Visual Plane
every picture states the point it is correct from
Field

Where to stand

A perspective picture is correct from exactly one point in the room. Every figure here computes where that point is, which turns the oldest complaint about wide-angle photographs into arithmetic.
the picture, 160 mm wide22 cm40°the eyefocal length 948 px22 cm at 160 mm wide

The point you have to stand at

How a picture's correct viewing distance follows from its focal length and displayed width, and what follows from nobody standing there.

 54 px69 px84° across27% wider at the edge

Wide angle is not distortion

Why wide-angle photographs stretch faces at the frame edge, why it is not a lens fault, and the viewing distance at which it vanishes entirely.

 eye, 74° offgrey: the word before the projectionblack: the same word, projected

Anamorphosis is only a viewpoint

Anamorphic construction as an ordinary projection with an oblique centre, the homography that produces it, and why straight strokes staying straight is the test.

 flat picture plane — straight lines stay straightcylindrical picture surface — even stretch, bowed linesone scene, two picture surfacesneither is the distorted one

When the picture surface is not flat

Cylindrical and spherical picture surfaces against the flat plane, what each preserves, and why a panorama bends the horizon.

 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

The oblique anamorph built by projecting an upright word onto the sheet from the eye, checked by re-projection, and shipped as a printable sheet at true scale.

 the sheet, seen from abovemirrorthe eye, 2.4 radii upwhat stands up in the mirrorthe light path reverses to the eye to 1e-15scale varies 7.2× across the design

The cylindrical mirror unbends it

Catoptric anamorphosis computed by tracing the reflection in a cylinder, checked by running the light backwards from the paper mark to the eye, with the unevenness of the map measured as a ratio of local area scales.

 the picture — identical at every viewing distanceplan: the true cube, and the solid depicteddepth × 2.12, width × 1.00correct from 18.8 cm at 160 mm wideread from 40 cm — depth × 2.12

Standing in the wrong place

What a perspective picture depicts when it is viewed from somewhere other than its station point — depth scaled by the ratio of the distances, transverse dimensions unchanged, and every image point exactly where it was.

 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

The map from an upright design to its floor marks is a homology whose axis is the ground line, whose centre is the eye's height plus its distance, and whose ratio is minus the distance over the height.

 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 fixed250 mm sidewayszero on the axis, 212.5 mm at the top

Where the anamorph still works

A wrong eye composes an anamorph with an elation or a homology on the ground line, and the departure is the step times the height over the eye's height.

 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

The marks name the place, not the height

An anamorph's marks determine the reader's position on the floor exactly and the eye's height only in the product height times aspect.

 barrel vaulteye · 1.62 mon the floor1e-12 mmon the vault529.4 mmworst miss of the best homography, log scalevault radius 4.0 m7.7% of the extent, against 4e-14%

The ceiling that is not a plane

The map from an intended picture to its painted marks is a collineation exactly while the receiving surface is flat, and the residual across that boundary is fifteen orders of magnitude.

 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

A forced-perspective set matches its intended row exactly from one eye, and the disparity two eyes read is a ratio of reciprocal depths rather than the depth compression the set was cut at.

 toward the designup10 mm14 mm across36 mm along the sight line

The room the eye may stand in

The region of eye positions from which a pavement anamorph still reads, measured as a solid rather than a radius, with its shape, its long axis, its volume and how all three scale with the tolerance allowed.

 toward the designup63 mm apart10 mm14 mm across36 mm along the sight line

An anamorph has one eye

What the second eye is told about a pavement anamorph, measured as a binocular disparity in arcminutes and set against the disparity the intended picture would have produced.

 eye level — the ray never comes downeye · 1.65 m up12.8 m of floor at 80% of eye heightand the top of the design has no mark at all

The design that outruns the floor

How much pavement an intended picture requires, why the growth is a pole at eye level rather than a slope, and what the stretch at the far edge does to a design that tries to include a horizon.

 wallthe intended picturethe joineyecontinuous across the cornerand 3.88× the scale on one side

The anamorph that crosses a corner

A design cast across two planes at once — two homologies sharing an axis, continuous at the corner and not smooth, with the scale jump computed from the two grazing angles alone.

 020406011.5022.503candidate height for the eye, in metreswhat the best rectangle leaves over, in millimetresa flat floor: every height fitsthe eye that made the markscurvature costs the collineationand buys the parameter back

The height a flat floor cannot give

Why the eye's height is unrecoverable from marks on a plane, why it becomes recoverable on a floor with any curvature at all, and what that trade says about the price of a projective description.

 the eye the picture is for1 faces no ray reaches9 steps, 18 planes, eye 1.65 m up17 used · 58.5% of the picture on risers

A stair does not use all its faces

The design of an anamorph on a flight of steps, the faces no ray from the design eye ever reaches, and why every stair mural is painted on the risers.

 horizoncorrect from 21 cm, at 160 mm wide8/21 faces · 73% of the surface

What an eye can paint

The paintable subset of a faceted object as a function of the eye, generalised from a flight of steps to four other objects, with the design band, the per-face shares and the depth ratio measured on each.

 horizonfaces the eye, gets nothingfaces the eye, gets nothingcorrect from 21 cm, at 160 mm wide4 facing faces unreached · 3.24 m²

Facing the reader is not being reachable

The difference between a face that turns toward the eye and a face the design reaches, measured as an area rather than as a count, with the height at which occlusion clears entirely.

 00.2500.5000.75012345how high the eye is (m)share of the object, and share of the design that landsof the object's surfaceof the design that landsa corridor with a doorway, 7 facesbest surface share 100% at 2.5 m

The eye that reaches the most

Sweeping the eye's height over four faceted objects and finding that the quantity a designer maximises decides the answer, with one object where the optimum is genuinely interior and three where it is at the end of the range.

 horizoncorrect from 21 cm, at 160 mm wide6/7 faces · 99% of the surface

A design that lands in two rooms

An anamorph whose design lands on surfaces at two very different depths, the scale jump at the edge of the opening, and why the design band on such an object has to be sampled rather than bisected.

 00.2500.5000.75010.2500.5000.75011.25the largest stretch a painter will accept, log₁₀share of the design that is inside ita flat floora cornera cluster of blocksa seating rakean ascending flighteye at 1.70 m, 2.4 m in front of the picturethe cap a painter will accept

The stretch decides the band

The local stretch of an anamorph's map, why it rather than the geometry should decide how much design an object is given, and what a cap on it leaves of each of six surfaces.

 where each reader's picture landsgoing up: risers67%going up: treads33%coming down: risersnothingcoming down: treads100%a flight of 9, eye at 1.65 m at each endthe two sets of faces do not overlap

One flight, two pictures

Whether one faceted object can carry an anamorph for each of two eyes, measured on a flight of stairs from its two ends, with the asymmetry that one reader sees both pictures and the other sees only theirs.

 02.5057.5010020406080how far the floor is folded from flat (°)what a 10 cm error in the height leaves (mm)two degrees: 0.58 mma design 1.4 m wide, the eye 1.62 m up, the crease at 1.2 mflat: 1.5e-13 mm · 90°: 9.3 mm

A fold names the height

Whether a single crease is enough to break the height ambiguity a flat floor leaves an anamorph recovery, measured against the angle of the fold and against the size of the error it is asked to detect.

 no single viewpoint — the rays miss by no distance at all — the rays are parallel4 pictures on 3 kinds of face

A flight that has ends

The step ends the earlier flight did not model, the count of disjoint pictures a real staircase supports, and the correction to a rule that counted the wrong thing.

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