture does not say where it is in the other; it removes one of the two degrees of freedom.1234512345a point is chosen hereand must be on this linematch to its own epipolar line: 1.4e-11 px5 of 44 correspondences drawn
Fig. 8 The mechanism by which a second view removes the ambiguity. A point in one picture is a whole line of possible world points; a second view cuts that line at one place. An anamorph relies entirely on the first sentence being all there is.

What the interocular distance decides

The disparity signal is proportional to the eye separation, so it is worth asking what changes it.

A child has eyes about 55 millimetres apart rather than 63, so gets about 13 per cent less signal — and is also shorter, which moves the eye off the design point along the tight axis by far more than that gain is worth.

Distance from the design. The disparity of anything falls as the square of the distance, so an anamorph designed to be read from ten metres gives a much weaker giveaway than one designed for three. Very large street paintings, read from a balcony, work considerably better in person than small ones read from standing height, and this is why.

Nothing about the design’s size. Making the painting bigger and moving the viewer back in proportion keeps the picture the same and weakens the disparity — so the illusion improves with scale, at constant angular size, purely because everything is further away.

That last one is a useful piece of advice that falls straight out of the arithmetic: for a better anamorph, build it bigger and stand further back. Nothing about the geometry changes and the strongest cue against it weakens as the square.

The room, direction by directionNinety-six directions, each bisected until the picture is 10 mm wrong. Along the line of sight the eye may move 36 mm; across it, 14 mm. The curve is a shape rather than a scatter, which is what makes "one viewpoint" a solid.010203020406080angle between the step and the line of sight, in degreeshow far the eye may go before the picture is 10 mm wrong2.6× longer than it is wideand the long axis is the sight line
Fig. 9 The other constraint working the same way. The room the eye may stand in is set by the tolerance and by the geometry; standing further back for the same angular design makes the room larger in absolute terms too, so both of the difficulties ease together.

The other place the second eye decides everything

There is a family relationship worth drawing out, because it explains why this collection has three separate essays that end in the same place.

A curved mirror has no eye at all — continue the lines of sight behind a mirror ball and they pass through no common point, so a photograph of one is a projection of nothing from anywhere. A picture taken through water is the same. A forced-perspective set has an eye and only one. An anamorph has an eye and only one.

In each case the question is how many centres of projection does this arrangement have, and the answers are none, one, and one. What varies is what that costs, and the cost is always paid by whatever expects there to be two.

A single camera is indifferent: it has one centre and asks for one. A person is not, and the amount they are inconvenienced is the amount their two centres disagree.

That is a tidier statement of the whole subject than “anamorphs are tricks”, and it has the advantage of being measurable. The disparity above is the disagreement, in arcminutes, between two centres of projection looking at an object built for one.

A mirrored cylinder, and the smear that stands up in itEvery mark on the paper is where a ray from the eye, reflected in the cylinder, lands. Run backwards from the mark, the light returns to the eye to 9e-16. One part of the design gets 15.5 times as much paper as another, which is what makes it unreadable flat.the sheet, seen from abovemirrorthe eye, 1.6 radii upwhat stands up in the mirrorthe light path reverses to the eye to 9e-16scale varies 15.5× across the design
Fig. 10 The catoptric case, where the same accounting runs. A design smeared on a table and read in a mirrored cylinder is correct from one place; a viewer with two eyes reading it in the mirror has the same complaint, and the same photograph fixes it.
Two rays, 0.31 mm apart, in the plane that contains bothThe ray from the left eye through its mark and the ray from the right eye through its. With the marks placed exactly they meet, to 1.1e-13 m. With the same marks read to 1 px they miss by 0.31 mm at a range of 7.29 m. Triangulation is not an intersection; the reported point is a choice about what to minimise, and the gap is the part a residual alone will not tell you.midpoint — 0.31 mm gapfrom the left eyefrom the right eyegap 0.31 mm at 7.29 mexact marks: 1.1e-13 m
Fig. 11 Two eyes treated as what they are — a stereo pair with a 63 mm baseline. The pair triangulates whatever is in front of it, and what is in front of it is a floor.
One reconstruction, drawn at its own scale and at 2.0×The courtyard recovered from two pictures, seen from above. Every pairwise distance ratio matches the world's to 9.3e-11, so the shape is exact. The size is not determined at all: the right-hand plan is the same reconstruction 2.0 times larger and fits the same two pictures equally well. One measured length in the scene — here 0.800 m — fixes it, and nothing in the pictures can.as recovered12.698 across× 2.0 — same two pictures25.397 acrossworst ratio deviation 9.3e-11size fixed only by a supplied 0.80 m
Fig. 12 And the general fact about pairs. Two views fix the shape of what is in front of them and not its size. An anamorph survives the second constraint — a scaled anamorph is another anamorph — and fails the first, because the shape two eyes recover is the shape of the floor.

The general point

An anamorph is usually described as a trick, and the description carries an implication that it is a trick on the visual system. It is not. It is a construction that is exactly correct for a stated viewpoint, and the visual system is not fooled by it because the visual system has two viewpoints and is entitled to use both.

The trick is on the camera, which has one. Everything about anamorphosis that reads as psychological — the illusion, the surprise, the failure in person — comes out of the difference between one centre of projection and two, and both halves of that are geometry.

The same sentence covers the built set, the pavement painting, and the trompe-l’œil ceiling. Each of them is exact from a point. Each of them is photographed rather than visited. And each of them is a projection through a centre that a person, having two, cannot occupy.

The same 60° view, projected onto a plane and onto a cylinderOn the plane every straight line stays straight (3e-13 px of bend) and the edges stretch; on the cylinder the stretch is even and straight lines bow by up to 1346 px.flat picture plane — straight lines stay straightcylindrical picture surface — even stretch, bowed linesone scene, two picture surfacesneither is the distorted one
Fig. 13 And a reminder of what the design is a design of. The intended picture is an ordinary flat picture standing at the ground line; the anamorph is only the map that puts it on the floor. Everything a flat picture suffers from, an anamorph suffers from more.

Where this goes next

eyefaint: the row it imitates · solid: the row that is builtbuilt row21.0 pxthe deep row28.4 pxdisparity across the row, two eyes 63 mm apartfar column cut to 0.300the eyes read 74.1%, not 22.2% A set cut for one eye Build a colonnade four metres deep and cut every column so that its picture is the picture of one eighteen metres deep. The taper is forced rather than chosen — height and width both scale as the real depth over the intended one — and the match from the design eye is exact to sixteen decimal places. What gives it away is the second eye, and not by the ratio anybody would predict. toward the designup10 mm14 mm across36 mm along the sight line The room the eye may stand in An anamorph is correct from one point, and one point is not a thing a person can occupy. Fix a tolerance on the picture and the set of eye positions that meet it is a solid — for a design 1.8 m wide and a ten-millimetre tolerance it is 36 mm long, 14 mm across and 31 cubic centimetres altogether, a spindle pointing along the line of sight. Ten times the tolerance is a thousand times the room. as recovered0.308 across× 3.4 — same two pictures1.046 acrossworst ratio deviation 6.3e-14size fixed only by a supplied 0.80 m Two views give shape and no size Every pairwise distance ratio in a courtyard recovered from two photographs matches the real one to fourteen digits. The courtyard's actual size is not merely uncertain — it is absent, and a reconstruction three and a half times larger fits the same two pictures exactly as well.

Shares its objects with

Essays that name at least two of the same things, and that neither author linked.

Named objects

A flat tag is an object no other essay names yet.

AnamorphosisBinocular disparityDepth cueForced perspectiveGround planeHomologyInterocular distanceStation pointStereoacuityVirtual image