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

The marks name the place, not the height

Run the site's own round trip on an anamorph — hand the machinery the floor marks and ask for the eye back — and it returns the spot on the floor to eleven decimal places with nothing assumed at all. It does not return the height. What the marks fix is the product of the eye's height with the design's aspect ratio, and no amount of looking at the floor separates the two.
20 min read 14 figures The round tripDetermined up to somethingOne picture, one length

Worth reading first: A floor anamorph is three numbers · Where the anamorph still works · Recovering the camera from the picture it drew.

This site has one move it makes in every field, and it is the reason the figures can be trusted. A construction is run forward from a stated camera, and then run backwards from the picture it produced, and the recovered camera is compared with the one that drew it. The foundation phase gets a focal length back to one part in 101510^{15} from twelve drawn edges. The metrology field gets a height out of a photograph. The multi-view field counts what a reconstruction cannot fix.

An anamorph invites the same move, and the invitation is unusually direct. The marks on the floor were cast from one eye; a reader standing in a gallery wants to know where that eye was. Nothing about the marks is hidden and nothing has been thrown away — so the recovery ought to be exact.

It is, in part. The part it is not exact in is the interesting one.

The eye, read back out of three numbersFour marks and the design's stated proportions, and the eye comes back 4.8e-12 mm from where it stood. Assume a different shape and the floor position is unchanged to 7.4e-11 mm while the height runs from 0.762 m to 2.592 m — the marks fix the product height × aspect, and it is 2.592.axis — the ground line, fixed pointwisecentreaxisthe ground linefixed pointwisecentre(0.150, 4.020)height + distanceratio-1.481481−distance / heightheight × aspect = 2.592, and neither aloneeye recovered to 4.8e-12 mmeye 1.62 m up, 2.40 m backthree numbers back to the eye: 2.2e-16 m
Fig. 1 Four marks, the design’s stated proportions, and the eye that cast them. With the shape known the recovery lands on the original to within picometres; assume a different shape and the spot on the floor is unchanged while the height moves by a factor of three.

The recovery, and what it is given

The setup is deliberately mean. The recovery is handed four floor marks and one sentence: the intended picture is a rectangle of aspect aa, standing upright with its base on the ground line. It is not told the eye’s height, its distance, or where it stood sideways.

Three unknowns. The residual it minimises has four components, and each is a statement about the preimage of the four marks on the upright plane: the corner is a right angle; the two width vectors agree as vectors, so the figure is a parallelogram; and the ratio of the sides is the stated one. Gauss–Newton from a deliberately poor start — an eye one metre up and one metre back, which is neither — converges in a few dozen iterations.

With the aspect right, the recovered eye is 4.8e-12 mm from the eye that cast the marks. That is not a tolerance; it is the solver hitting the double-precision floor and stopping.

The design, the eye, and where the rays landA 0.90 m design standing on the ground line, an eye 1.62 m up and 2.40 m back, and the marks the rays leave on the floor. Above: the section, with the ray through the top of the design reaching 3.00 m away. Below: the marks themselves, in plan.floorpicture planeeye level — no mark above thiseye · 1.62 m up, 2.40 m backthe ground line, seen from abovethe mark runs to 3.00 ma point 1.62 m up casts no mark at all
Fig. 2 The forward construction the recovery inverts. Everything it is given is in the lower panel — the marks, and where the ground line is — and everything it must produce is in the upper one.

The claim that was made here and was wrong

The first version of this machinery carried a docstring saying that the system was over-determined by one: four conditions on three unknowns, with a residual that the recovery could fail, which would make the residual a test.

The site’s own habit is to check a claim of that kind by trying to break it. Feed the recovery a deliberately wrong aspect — tell it a rectangle nearly three and a half times as wide as it is tall, when the truth is 1.6 — and it should fail to drive the residual down.

It returned 5.9e-16 and a different eye.

That is what a confident wrong answer looks like, and it is the failure mode this site has recorded twice before in other fields: a sensitivity computed in a variable nobody perturbs, and a cross-ratio evaluated at the one input where it cannot fail. Here it is a count of conditions that was made by counting rather than by trying.

Moved 250 mm vertical, the picture becomes a homologyThe 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 212.5 mm at 1.38 m up.ground line — right from anywherefaint: intended · solid: seendeparture (mm) against height (m)01.38the error map is a homology, with the ground line as its axischaracteristic ratio 1.154321250 mm verticalzero on the axis, 212.5 mm at the top
Fig. 3 The reason, in one picture. An upward step is a homology whose centre is at infinity: it stretches the reconstructed picture’s height and leaves its width alone. So the family of eyes above one spot on the floor produces rectangles of every aspect, and every one of them is a perfectly good rectangle.

What the marks actually fix

Once the reason is visible the true statement follows immediately, and it is sharper than the false one.

Raising the eye by δ\delta multiplies the reconstruction’s height by (1+δ/ey)(1 + \delta/e_y) and leaves its width alone. So along the vertical family the aspect runs continuously through every positive value, and each member of the family is a rectangle. There is no residual to fail because there is nothing inconsistent about any of them.

What the family does not change is where it starts. Handing the same four marks to the recovery with four different assumed aspects gives four different eyes, and their positions on the floor agree to 7.4e-11 mm — the same double-precision floor as before. The lateral position and the distance come back with nothing assumed about the design at all.

And the heights are not arbitrary. They run from 0.762 m to 2.592 m across the four assumptions, and

height×aspect=2.592\text{height} \times \text{aspect} = 2.592

in every one of them, to nine digits. The marks fix that product and neither factor.

The four rows

The whole result fits in a table, and it is worth reading as one because the pattern in the last column is the finding.

assumed aspect recovered height height × aspect
1.0 2.592 m 2.592
1.6 1.620 m 2.592
2.4 1.080 m 2.592
3.4 0.762 m 2.592

Four different beliefs about the design, four different eyes, one number in common — and the lateral position and the distance are identical in all four rows to eleven decimal places. The height runs over a factor of 3.4 across those assumptions, which is the difference between a reader kneeling and a reader on a ladder. Nothing in the marks distinguishes them.

The true eye is the second row, and there is nothing in the four floor marks that says so.

Why the solver is numerical

There is a closed form for this. The three numbers of the forward map determine the eye, so a route through them exists, and it would be shorter than the iteration.

It is deliberately not used, and the reason is the one the site applies to every recovery it builds: a closed form reports an answer and hides how firmly the answer is held. Running Gauss–Newton with Levenberg damping from a start that is wrong in all three coordinates makes the conditioning visible, because the step the solver is willing to take in each direction is the thing the damping is negotiating with. The vertical direction is where the residual surface is flat, and a solver walking on it says so by walking a long way for nothing.

That flatness is the result. A closed form fed a wrong aspect would return a wrong height with the same confidence and no sign that anything was undetermined; the iteration at least shows where the ground is level.

The sentence this is a case of

The site has been making a version of this statement since its second phase, and it is worth putting the three side by side because they look like different results and are one.

One view supplies every ratio and no size. A world 137×137\times larger photographed from 137×137\times further away gives an identical picture, so a single photograph cannot say how big anything is until one length in it is named.

Two views give shape and no size. A stereo pair reconstructs the scene up to a scale factor, and the scale comes from a measured baseline or a known object.

And here: an anamorph’s marks give a place and not a height. The unmeasured quantity is not a length in the scene; it is the reader’s own eye, and the thing that would pin it is the picture’s proportions. Same structure, one rung further in: the free parameter has moved out of the depicted world and into the viewer.

Two scenes 137× apart, and the one picture they both makeEverything in the second plan — the room, the eye's distance, the eye's own height — is 137 times the first. Every projected vertex agrees to 1e-13 px. A single photograph has no scale, and this is what that means.a room 2.8 m across, eye 1.6 m up1 mthe same plan, 137× bigger137 midenticalpicturesthe picture — both scenes, drawn twice, one on top of the otherlargest disagreement 1e-13 px over 8 verticesone length has to come from outside the picture
Fig. 4 The metrology field’s version. Two scenes differing by a factor of 137, photographed from distances differing by the same factor, produce the same picture to arithmetic noise — so one view determines every ratio and no absolute size.
One reconstruction, drawn at its own scale and at 3.4×The courtyard recovered from two pictures, seen from above. Every pairwise distance ratio matches the world's to 6.3e-14, so the shape is exact. The size is not determined at all: the right-hand plan is the same reconstruction 3.4 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 recovered0.308 across× 3.4 — same two pictures1.046 acrossworst ratio deviation 6.3e-14size fixed only by a supplied 0.80 m
Fig. 5 And the two-view version. The reconstruction is right in every angle and every ratio, and its overall size is a free parameter that no amount of further looking removes.

Why the floor position survives and the height does not

The asymmetry is not an accident of which coordinates were chosen, and the three moves of the rung below say why.

A sideways step shears the reconstructed picture. A sheared rectangle is not a rectangle, so the requirement that the preimage be a rectangle rules out every lateral position but one. The lateral coordinate is pinned by a condition that a wrong value visibly violates.

A step in or out scales the picture about a point inside it — a similarity in neither direction, and in fact a projectivity that changes the ratio of the design’s two halves. Again a wrong value produces something that is not a rectangle with parallel sides, and the distance is pinned.

A vertical step stretches the height and nothing else. A stretched rectangle is a rectangle. There is no condition to violate, and the coordinate floats.

So it is the rectangle’s own symmetry that hides the height: a shape whose defining property survives one of the three moves cannot pin the coordinate that move belongs to.

Three numbers, and the whole mapThe rabatted design maps to the floor marks by a homology: the ground line is fixed pointwise, one point off it is fixed, and one ratio does the rest. Rebuilding every mark from those three misses by 2.2e-15 m.axis — the ground line, fixed pointwisecentreaxisthe ground linefixed pointwisecentre(0.150, 4.020)height + distanceratio-1.481481−distance / heightevery mark rebuilt to 2.2e-15 meye 1.62 m up, 2.40 m backthree numbers back to the eye: 2.2e-16 m
Fig. 6 The forward map’s three numbers again. The centre carries the sum of height and distance, and the ratio their quotient — so the pair determines both. That is a statement about the map, which the recovery does not have; the recovery has four marks.

What would fix the height

Four things would, and listing them is the practical content of the result.

A stated length. Telling the recovery that the design is 1.38 m tall fixes the aspect and therefore the height, and this is why the printable anamorph is the one figure on this site that quotes a viewing distance unconditionally. The sheet is printed in millimetres, so the design’s scale is not an assumption; it is a fact about the paper.

Any mark off the ground plane. A single point of the design at a known height, or a second design standing on a second ground line, breaks the vertical family at once, because the family’s stretch would move it.

A non-rectangular design. The float exists because a vertical stretch preserves the property being asserted. A design containing a circle does not have that problem: a stretched circle is an ellipse, and one condition is enough.

The construction’s own three numbers, if they can be obtained. The centre at ey+eze_y + e_z and the ratio −ez/ey-e_z/e_y determine the height and the distance separately, and getting them needs the map rather than four marks — which in practice means knowing the design, which is the assumption again.

An anamorph, and the sheet it can be checked onThe word stands 18 mm high on a 150 × 105 mm sheet; each corner is projected onto the paper from an eye 52 mm above it. The marks run 2.5 times as deep as the letters are tall, and from that eye they cover the letters to 1e-13 px.the sheet, 150 × 105 mmthe eye, 52 mm up and 80 mm backgrey: the word standing upright · black: the same word on the paper142 mm from the sheet's middle
Print at 100%. Put one eye 52 mm above the sheet, 80 mm beyond the near edge, on the dashed centre line, and read.
Fig. 7 The sheet that closes the loop, and the reason it can. Its design’s size is stated in millimetres on the paper, so the free parameter is filled in by the printer rather than assumed by the reader.
Two points, and everything metric followsThe imaged circular points are where the horizon meets the image of any circle in the plane, and they are a conjugate pair — the first coordinate here is 169.5 − 446.0i. A rectification built from them and nothing else returns the world's angles to 2.2e-13° and its length ratios to 5.6e-15, and no length at all.horizonthe horizon does not cut the circle — the pair is complexrectified from the two points aloneangles: 2.2e-13°ratios: 5.6e-15length: —circle of radius 1.05 ma dash is a quantity two points cannot buy
Fig. 8 The foundations field’s version of the same repair, one field away. A rectification built from a plane’s two circular points recovers every angle and every ratio and refuses to name a length, and prints a dash where the length would go rather than a number.
The ground, turned into a planFour corners of a rectangle of known proportions fix the homography. Three lengths it was never given come back to 1e-15 relative — so every measurement on that plane is available, in units of the rectangle's own width.the picturethe same plane, rectifiedacross — 0.6000 widths (true 0.6000)along — 0.7667 widths (true 0.7667)diagonal — 1.2023 widths (true 1.2023)worst error 1e-15 relativethe probes were not used to build the map
Fig. 9 And the rectification itself. A photographed plane is brought back to a similarity of the real one — angles right, ratios right, size free — which is exactly the ladder this essay’s result sits on.

What a visitor can do with this

Three practical consequences, and each is a statement about what a reader can and cannot infer from a painted floor.

The spot is findable without knowing anything about the picture. A reader who can see the marks and can see where the design’s ground line is can work out where on the floor to stand, exactly, with no assumption about what the picture depicts. That is a stronger statement than it sounds: it means the plan position of the design viewpoint is a property of the marks, recoverable by anybody, and it does not depend on recognising the subject.

The height must be guessed, and the guess is a guess about the picture. A reader who assumes the depicted thing is roughly as tall as it is wide will stand at one height; a reader who assumes it is a tall narrow panel will stand at another; both are consistent with everything on the floor. The usual resolution — stand with the eye at the height of an adult — is an assumption about who the picture was made for rather than a reading of it.

A wrong height gives a picture that is wrong in a specific way and not in a detectable one. The reconstruction from a too-high eye is the intended design stretched vertically. Every straight line is still straight, every crossing still crosses, and only the proportions have gone. So there is no internal evidence in the resolved picture that anything is wrong, which is exactly the sense in which the marks do not carry the height.

The conditioning, read the other way

There is one more thing the recovery says, and it is the tolerance essay’s result arriving from the opposite direction.

The direction the recovery is least sure of is the vertical one, and the direction the anamorph most forgives is not the vertical one — a vertical step costs exactly as much as a sideways step of the same size. Those two facts sit together without contradiction and it is worth seeing why, because the natural expectation is that a badly determined direction is a forgiving direction.

The recovery is badly determined vertically because the shape it is asked about does not change under a vertical stretch. The picture very much does change: everything gets taller. A reader looking at the anamorph from too high sees a stretched version of the intended design and would notice at once. What they could not do is prove the design was not meant to be that shape.

So the anamorph’s tolerance and the recovery’s conditioning measure two different things — how wrong the picture looks, and how much the marks constrain — and the two need not agree. Conflating them is easy and this essay is one of the places it would have been easy.

Projective, affine, metric — what each stage buysThe photograph fixes the plane only up to a projectivity: the midpoint of a receding side lands 0.3970 of the way along. Supplying the plane's vanishing line buys the midpoint back exactly and nothing else. Supplying the image of one circle buys the last three numbers, at which point the right angle is 90.000° and two equal sides measure 1.000000. The cross-ratio is 1.333333 in all three, because it was never lost.projectiveaffinemetricmidpointtwo equal sidesa right anglecross-ratioprojective———1.333333333affine0.500000——1.333333333metric0.5000001.00000090.000°1.333333333— means the stage does not determine it at allcross-ratio 1.333333 throughout
Fig. 10 The ladder all of this sits on. Each rung buys a quantity with a piece of information, and prints a dash for the quantities that rung does not determine — the discipline that makes the missing height a result here rather than an omission.
A box drawn from a known camera, and the camera recovered from the drawingThree vanishing points found from the twelve drawn edges alone give back the focal length to 2e-15 relative.recovered principal pointused to drawrecoveredgapfocal length947.88947.882e-15principal x345.0345.02e-12angle40.0°40.0°—correct from 22 cm, at 160 mm wide40° across
Fig. 11 The site’s own round trip, for comparison: twelve drawn edges give the focal length back to one part in 101510^{15}, with nothing assumed. The difference is that a box has three directions and a rectangle standing on a ground line has one symmetry too many.
Four constructions, three of them the same mapA shadow, a floor anamorph, a mirror and a rectification, each decomposed into its fixed points and lines. Three are central collineations with a line of fixed points; the fourth is not, and that is the difference between changing a picture and changing where it is seen from.constructionfixed structurea shadow, ground to floorhomology · ratio 0.6719a floor anamorphhomology · ratio -1.4815a mirror in a vertical planehomology · ratio -1.0000a rectificationgeneral · three fixed points3 of 4 are centrala line of fixed points is what they share
Fig. 12 And the map the recovery is inverting, in company. It is the anamorph’s row that matters here: a homology has five numbers and the recovery is given four marks, which is why the accounting has to be done rather than assumed.

One more thing the marks do carry

The product height×aspect\text{height} \times \text{aspect} is not a bookkeeping artefact of the parameterisation, and it has a reading.

Multiply out what the vertical family does. A design of true height hh standing on the ground line, seen from an eye at eye_y, casts its top edge at a depth that depends on the ratio h/eyh/e_y and on nothing else about either. Fixing the marks fixes that ratio; the width of the design is fixed independently by the lateral positions, which the marks do determine. So the product the recovery reports is, up to the design’s width, the fraction of the way to the horizon that the design’s top edge reaches — one number that the floor plainly shows, since the marks stop where they stop.

That is why the product is what survives. It is the one quantity a reader can measure with a tape on the floor: how far the marks run. Everything else the recovery reports about the vertical direction is a re-description of it, and the height and the aspect are two ways of dividing one measured length between the picture and the person looking at it.

A 3.4 m object measured from one picture, 18 m awayThe base, the horizon crossing, the top and the vertical vanishing point have a cross-ratio of 1.9101. With the eye at 1.62 m that gives 3.400 m, against a true 3.4 m. The camera is not consulted.horizon — the eye's own heightbase — 0 mhorizon crossing — 1.62 mtop — 3.40 m recoveredthe vertical vanishing point is 8586 px above this framerecovered 3.400 m · true 3.400 m3.0 cm per pixel of click error
Fig. 13 The same division in the metrology field, run the other way: a height comes out of a photograph once one length in the scene is named, and the cross-ratio along the vertical is what carries it.
Where the reader has to be for a 40° picture to be correctShown 160 mm wide, this picture is a correct projection only from 22 cm away. Drawn to scale.the picture, 160 mm wide22 cm40°the eyefocal length 948 px22 cm at 160 mm wide
Fig. 14 And the quantity this whole essay is about, in the form the site states it everywhere else: the point a picture is correct from, computed rather than recommended. For an anamorph two of its three coordinates come out of the marks and the third comes out of the picture.

The short version

Hand an anamorph’s floor marks to a recovery and it returns the reader’s position on the floor exactly, with nothing assumed about the design. It returns the eye’s height only in the product of that height with the design’s aspect ratio, and any of the infinitely many pairs with the right product satisfies every condition the marks impose.

The reason is that raising the eye stretches the reconstruction vertically and leaves it a rectangle. A shape that survives one of the three ways an eye can move cannot fix the coordinate belonging to that move.

An anamorph therefore names the place to stand. How high to hold the eye is a statement about the picture’s proportions, and the picture is the one thing the marks do not carry.

Where this goes next

floorpicture planeeye level — no mark above thiseye · 1.62 m up, 2.40 m backthe ground line, seen from abovethe mark runs to 3.00 ma point 1.62 m up casts no mark at all A floor anamorph is three numbers An anamorph is described everywhere as a picture stretched until only one viewpoint can read it, which says what it looks like and nothing about what it is. Cast one onto a floor and fit a map to the marks, and the map turns out to be a planar homology — a line of fixed points, one point off it, and a single ratio. Those three numbers are not a description of the eye. They are the eye. recovered principal pointused to drawrecoveredgapfocal length853.90853.904e-15principal x345.0345.02e-12angle44.0°44.0°—correct from 20 cm, at 160 mm wide44° across Recovering the camera from the picture it drew Draw a box from a known camera, forget the camera, and get it back from the twelve drawn edges alone. Agreement to one part in 10¹⁵ is a statement about the geometry, because the only thing that crossed between the two halves was a list of line segments. a room 2.8 m across, eye 1.6 m up1 mthe same plan, 137× bigger137 midenticalpicturesthe picture — both scenes, drawn twice, one on top of the otherlargest disagreement 1e-13 px over 8 verticesone length has to come from outside the picture The one thing a single view cannot give Make the world a hundred and thirty-seven times larger and move the eye a hundred and thirty-seven times further away, and the picture does not change by a measurable amount. Every ratio in a scene is recoverable from one photograph and no size is, and that is not a caveat about the method — it is the shape of the method.

What links here

Computed from the collection, not written here: the essays that point at this one.

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.

AnamorphosisCharacteristic ratioConditioningdegrees of freedomDemonstrationFixed pointFree parameterPlanar homologyRabatmentreconstruction ambiguityReference lengthscale ambiguitysingle-view metrologyStation pointViewing position