Visual Plane
every picture states the point it is correct from
Systems that kept the measure

A parallel floor under a perspective room

Draw the floor without diminution and the people on it with it, and the picture has a centre at infinity glued to a centre in the room. The same map absorbs it — but there is no shear this time, and there cannot be: bringing a point in from infinity is not something an affine map does, so the room the picture is equally a picture of has its midpoints moved as well as its angles.
24 min read 23 figures Not a projection at allOne machine used twiceDetermined up to something

Worth reading first: A picture with two eyes in it · A picture with no size–distance signal · The eye taken to infinity.

A great many drawing traditions do the same thing with a floor. The tiles do not converge — the pattern runs to the top of the picture at the same width it has at the bottom — and the people standing on the floor are drawn smaller when they are further away.

That is a parallel projection glued to a perspective one, and it is a picture with a centre at infinity in one half and a centre in the room in the other.

The same picture, from one eye, of a different roomThe thick outlines are the far boxes as the second eye drew them, by parallel projection along a fixed direction. The thin outlines are the **first** eye's picture of a moved copy of those boxes — moved by one projective map of space, the one that holds the picture plane still point by point and carries the second eye onto the first. The two lie on top of each other to 6.4e-11 px, over 16 corners. Without the map the same first eye would have drawn those boxes 2567 px away. So the two-eyed picture is a one-eyed picture, of a room that is not the room.apart by 1.9e-11 pxcorrect from 17 cm, at 160 mm wideone eye again
Fig. 1 The composite, and its absorbed reading. The thick outlines are the far half drawn by parallel projection along a fixed direction; the thin ones are the first eye’s picture of a moved copy, and the two agree to six parts in a hundred thousand million million of a pixel.

A centre at infinity is a centre

The construction that absorbs a second eye takes the centre as a homogeneous point, so a direction is an ordinary input rather than a special case. Nothing in the algebra changes: an identity plus one outer product, a picture plane held still point by point, and the second centre carried onto the first.

A parallel projection is a perspective one with the eye taken awayThe four small boxes are photographs from 2.5 m, 6 m, 20 m, 200 m with the lens lengthened to match. The isometric drawing is the limit, and its bundles stay parallel to 0e+0 radians.isometric — the limit2.5 m6 m20 m200 msame box, same drawn sizethe eye recedes
Fig. 2 Why a direction is a centre. Take the eye back and the projection approaches a parallel one, the vanishing points run out, and the limit is a projection whose centre is a point at infinity.
Moving the object, in both familiesThe same box, in place and translated 2.4 m across the world. In the parallel drawing the second image is the first translated by 98.6 px and nothing else — every edge the same length to 3e-14 px. In the perspective drawing the edge lengths change by up to 77.0%, because the direction from the eye has changed and a projection through a centre depends on it.horizonisometric — the same drawing, movedperspective — a different drawingcorrect from 23 cm, at 160 mm wideparallel: 3e-14 px · perspective: 77.0%
Fig. 3 The limit already taken. A parallel projection is a stated map from a world to a page, with no station point and no diminution, and it is not primitive perspective.

So the composite has the same description as the previous ones: it is a one-eyed picture of a scene that has been moved by a map fixing the glass. What differs is the map.

And this time there is no shear

The family of absorbing maps has one member that goes to the identity as the two centres come together. When both centres are finite, that member’s own centre is a point at infinity, which makes it affine — a shear, growing with depth.

A centre at infinity has no such member and cannot have one. An affine map takes points at infinity to points at infinity. The whole job here is to bring one in from infinity to a finite place, so no affine map does it, and every member of the family is properly projective.

The room, and the room the picture is equally a picture ofBoth rooms, from somewhere else. The thick boxes are where the far half really is; the thin ones are where the map puts them, and the first eye's picture of the second set is the second eye's picture of the first set, mark for mark. The map is a **homology of space** — an identity plus one outer product — whose axis is the sheet of glass and whose centre is on the line joining the two eyes. Absorbing a centre that is **at infinity** has no affine member and cannot have one: a parallel projection's centre has to be brought in from infinity, and no affine map moves a point at infinity to a finite one. So this map is properly projective, midpoints move — by 5.02% of an edge here — and the only thing left exact is flatness, at 3.5e-16 m.correct from 18 cm, at 160 mm wideprojective, not affine
Fig. 4 The two rooms. Absorbing a centre at infinity moves midpoints — by five per cent of an edge here — and the only thing left exact is flatness.

That is a real difference and it is worth saying what it costs. When two finite eyes are absorbed, parallel walls stay parallel and every midpoint stays a midpoint; the room comes out skewed and remains, in a recognisable sense, the same room. When a parallel half is absorbed, midpoints move, so a wall’s centre is no longer its centre and a floor’s tiles are no longer evenly spaced in the absorbed reading.

The midpoint of one segment, under both familiesThe parallel projection places it exactly halfway (0e+0 px out). The perspective projection places it 22 px away from halfway, 7% of the drawn length.halfway along the drawn linethe actual midpointcorrect from 26 cm, at 160 mm wide22 px apart
Fig. 5 The quantity in question. A projection through a centre does not keep a midpoint; an affine map of space does; and the map that absorbs a parallel half is neither of those things.

Which is the same statement as the convention’s own point

The convention exists because a parallel floor is measurable. Its tiles are evenly spaced on the page, so a reader can count them and know how far into the room something is.

A 1.7 m figure, drawn at every distance, by two systemsThe falling curve is a pinhole: f·H/Z, dropping -4.06 px per metre on average and 20.0× overall across the range. The flat one is an oblique system, whose slope is 0e+0 px per metre — zero, not nearly zero. A system with no diminution has no size–distance signal at all, so depth has to be carried by something else, and the something else is where the figure stands on the page.0100200204060distance of the object, metresits drawn height, pxa pinholean oblique systemdrawn height of a fixed object against its distance20.0× against 1.000
Fig. 6 What a system with no diminution does instead, measured. The drawn size falls at exactly zero pixels per metre, so nothing in the picture says how far away anything is, and depth is carried by height on the page.

The absorbed reading destroys exactly that. Its midpoints have moved, so the evenly spaced tiles of the parallel half are unevenly spaced in the room it is a picture of — which is precisely the property the convention was drawn to have.

So the two descriptions are not equivalent in what they make available to a reader, even though they are equivalent in what they put on the page. The composite reading says “the tiles are even and there is no single eye”; the absorbed reading says “there is a single eye and the tiles are uneven”. Both are true of the same marks.

Where a depth range lands on the pageEach rule is one depth, evenly spaced from 3 m to 60 m, drawn at the height its system puts it. The shaded band is the top tenth of each strip. A pinhole files 69% of the whole depth range into it; a linear depth map puts exactly 10% there, because a tenth of a page is a tenth of anything under a linear map. That band is what a horizon is.a pinhole69% of the rangean oblique system10% of the rangeevenly spaced depths, drawn where each system puts themthe shaded band is the top tenth of the stripa horizon is a band, not a line
Fig. 7 And what the alternative costs: a perspective floor files its whole far half into the last tenth of the page, which is the readability the convention is buying.

The two halves, as maps

It helps to write the two halves side by side, because the difference between them is one entry in a matrix.

A projection through a finite centre divides by depth: the drawn position of a point is its lateral offset over its distance along the axis. A parallel projection does not divide at all: the drawn position is the lateral offset, full stop.

In the matrix form this collection uses for a rendering pipeline, that is a single row. Set the bottom row of a projection matrix to a row of zeros ending in one and the same matrix draws a parallel projection; leave it alone and it draws a perspective one.

A frustum is four numbers and a windowThe camera's own section, looking down on it. The two slanted lines are the left and right sides: divide either by the near plane and it is a direction, -0.6285 and 0.3041, which between them carry the focal length 739.9 px and the principal point 465. The two crossbars are the near and far planes, and moving either of them moves no mark in the picture at all.the eyenear 0.90 mfar 4.20 mfocal 740 pxprincipal 465, 210
Fig. 8 The one entry that decides it. Flattening the bottom row preserves midpoints to six parts in a hundred million million of a pixel where the unflattened matrix misses them by thirty-three.

So the mixed picture is a picture made by two matrices differing in one row, applied to different parts of the same scene, drawn onto the same glass. Said that way it is a small thing to do and the arithmetic of undoing it is correspondingly larger, which is a common enough pattern.

Clip space: the box the frustum becomes, and the plane the segment stops atA plan section through the frustum. The segment runs from 6.60 m in front of the eye to 2.99 m behind it, and the clipper stops it on the near plane at 0.40 m. Clipping happens here, before the divide, because both ends are still finite numbers at this stage.near 0.4 mtoward the far planethe eyein front3.0 m behind the eyeclipped hereeverything below the eye's line has a negative w, and dividing by it flips both signsplan section · near 0.4 m, far 26 mthe segment crosses the eye plane at t = 0.647
Fig. 9 The pipeline the two matrices live in. What the two halves of the composite share is everything except that one row.

The seam, which is where a mixed picture shows

A composite of two finite eyes has a seam whose disagreement is a parallax — it needs depth to show and it vanishes on a flat scene. A composite of a parallel half and a perspective half is different: the two halves disagree about diminution, which is a fact about size rather than about position.

The same picture, from one eye, of a different roomThe thick outlines are the far boxes as the second eye drew them, by parallel projection along a fixed direction. The thin outlines are the **first** eye's picture of a moved copy of those boxes — moved by one projective map of space, the one that holds the picture plane still point by point and carries the second eye onto the first. The two lie on top of each other to 4.0e-11 px, over 16 corners. Without the map the same first eye would have drawn those boxes 3776 px away. So the two-eyed picture is a one-eyed picture, of a room that is not the room.apart by 4.0e-11 pxcorrect from 17 cm, at 160 mm wideone eye again
Fig. 10 A wider mismatch. Without the map the same first eye would have drawn those boxes more than two thousand pixels away, which is several times the picture’s own width.

So the discrepancy is much larger than a two-eye seam of comparable geometry, and it shows in a way a reader recognises without any geometry at all: a thing that ought to be smaller is not.

That is presumably why the convention is applied to floors and pavements and rarely to figures. A floor’s tiles have no expected size, so a reader has nothing to be surprised by; a person does.

A 46° picture, 160 mm wide, read from various distancesThe picture is correct from 18.8 cm. Read from an ordinary reading distance of 40 cm it depicts a scene 2.12× deeper than the one it was made from, and no mark on the page has moved.02420406080100how far the reader's eye is from the page (cm)how much deeper the depicted scene becomescorrect at 18.8 cm40 cm → × 2.12at 160 mm wide× 2.12 at 40 cm
Fig. 11 The signal being suppressed, from the other side. The rate at which a projection compresses depth, which a parallel projection sets to a constant.

What a reader can actually measure off it

The interesting practical question is what a person holding such a picture is entitled to compute, and the two halves give different answers.

Off the parallel half: ratios of distances along the floor, exactly. That is what a system with no diminution buys and it is a strong result — a reader counting tiles is measuring the room, with no camera, no focal length and no station point.

What each parallel system does to the three axesThe smallest axis scale is plotted; the note says how many of the three coincide and whether the system is an orthographic projection or an oblique construction. cavalier and isometric both make all three the same — but isometric is the only orthographic one that does, at 0.8165, and the oblique route gets there by drawing depth at true length.elevation0.0000two equal, orthographic ←cabinet0.5000two equal, obliquecavalier1.0000all three equal, obliquedimetric0.4714all three different, orthographicisometric0.8165all three equal, orthographictrimetric0.5479all three different, orthographicsmallest of the three axis scalesmeasured from each projection
Fig. 12 The battery, with the column that matters here. True measure is exactly what a parallel system keeps and a central projection destroys.

Off the perspective half: cross-ratios, and everything the site’s ordinary single-view machinery extracts from them — which is more than ratios but needs a horizon, and the horizon of the parallel floor is at infinity and unavailable.

And across the seam: nothing. The two halves are pictures of the same room by different rules, so a length read from one and a length read from the other are not comparable, and there is no factor that makes them so — the ratio between them depends on depth, which is precisely what one half has deleted.

Where a depth range lands on the pageEach rule is one depth, evenly spaced from 3 m to 60 m, drawn at the height its system puts it. The shaded band is the top tenth of each strip. A pinhole files 69% of the whole depth range into it; a linear depth map puts exactly 10% there, because a tenth of a page is a tenth of anything under a linear map. That band is what a horizon is.a pinhole69% of the rangean oblique system10% of the rangeevenly spaced depths, drawn where each system puts themthe shaded band is the top tenth of the stripa horizon is a band, not a line
Fig. 13 Why no single factor exists. The perspective half’s compression varies with depth over orders of magnitude; the parallel half’s does not vary at all.

That last point is the practical content of the whole essay. A mixed picture is two measuring instruments on one sheet, each of them exact, with no conversion between them.

What the family looks like here

The absorbed scene is still not one scene — the family is still one-parameter, for the same reason as before — and its members here are more different from each other than in the finite case.

Not one room. A one-parameter family of themThe map that absorbs the second eye is not unique: composing it with any map that already holds the picture plane still **and** holds the first eye still gives another one, and that is a one-parameter family. Three members are drawn, 1.64 m apart at the widest, and all three draw the identical picture from the one remaining eye — to 7.6e-13 px. So the honest statement is not that a two-eyed picture is a picture of one particular other room; it is that it is a picture of a whole family of them, and the picture has nothing to say about which.1.64 m apart, one picturecorrect from 17 cm, at 160 mm widethree of a family · identical to 7.6e-13 px
Fig. 14 Three members of the family for a finite second eye, drawn for comparison. All three draw the identical picture, and the spread between them is what the picture does not contain.

The reason is the pole. A projective map of space has a plane it sends to infinity, and for the maps that absorb a parallel half that plane sits somewhere in the room rather than out beyond it. Points near it are thrown a long way, so members of the family that differ slightly in parameter differ enormously in where they put the far parts of the scene.

How wide the picture gets as the field of view opensOn a flat plane the picture's half-width is tan(θ/2): it multiplies by 6.6 between 120° and 170° and is unbounded at 180°. On a cylinder it multiplies by 1.42 over the same range and keeps going past 180° without incident.0246850100150field of view across the picture (degrees)half-width of the picture, in focal lengthsplanecylinderstereographicequidistantcut off at eight focal lengthsthe plane crosses it at 166°
Fig. 15 The pole in its familiar form. A point approaching the plane through the eye images further and further out, and a projective map of space has the same behaviour at its own vanishing plane.

Which is a practical warning as much as an observation: the absorbed reading of a mixed picture is well behaved near the glass and badly behaved far from it, and quoting a single absorbed room without saying which member of the family is doing the quoting more work than it can bear.

The scroll is the continuous version

One system this collection already models does exactly this along one axis and continuously.

A handscroll is orthographic along the roll and perspective across it: a mile of river holds its scale as the picture unrolls while a single pavilion still recedes. That is a parallel projection in one direction glued to a central one in the other — not two halves of a picture but two axes of the same map.

A scroll keeps the midpoint along its length and loses it acrossLeft, a segment lying along the roll: the image of its midpoint and the midpoint of its image are the same point to 0e+0 px. Right, a segment running away from the eye: the two are 21.9% of the segment apart. One projection, two answers, because the eye is at infinity in one direction and seven metres away in the other.along the rollthe two midpoints coincide — 0e+0 pxone mark, drawn twiceacross itthey separate by 21.9%the image of the midpointthe midpoint of the imageno single viewpoint — the rays miss by 6.9 m21.9% of the receding segment
Fig. 16 The consequence measured: the midpoint survives along the roll and not across it. One picture, two rules, split by direction rather than by region.
A straight line, drawn by a scroll, sags 9.6 pxAbove: a straight world line running from 2 m to 15 m of depth. Its image is a hyperbola — the algebra says a Möbius function of the paper coordinate, and the sampled projection agrees with that closed form to 3e-14 px. Below, the control: the same line held at constant depth images straight to 0e+0 px. What bends a line in a scroll is changing depth, and nothing else.a receding straight line, and the chord it is not9.59 px of sagthe control — the same line at constant depth0e+0 pxno single viewpoint — the rays miss by 6.1 ma straight line's image is a hyperbola
Fig. 17 And what it does to a straight line, which is the signature of the split: a straight line in the world images as a hyperbola, because the two axes are governed by different denominators.

So the mixed floor and the handscroll are the same trade made along different cuts of the picture. The scroll cuts by direction and keeps every column a projection; the floor convention cuts by region and keeps every part a projection. Both buy measurability in one place with a station point everywhere.

The same row of 5 pavilions, by a pinhole and by a scrollAbove, one pinhole at 62°: the row is oblique to the eye, so the far pavilion is drawn 2.38× smaller than the near one. Below, the scroll: every pavilion is drawn at the same height to twelve decimal places, because along the roll the eye is at infinity. Neither is a defect — they are answers to different questions.one pinhole, standing backfar end 2.38× smallera scroll, unrolledevery end the same sizeno single viewpoint — the rays miss by 9.4 m across the lower stripthe upper strip is one pinhole at 62°
Fig. 18 The scroll’s own composite, drawn against an ordinary photograph of the same row. The two rules produce visibly different pictures of one thing.

What this does not say

It says nothing about what any tradition intended. The convention is named as the source of a rule being modelled, and everything computed is a computation on a map.

It says nothing about the parallel half being a bad choice. It is a good one for what it is for, and this collection’s own table says why: a system with no centre keeps true measure, and a system with a centre gives up true measure to get a station point. There is no system with both.

What each parallel system does to the three axesThe smallest axis scale is plotted; the note says how many of the three coincide and whether the system is an orthographic projection or an oblique construction. cavalier and isometric both make all three the same — but isometric is the only orthographic one that does, at 0.8165, and the oblique route gets there by drawing depth at true length.elevation0.0000two equal, orthographic ←cabinet0.5000two equal, obliquecavalier1.0000all three equal, obliquedimetric0.4714all three different, orthographicisometric0.8165all three equal, orthographictrimetric0.5479all three different, orthographicsmallest of the three axis scalesmeasured from each projection
Fig. 19 The exclusion, on one battery. No system answers yes to a centre and a true measure at once, which is why a picture that wants both takes two centres.

And it does not say the mixed picture is a worse composite than the two-eye one. It is a more distorted one when absorbed, which is a different claim: a bigger seam and a projective rather than affine map. Whether that matters depends on what the picture is for, and the geometry prices it without preferring.

The vanishing plane, and where it sits

The map that absorbs a parallel half is projective rather than affine, which means it has a vanishing plane — a plane of space it sends to infinity — and where that plane sits decides how well behaved the absorbed reading is.

For an affine map there is no such plane in the room: the plane at infinity goes to itself, which is what affine means. For the maps here it is somewhere finite, and points near it are thrown arbitrarily far.

How wide the picture gets as the field of view opensOn a flat plane the picture's half-width is tan(θ/2): it multiplies by 6.6 between 120° and 170° and is unbounded at 180°. On a cylinder it multiplies by 1.42 over the same range and keeps going past 180° without incident.0246850100150field of view across the picture (degrees)half-width of the picture, in focal lengthsplanecylinderstereographicequidistantcut off at eight focal lengthsthe plane crosses it at 166°
Fig. 20 The behaviour in its familiar form. A point approaching the plane through the eye images further and further out; a projective map of space does the same at its own vanishing plane.

That gives a practical rule about which part of the picture the absorbed reading can be trusted for. Near the glass it is tame; far from it, and especially near wherever the vanishing plane falls, the absorbed positions run away and small changes in which family member is used produce large changes in the answer.

The room, and the room the picture is equally a picture ofBoth rooms, from somewhere else. The thick boxes are where the far half really is; the thin ones are where the map puts them, and the first eye's picture of the second set is the second eye's picture of the first set, mark for mark. The map is a **homology of space** — an identity plus one outer product — whose axis is the sheet of glass and whose centre is on the line joining the two eyes. Absorbing a centre that is **at infinity** has no affine member and cannot have one: a parallel projection's centre has to be brought in from infinity, and no affine map moves a point at infinity to a finite one. So this map is properly projective, midpoints move — by 5.01% of an edge here — and the only thing left exact is flatness, at 1.1e-15 m.correct from 18 cm, at 160 mm wideprojective, not affine
Fig. 21 A gentler mismatch, whose absorbed room stays compact. The badly behaved cases are the ones with a large mismatch and a lot of depth.

Where else a mixed system appears

Worth listing, because the pattern is more common than the floor convention alone suggests.

A removed roof. The interior is drawn as though from above and the walls as though from the side, which is the same trade with the parallel half applied to the plan.

The same three rooms, reached by a parallel system and by one eyeFilled dots are floor samples whose sightline clears the walls with the roofs off. Above, a parallel system at 52°: 55.6% of every room, identical to the last sample, because the strip a wall hides is h/tan θ wherever that wall stands. Below, the eye that frames the same building from 14.5 m: 49%, 56%, 49% — a spread of 6.2 points across rooms that are identical.parallel, 52° above the ground56% · 56% · 56%one eye, 14.5 m away49% · 56% · 49%a square metre of floor varies 1.000× against 1.535×spread across rooms: 0 against 6.2 points
Fig. 22 The convention measured, with the coverage it buys against the elevation it is drawn at.

And a rendering pipeline that mixes projections. A parallel projection for a shadow map and a perspective one for the camera is exactly this composite, done by a machine, with the two matrices differing in one row. Nothing about that is a convention and the geometry is identical.

Clip space: the box the frustum becomes, and the plane the segment stops atA plan section through the frustum. The segment runs from 6.60 m in front of the eye to 2.99 m behind it, and the clipper stops it on the near plane at 0.40 m. Clipping happens here, before the divide, because both ends are still finite numbers at this stage.near 0.4 mtoward the far planethe eyein front3.0 m behind the eyeclipped hereeverything below the eye's line has a negative w, and dividing by it flips both signsplan section · near 0.4 m, far 26 mthe segment crosses the eye plane at t = 0.647
Fig. 23 Where the two matrices live. A pipeline that uses both is making the same composite the drawing traditions make, for the same reason: one of them measures and the other looks right.

So the mixed picture is not a historical curiosity. It is what happens whenever one part of a picture is wanted for measuring and another for looking at, and that want is old and current at once.

The transferable form

A limit case is usually the same construction with an ordinary input, and where it is not, the reason is a property the limit does not have — which is more informative than the construction working.

The absorbing map takes a centre at infinity without a special case. What it does not take is the affine member, and the reason is one sentence: an affine map cannot bring a point in from infinity. That sentence explains the whole difference between the two rungs — why the two-eye composite keeps midpoints and this one does not, why the family here is worse behaved, and why the seam is larger.

This collection has met the same pattern before, and it is worth noticing that it usually goes the other way. The eye taken to infinity, the parallel projection as a limit of a perspective one, the infinite far plane of a frustum: in each of those the limit turned out to be an ordinary member of a family rather than a separate case. Here it is a separate case, and the thing that separates it is not the limit itself but a property — affineness — that the family only has when nothing is at infinity.

Where this goes next

apart by 4.9e-13 pxcorrect from 17 cm, at 160 mm wideone eye again The second eye is a shear A picture drawn from two eyes is a picture drawn from one, of a different room. The map that puts the second eye away holds the picture plane still point by point and carries one centre onto the other, and the member of its family that matters turns out to be affine — a shear along the line joining the eyes, growing with depth, which is the same operation an oblique drawing performs. isometric — the limit2.5 m6 m20 m200 msame box, same drawn sizethe eye recedes The eye taken to infinity A parallel projection is a photograph from infinitely far away with the lens lengthened to match. That is not an analogy — it is the limit, it can be watched happening, and it explains why a long lens flattens a scene and why an isometric drawing has no viewing distance to state. 0100200204060distance of the object, metresits drawn height, pxa pinholean oblique systemdrawn height of a fixed object against its distance20.0× against 1.000 A picture with no size–distance signal In a system with no diminution the drawn size of an object falls at exactly zero pixels per metre, so nothing in the picture says how far away anything is. Depth has to be carried by something else, and what carries it is height on the page — linearly, and without a horizon.

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.

Affine mapcentre of projectionDemonstrationDiminutionDrawing systemFree parameterHomologyParallel projectionpoint at infinityShear