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

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.
18 min read 12 figures The viewer is in the geometryProjected, not constructedThe round trip

Worth reading first: Anamorphosis is only a viewpoint · The point you have to stand at.

Every account of anamorphosis begins with the effect. A picture is drawn so distorted that it means nothing from the ordinary place a picture is looked at, and resolves into a scene from one particular point — a doorway, a peephole, a mark on the floor. That is a true description of what a viewer experiences and it is not a description of anything at all, because it names no quantity. It does not say what the distortion is a function of, and it does not say what a viewer would have to know to undo it.

This site has had three anamorphosis essays since its second phase and each of them draws one anamorph. The first makes the general claim, the second prints one at true scale in millimetres, and the third bends the light once on the way. None of the three says what the map is.

It is a planar homology. That statement is short, it is exact, and everything else in this essay follows from it.

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. 1 The construction, in section above and in plan below. A design stands upright on a line drawn across the floor; an eye is placed at a stated height and a stated distance in front of it; and each point of the design is carried along the ray from the eye to where that ray meets the floor. The marks are what gets painted.

The construction, and the one thing it refuses

The frame is the one Alberti’s section already uses on this site, deliberately, so the two constructions can be laid over each other without changing coordinates. The floor is the plane y=0y = 0. The intended picture stands upright on the plane z=0z = 0, with its foot on the line where the two planes meet — the ground line. The eye is in front of the picture at positive zz, at height eye_y and distance eze_z.

A point of the design at (x,y)(x, y) goes to the floor along the ray from the eye. Setting the height to zero gives the parameter

t=eyey−yt = \frac{e_y}{e_y - y}

and the mark lands at (ex+t(x−ex),  ez(1−t))\big(e_x + t(x - e_x),\; e_z(1 - t)\big).

The interesting part of that formula is where it stops working. For y<eyy < e_y the parameter is greater than one and the mark lands beyond the picture plane, which is where a floor anamorph lives. At y=eyy = e_y the denominator vanishes: a point of the design exactly at the eye’s own height casts a ray parallel to the floor, and it lands nowhere. Above that height the parameter goes negative and the algebra returns an intersection behind the viewer, which is not a mark and must not be drawn as one.

So an anamorph on a floor can carry only the part of the intended picture below the horizon. The sky in any picture with a sky has no anamorph. That is not a limitation of the drawing; it is the construction reporting its own domain, and the machinery refuses rather than returning a point.

The design, the eye, and where the rays landA 0.90 m design standing on the ground line, an eye 2.20 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 1.66 m away. Below: the marks themselves, in plan.floorpicture planeeye level — no mark above thiseye · 2.20 m up, 2.40 m backthe ground line, seen from abovethe mark runs to 1.66 ma point 2.20 m up casts no mark at all
Fig. 2 The same design from a higher eye. Raising the eye lowers the parameter at every height of the design, so the marks pull in toward the ground line and the anamorph gets shorter — and the height at which the construction refuses rises with the eye, because that height is the eye’s.

Rabatment: putting both planes in one plane

To ask what kind of map this is, both ends of it have to be in the same plane. The floor marks already are. The design is on a vertical plane, and the classical way to bring it down is the one every perspective treatise uses under the name rabatment: fold the upright plane about the line it shares with the floor, until it lies flat.

Fold it forward, toward the viewer, so that a design point at height yy becomes a floor point at depth +y+y on the near side of the ground line. Now the map runs from one plane to itself: rabatted design in, floor marks out.

Fit a homography to four of the correspondences and decompose it, and the classification is unambiguous.

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. 3 The floor marks, the ground line, and the one point off it that the map leaves alone. Every mark lies on a line through that point, and each moves along its own line by the same ratio — which is the definition of a homology and the whole content of the map.

What a homology is

A projectivity of the plane has fixed points: the eigenvectors of its matrix. What those fixed points do decides everything the map can be used for, and there are exactly three cases.

A general projectivity has three distinct fixed points forming a triangle. Nothing slides along anything; the map has to be given as a matrix and read as one.

A homology has a repeated eigenvalue whose eigenspace is two-dimensional. A two-dimensional eigenspace is a line of fixed points, called the axis, every point of which stays where it is. The remaining eigenvector is a single fixed point off the axis, the centre. Every other point of the plane moves along the line joining it to the centre — never off that line — and the ratio in which it moves is one number, the same for every point in the plane. A homology is therefore an axis, a centre, and a ratio: two degrees of freedom for the line, two for the point, one for the ratio. Five numbers, where a general homography has eight.

An elation is a homology whose centre has fallen onto its own axis. The axis is still fixed pointwise, no point off it is fixed, and there is no ratio to state.

A homology: an axis, a centre, and one ratioEvery point moves along the line joining it to the centre, by the same ratio 2.4000; every point of the axis stays where it is. Three numbers, and the arrows are all that is left to draw.centrefaint dots: before · solid: afterhomologyratio 2.4000
Fig. 4 A homology drawn as what it does. The line across the drawing is the axis and nothing on it moves. Every other point slides along its own line to the centre, and the distances all scale by one number.
A general projectivity: three fixed points and no line of themThree fixed points, no two of them joined by a line of fixed points. Nothing slides along anything.faint dots: before · solid: aftergeneralno ratio
Fig. 5 For comparison, a general projectivity of the same plane. Three fixed points, no line joining two of them fixed pointwise, and no direction anything slides along.

The three numbers, in closed form

The decomposition of the rabatted anamorph returns the same three things at every eye, and each of them is a short expression in the eye’s own coordinates.

The axis is the ground line. That is immediate once it is pointed at: a design point at height zero is already on the ground line, its ray from the eye meets the floor after travelling exactly the distance to the picture plane, and the mark lands on the point itself. Every point of the ground line is fixed, and the ground line is therefore the axis.

The centre sits at

(ex,  ey+ez)\big(e_x,\; e_y + e_z\big)

in floor coordinates — directly in front of the eye’s lateral position, at a depth equal to the eye’s height plus its distance.

The characteristic ratio is

μ=−ezey\mu = -\frac{e_z}{e_y}

minus the eye’s distance over its height. It is negative because the map carries points across the axis rather than along one side of it, which is what makes a floor anamorph look inside out rather than merely stretched.

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 3.8e-15 m.axis — the ground line, fixed pointwisecentreaxisthe ground linefixed pointwisecentre(0.150, 5.020)height + distanceratio-2.098765−distance / heightevery mark rebuilt to 3.8e-15 meye 1.62 m up, 3.40 m backthree numbers back to the eye: 2.2e-16 m
Fig. 6 The same design and a more distant eye. The centre moves back by exactly the change in distance and the ratio grows in proportion; nothing about the design enters either number.

Two of those three carry the eye’s height and distance separately. The centre’s depth is ey+eze_y + e_z and the ratio is −ez/ey-e_z/e_y: a sum and a quotient, two equations in two unknowns, and solving them is arithmetic rather than fitting. The centre’s other coordinate is where the eye stands sideways. So the three numbers of the map are three numbers of the eye, and going back from one to the other loses nothing measurable.

Why this is a claim and not a re-description

There is an easy way to make a statement like this and mean nothing by it. A homography fitted to four correspondences reproduces those four exactly, whatever the underlying map is; a classification of that homography is then a statement about four points rather than about the construction.

So the check runs the other way. The three numbers are computed from the eye in closed form, a homology is built out of them with no reference to the design at all, and every point of the design is pushed through that homology and compared with the mark the ray actually cast. The design here is twenty-five points and the fit used four of them. Rebuilding the other twenty-one from three numbers misses by 2.2e-15 m.

That is what makes it a map rather than a fit. Three numbers, at metre scale, reproduce marks spread over several metres of floor to within a few femtometres, which is the double-precision floor and not a tolerance anybody chose.

The construction a perspective treatise already knows

Anyone who has met the classical perspective constructions will have recognised the centre by now.

The rabatted eye is what every treatise since Alberti calls the distance point. In the classical layout the eye is folded into the picture plane and marked on the horizon at a distance from the centric point equal to the viewing distance, and the diagonals run to it. Here it has been folded the other way, into the ground plane, and it sits at ey+eze_y + e_z rather than at eze_z because both folds have happened at once: down about the ground line, from a point that was already up at eye_y.

The distance point at 620 px — a picture correct from 14 cmThe orthogonals go to the centric point and the diagonal goes to the distance point; the transversals are where they cross. The distance point's offset is the viewing distance, so moving it moves the reader, and the drawing gives no sign that anything has changed.centric pointdistance point, 202 px off the sheet →620 pxcorrect from 14 cm at 160 mm wide33° across
Fig. 7 The classical pavement laid out with a distance point. The same fold, on the same ground line, from the same eye — and the free parameter the recipe never names is the viewing distance, which is one of the two numbers the homology’s ratio and centre carry between them.

This is the sense in which an anamorph is not an oddity. The construction that lays out a correct pavement and the construction that lays out an unreadable smear are one map at two settings of one ratio. A perspective drawing has an eye far enough away and high enough that the ratio is near one; an anamorph has an eye so close and so low that it is not. Nothing changes kind on the way between them.

Halving a receding rectangle two waysThe diagonals cross at the image of the rectangle's centre, 8e-14 px from it — the construction is exact at every camera because it uses only which lines meet where, and that is what a projection keeps. Halving the drawn side with a ruler instead lands 22.7 px from the image of the side's midpoint.the diagonals against a ruler, at 3.2 mthe diagonals — exactthe ruler — 22.7 px outcorrect from 23 cm, at 160 mm wideharmonic set -1.000000 · 8e-14 px
Fig. 8 The straightedge construction that finds the image of a rectangle’s centre. It survives a projection because it names only which lines meet where, and a homology maps lines to lines — so it survives an anamorph too, and works on the floor marks unchanged.

The classical construction is the same map at a different ratio

It is worth saying what that identification licenses, because it is more than a pleasing remark. Alberti’s lateral section computes where a pavement’s transversals fall by rabatting the eye into the drawing and reading off intersections; this site checks it against a pinhole camera and finds the two agree to within a tenth of a picosecond of drawing precision. The construction is a homology executed with a straightedge.

So the entire classical apparatus transfers to an anamorph unchanged. The diagonal that lays out the next bay of a receding fence lays out the next bay of an anamorphic one. The measuring point that puts a post at three metres puts an anamorphic post at three metres. Nothing has to be re-derived, because a homology is a homology whatever its ratio, and the ratio is the only thing an anamorph changes.

What does not transfer is the eye’s comfort. At a ratio near one the construction produces a picture a reader can stand almost anywhere to look at; at −1.48-1.48 it produces one they cannot. That is a fact about the number rather than about the method, and it is the whole difference between a pavement and a pavement anamorph.

What the axis being the ground line buys

An axis of fixed points is a strong property and it has a consequence a visitor can check without any of this machinery.

Every mark on the ground line is right from everywhere. The bottom edge of a pavement anamorph — the part painted along the line where the design stands — is not distorted at all, because the map fixes it pointwise. Move away from the design viewpoint and the picture falls apart upward from that line, and the line itself stays correct.

That is why a floor anamorph painted against a wall, or against the far edge of a room, holds together better than one painted in the middle of an open space: the axis is somewhere the eye naturally reads, and the departure grows from it. The next rung measures how fast.

Moved 250 mm in, 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 32.3 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 0.905660250 mm inoutzero on the axis, 32.3 mm at the top
Fig. 9 The design and the picture a viewer standing a quarter of a metre too far back actually gets. The two coincide exactly along the ground line — the axis is fixed pointwise for any wrong eye, not merely for the right one — and separate upward from it.

The census this belongs to

A homology is not a rare object on this site. It is what several constructions built in different fields and different phases turn out to be, and none of them was written knowing about the others.

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. 10 Four maps, decomposed. A shadow, a floor anamorph and a mirror all have a line of fixed points; a rectification does not. The first three change where a plane’s picture is cast from, and the fourth changes the plane’s picture into another picture.

The shadow a plane figure casts is a homology, with the line where the two planes meet as its axis and the light’s foot as its centre. A mirror is a homology with ratio exactly −1-1. Rectification is not: it is a general projectivity, and that difference is the difference between moving the point a picture is seen from and turning one picture into another. The essay that sets the census out is in the foundations field, because the fact is about maps rather than about anamorphs.

What is not a homology

Three qualifications, and the third is the one that matters most.

The map is a homology in the rabatted picture’s own coordinates, at the design’s own scale. Rabat at half scale and the composition is a general projectivity; the homology appears when the fold is at true size, which is the same statement as saying the fold is a fold.

The map is a homology of a plane. The receiving surface has to be flat for a collineation to exist at all, and the moment it curves nothing of this survives — not the axis, not the centre, not the ratio, not the possibility of any homography at all. The vault is where that boundary is measured, and the number there is the phase’s central one.

And the map is a homology between marks and a design, not between marks and a viewer. Knowing that the map is a homology does not tell a reader standing in a gallery where to go. Getting the eye back out of the marks is a different operation with a different answer, and the answer is not what it looks like: the marks name where to stand exactly, and how high to look only in a product with the picture’s own proportions.

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. 11 The site’s own claim, from its first phase: a picture computes the point it is correct from. An anamorph is the case where that point stops being a recommendation, and the three numbers are how it is stated.
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. 12 The printable version, in millimetres. It is the one figure on this site that states a viewing distance unconditionally, and the reason it can is that the sheet fixes the design’s scale — which is exactly the quantity the rabatment assumes and the recovery cannot supply.

The short version

An anamorph on a plane is a planar homology. Its axis is the line the picture stands on, and everything there is right from anywhere. Its centre is directly in front of the eye at a depth of the eye’s height plus its distance. Its characteristic ratio is minus the eye’s distance over its height. Those three quantities are the whole map, they rebuild every mark to fifteen digits, and they are three statements about where somebody is standing.

The effect that gets described — the picture that means nothing until a viewer finds the spot — is the ratio being far from one. It is not a different kind of picture. It is a perspective construction with the eye brought close enough that the number shows.

Where this goes next

eye, 74° offgrey: the word before the projectionblack: the same word, projected Anamorphosis is only a viewpoint A smear on a wall that resolves into a skull when the viewer stands in one particular place off to the side. It is not a trick added to perspective — it is perspective, with the centre of projection put somewhere nobody would naturally stand, and it is the clearest demonstration there is that every picture has such a point. 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 An anamorph at true size, on paper An anamorph is the shadow of the intended picture, cast from the reader’s own eye. Every claim on this site is quoted against an assumed display width because the site cannot know how wide its figures really are; this one is not, because it ships a sheet in millimetres and states where to put an eye. centric pointdistance point, 12 px off the sheet →430 pxcorrect from 10 cm at 160 mm wide47° across The distance point is the viewing distance, drawn There is exactly one place in the whole classical apparatus where the distance from the eye to the picture appears as a length on the page, and it is the offset from the centric point to the distance point. Everything this site exists to compute was drawn in the fifteenth century, on the horizon, and nobody said what it was.

What links here

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

Reads more easily once this is understood

Essays that name this one as worth reading first.

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.

AnamorphosisCentral collineationCharacteristic ratioCollineationDemonstrationDistance pointFixed pointGround lineHomographyPicture planePlanar homologyProjective mapRabatmentReceiving surfaceStation pointViewing distance