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

Counting the eyes needs the room

How many eyes made a picture is not a question the picture can be asked. Told what the room really measures, the rays refuse to meet and a second eye has been caught; told instead that the room is the one the picture is consistent with, the same rays meet exactly, at the first eye. The refusal is real and it belongs to the room.
25 min read 23 figures Not a projection at allDetermined up to somethingFitted, not assumed

Worth reading first: A picture with two eyes in it · A centre and a measure are exclusive · The one thing a single view cannot give.

Two rungs have measured a two-centre picture and then taken the second centre away. Both are true and they read as contradictory, so it is worth settling what has actually been shown.

Only a known room counts the eyesThe same picture, fitted for a single centre twice. Told what the room really measures, the rays miss their best point by 0.665 m — which is a refusal, and a picture with two eyes has been caught. Told instead that the room is the one the picture is consistent with — the absorbed reading, a perfectly ordinary set of boxes with flat faces and straight edges — the same rays meet to 1.5e-15 m, at the first eye exactly. Counting the eyes in a picture is not something the picture can be asked; it is something the room is asked.the room, as it ismisses by 0.665 mthe room it is consistent withmeets to 1.5e-15 mdecimal places the rays agree tothe picture cannot be askedthe room can
Fig. 1 The same picture, fitted for a single centre twice. Told what the room really measures, the rays miss their best point by two thirds of a metre. Told that the room is the one the picture is consistent with, the same rays meet to two parts in a thousand million million of a metre — at the first eye exactly.

The marks are identical in both. Nothing in the drawing changed between the two answers.

What the fit is actually given

Every measurement in a picture with two eyes in it takes a pair: a drawn mark, and the world point it is a mark of. From the pair it builds a line in space, and it asks for the point all the lines pass through.

The world points are the part the picture does not contain.

Two eyes, one sheet of glassThe arrangement the picture is made in, seen from somewhere else. The rectangle is the picture plane — a real plane in the room, not either eye's pixels — and the two eyes both draw onto it. Every mark on it is where a line from one of the eyes to a point of the room crosses the glass. The two eyes are 1.41 m apart. Nothing in the drawing they make between them records which of them drew which mark.correct from 19 cm, at 160 mm widetwo eyes 1.41 m apart
Fig. 2 The arrangement the pairs come from. Each ray is a join between a world point and a place on the glass, and the picture is the second half of each pair.

So the fit is not reading the picture. It is reading the picture and a claim about what is in front of it, and the residual it reports is the disagreement between the two.

How far from having a centre, and it depends on knowing the roomTake every mark in the two-centre picture with the world point it is a mark of, join the two, and ask for the point all those lines pass through. There is none: at 2.40 m of separation the best point misses them by 1.118 m on average and 1.565 m at worst, and the miss falls to nothing as the eyes come together. That is the measurement — and it needs the room. Given the same picture and the room the picture is *consistent with*, the same fit returns a residual of 1.2e-15 m at every separation on this plot.00.500100.50011.502distance between the two eyes (m)how far the rays miss their own best point (m)worst raythe room it is consistent with1.12 m at 2.4 m apartzero for the absorbed reading
Fig. 3 The miss against the separation of the eyes, with the absorbed reading of the same picture alongside it. One curve rises and the other is flat at the arithmetic floor, and both are computed from the same drawn marks.

The three questions, which are different

Untangling this needs three questions kept apart.

Was this picture drawn from one centre? A question about the making of it. Unanswerable from the picture, and answerable if the scene is known.

Is this picture a projection of a stated scene from one centre? Answerable, and it is what the fit answers. The refusal is a refusal about the pair.

Is this picture a projection of anything from one centre? Answerable, and the answer is nearly always yes — because the absorbed scene exists and is an ordinary room.

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. Its centre turns out to lie at infinity, so it is **affine** — a shear along the line joining the two eyes, displacing every point by 1.1396 of its depth beyond the glass. It is not a rigid motion: a right angle in the room comes out at 94.8°, 138.4°, 103.2° in the moved copy. What it keeps it keeps exactly: four coplanar points stay coplanar to 8.8e-17 m, and the midpoint of an edge is still the midpoint to 1.0e-17 of the edge.correct from 18 cm, at 160 mm widea shear of 1.140 per metre of depth
Fig. 4 The room the answer to the third question is about. Flat faces, straight edges, parallel walls parallel, and corners out of square.

Once separated, the apparent contradiction goes away. The previous two rungs answer the second question and the third, and neither of them answers the first.

What “nearly always” is hiding

The third answer is not unconditionally yes, and the exceptions are worth having.

A picture with no depth in it cannot be caught at all. A plane seen from two centres differs by a homography of the picture, and a homography of a picture of a plane is a picture of the same plane from somewhere else. So a two-centre picture of a flat thing is a one-centre picture of that flat thing, with no shearing of anything, and the second eye leaves no trace whatever.

The picture plane tilted 14°Pointing the camera up tilts the picture plane with it, and three things happen at once: the verticals converge — 3.59° between the outer two — the horizon drops 213 px below the middle of the frame, and the vertical vanishing point arrives at 3425 px from the principal point. They are one fact: the product of those two offsets is f².correct from 20 cm, at 160 mm wideverticals converge 3.59° · horizon 213 px off centre
Fig. 5 The result that says so. One eye and two picture planes give a homography; two eyes and one plane in the scene give a homography too, and a homography of a picture is a picture.

A picture whose parts do not overlap in depth is harder. The shear grows with depth, so parts that sit at nearly the same distance are displaced by nearly the same amount, and the absorbed scene is nearly a rigid copy of the real one.

What the map keeps, and what it does notThe natural absorbing map is a shear, and a shear is affine, so what it keeps it keeps exactly. At 0.36 m of separation a right angle of a box comes out up to 19.39° from square in the moved copy — and the midpoint of every edge is still the midpoint, to 1.5e-15 of the edge, at every separation on this plot. The second curve is the control: a different member of the family absorbs the same eye and draws the same picture, and it moves a midpoint by 6.60%. Flatness holds for both, to 2.9e-16 m. A room is still a room in the absorbed reading, with its walls still parallel; it is not the same room.02040600.50011.5022.50distance between the two eyes (m)degrees off square, and % off the midpointdegrees off square% off the midpoint, another memberthe shear keeps every midpointworst angle 19.39°midpoints kept exactly
Fig. 6 The distortion at a small separation. A right angle comes out a few degrees from square, which is a room a person would accept as a room.

And a picture of something whose shape is known is caught immediately. That is the case the fit is testing, and it is why the answer to the second question is a refusal: the room’s measurements are the extra information, and the moment they are supplied the two-centre picture has nothing left to hide behind.

What the map keeps, and what it does notThe natural absorbing map is a shear, and a shear is affine, so what it keeps it keeps exactly. At 2.16 m of separation a right angle of a box comes out up to 58.08° from square in the moved copy — and the midpoint of every edge is still the midpoint, to 1.5e-15 of the edge, at every separation on this plot. The second curve is the control: a different member of the family absorbs the same eye and draws the same picture, and it moves a midpoint by 6.88%. Flatness holds for both, to 2.9e-16 m. A room is still a room in the absorbed reading, with its walls still parallel; it is not the same room.02040600.50011.5022.50distance between the two eyes (m)degrees off square, and % off the midpointdegrees off square% off the midpoint, another memberthe shear keeps every midpointworst angle 58.08°midpoints kept exactly
Fig. 7 At a wide separation the absorbed room is grossly non-rectangular. Anyone who knows the room is rectangular has caught it; anyone who does not has an unremarkable picture of a peculiar room.

How much room-knowledge it takes

“Told the room” is doing a lot of work in the sentences above, and it can be made quantitative.

The fit does not need the whole room. It needs enough of it that the shear cannot be absorbed into a rigid motion, and that is a small requirement: the shear is three parameters — a direction and a rate — so any three independent metric facts about the scene that the shear would violate are enough to catch it.

One right angle spanning the seam is one such fact. Two lengths in different directions are two. A single known solid with three edge directions is three and to spare.

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 4e-15 relative.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
Fig. 8 The most convenient version: a box, which supplies three orthogonal directions at once and is the object this site has used to recover a camera since its first commit.

What is not enough is a great deal of information about a single plane. A floor covered in tiles of known size, however many, says nothing about the number of eyes, because the whole plane is absorbed by a homography and homographies of a plane are pictures of it.

A façade, flattened out of the photographFour corners of a rectangle of known proportions fix the homography. Three lengths it was never given come back to 4e-16 relative — so every measurement on that plane is available, in units of the rectangle's own width.the picturethe same plane, rectifieddoor — 0.5068 widths (true 0.5068)sill — 0.3026 widths (true 0.3026)diagonal — 1.1766 widths (true 1.1766)worst error 4e-16 relativethe probes were not used to build the map
Fig. 9 A plane rectified from a photograph. Everything recoverable here is recoverable from a two-centre picture as well, because a plane cannot tell the two apart.

So the useful summary is a shape rather than a count: catching a second eye needs metric knowledge that spans depth, and no quantity of knowledge confined to one surface will do it.

What a wall hides is h/tan θ, wherever the wall standsThe smooth line is 2.3 m of wall divided by the tangent of the elevation, subtracted from the 4.2 m room. The steps are what the sightline test actually finds over the floor grid. They agree to one row of samples across the whole range, which is what says the occlusion test is measuring the geometry and not the grid.02550754050607080elevation of the parallel view, degreesshare of each room's floor reached, %h / tan θthe sightline testwalls 2.3 m, rooms 4.2 m deeptwo independent routes
Fig. 10 And the reason depth is where it lives: what a picture hides is a matter of what is behind what, and a picture with nothing behind anything hides nothing and reveals nothing.

The general shape this belongs to

This is not a special situation. It is the standing shape of every single-view question on this site.

A single view fixes the geometry of a scene up to a similarity, so the one thing a single view cannot give is a length — and the way that is shown is to produce two scenes at different scales that draw the identical picture.

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. 11 The oldest version of the same move. Two scenes, one picture, and the difference between them is exactly what the picture does not contain.

Two views fix shape and not size, and two views give shape and no size exhibits the family rather than asserting it. Many views leave seven directions untouched, and seven numbers no picture can name walks along them.

Seven moves that change no picture, and three that change every oneThe whole reconstruction shifted by up to 1.3 m, turned by up to 0.55 rad and scaled by 2.7: every picture stays where it was, to 2.6e-11 px. Move one camera by 50 mm or one point by 50 mm and the pictures move by 0.63 px or more. The seven are not small effects that could be measured with better data; they are exactly zero, and the 2e+10× between the two groups is what makes that a claim rather than a tolerance.shift x1.7e-11 pxshift y1.7e-11 pxshift z2.6e-11 pxturn x3.3e-12 pxturn y1.7e-11 pxturn z7.9e-12 pxscale ×2.71.8e-11 pxturn camera 0 by 0.01 rad0.940 pxmove camera 0 by 50 mm2.138 pxmove point 0 by 50 mm0.626 pxchange in reprojection error, log scalebelow: not gauge directionsflat to 2.6e-11 px · stiff from 0.63 px2e+10× apart
Fig. 12 Seven free directions, walked along. Every number in the reconstruction changes and no reprojection does, which is the only honest way to show a freedom is real rather than small.

The number of centres is one more entry on that list. It is free, the family is a one-parameter group of shears, and what closes it is knowledge from outside the picture.

What each drawing system preservesEvery cell is read out of the system's own map rather than typed: the centre is fitted from its rays, measure is the site's own midpoint test, diminution is the drawn size of a fixed object at two depths, bounded depth asks whether infinite depth lands in finite page, and straightness is the sag of a receding line. Only perspective has a centre, and it is the one system with no true measure. No row has both.a centretrue measurediminutionbounded depthstraight linesperspectivehandscrollisometricdimetrictrimetriccavaliercabinetelevationmilitaryfilled means the system keeps itmeasured from each system's own projectionno row is filled in both of the first two
Fig. 13 And the battery this site measures a drawing system on, which is the same idea applied to conventions rather than to reconstructions: what each keeps, rather than how far each falls short.

What it takes to close it, in practice

Three things, in the order a person would reach for them.

A rigid object of known shape in the picture. A rectangle, a box, anything with a right angle. If it spans the seam it is caught at once; if it lies entirely inside one part it says nothing, because that part is an ordinary projection.

The taught two-point cube, with the two far edges placed 8 points apartThe corner angles are 90° because the method forces them. The side ratio is 0.719, so this picture depicts a box whose depth is 1.39× shallower than its width.horizoncorner angles90.000° — forced by the methoddepicted side ratio0.7195lens this drawing implies60° acrossdrawn exactly as the method prescribesthe free step is where the far edges go
Fig. 14 The kind of object that does the catching, and the reason it has to span the seam: a box drawn from one centre is a box, whatever the rest of the picture was drawn from.

A second picture of the same scene. Two views of a scene drawn from two centres each are four rays per point and no consistent reconstruction, and the inconsistency shows without any prior knowledge of what is in the scene.

Two rays, 2.53 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 3.3e-14 m. With the same marks read to 1 px they miss by 2.53 mm at a range of 7.45 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 — 2.53 mm gapfrom the left eyefrom the right eyegap 2.53 mm at 7.45 mexact marks: 3.3e-14 m
Fig. 15 The two-view version of the same test. Rays that should meet and do not, measured in millimetres of the world.

Or a straight line spanning the seam. A line drawn from one centre is a line; a line whose near half was drawn from one centre and far half from another is two lines with a kink in them, and the kink is visible.

That last one is the practically important one and it is why the seam matters. A two-centre picture is undetectable as long as nothing crosses the seam, and it is caught the moment something does — which is exactly the compositional rule the traditions that use the convention observe.

A shadow across the creaseOne straight rod, one lamp, two receiving planes. Each piece of the shadow is dead straight — 7e-16 m and 1e-15 m from the line through its own ends — because each is a plane projectivity of the rod, and a projectivity takes a line to a line. They meet at 35.08°, and the corner is the image of the crease rather than anything about the rod.35.08°correct from 16 cm, at 160 mm widetwo maps, meeting at 35.08°
Fig. 16 A seam of a different kind, where two receiving surfaces meet. What crosses it is what reveals it, in both cases.

The convention’s side of it

A tradition that draws a floor from above and its figures from in front is not attempting a photograph and failing. It is making a choice, and this rung says what the choice costs in the only currency this site deals in.

It costs the relation between the parts. Each part, on its own, is an ordinary projection of an ordinary thing: the carpet is a correct picture of a carpet, the figures are correct pictures of figures. What is not correct is where they stand relative to each other, and that is exactly the quantity a shear moves and a rigid motion does not.

The carpet and the people want optical axes 90° apartLeft, a camera on the carpet's normal: the carpet is a true square and a figure standing under the eye is drawn at exactly no height at all, whatever its height — off the axis its drawn length is proportional to how far off it stands, not to how tall it is. Right, a level camera: the people are right and the carpet is a 56 px band against its 271 px width. The convention takes the carpet from the left picture and the people from the right, and there is no camera that supplies both.a camera looking downthe carpet is true, the people are nota camera looking levelthe people are true, the carpet is notthe two views a miniature is assembled from90° apart, exactly
Fig. 17 The convention, with the two requirements it makes of a single camera and the right angle between them.
Seen from straight above, a figure's drawn height is its positionIdentical 1.7 m figures, drawn from a camera 9 m directly overhead. The one beneath the eye is drawn at 0e+0 px — not small, absent — and the rest lie on a straight line through the origin to 1e-13 px. So a plan view does not shorten height; it replaces it with position, and there is nothing left in the picture to recover a height from.0204060800123how far the figure stands from the point under the eye, metresdrawn length of a 1.7 m figure, pxa figure under the eye is drawn at zeroidentical figures at increasing radiusthe fit is exactly linear
Fig. 18 And the specific exchange it performs: a plan view does not shorten a standing figure, it replaces its height with its distance from the point under the eye.

Which is a good trade if what the picture is for is identifying what is present and how the floor is laid out, and a bad one if what it is for is measuring where anything stands. That is a statement about what a picture is for, and the geometry can price it and cannot decide it.

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 whole comparison, on one battery. No system answers yes to a centre and a true measure at once, and a system that wants both takes two centres and pays for it here.

What this does not say

It says nothing about whether a viewer notices. A composite picture is read by people as a picture and the geometry is not what they are reading. That is a fact about seeing and this site has no standing on it.

It says nothing about the order the parts were drawn in, or which eye came first. The construction picks one centre and absorbs the others onto it, and picking a different one gives a different absorbed scene drawing the identical picture. There is no first eye in the geometry; there is a first eye in the choice of which one to keep.

It says nothing about detection algorithms. What is described here is what information is present, not how to extract it, and a method for extracting it from an image would be a claim about image data — which this site’s own rulings put outside its scope.

And it does not say the absorbed room is a good description. It is a consistent one, which is a much weaker property, and the whole point of the previous rung’s measurements of what the shear costs is that consistency is cheap.

The refusal is real, and it is worth saying which one

Two different things in this essay are called refusals and they should not be confused.

The fit’s refusal — the rays missing by two thirds of a metre — is a statement that a stated pair of scene and picture is inconsistent with any single centre. It is exact, it is measured in metres of the world, and it is what the previous rung’s slider drives.

The collection’s refusal — the strip that says a figure has no viewpoint — is a statement about a figure, and it is under control rather than available on request: exactly the figures on a named list may print it, every one of them must, and each one’s configuration has to be shown to have no centre by this same solver before it is allowed to.

The near half from one eye, the far half from anotherThe floor and the near box are drawn from the first eye; the two far boxes are drawn from a second eye 1.30 m away from it. The faint outlines are where the first eye would have put those same boxes, and the gap between the two answers about one world point runs to 902 px. Taking every drawn mark with the world point it is a mark of and asking for the one point all those lines pass through, the best answer misses them by 0.539 m — so this picture is a projection of this room from nowhere at all.two eyesno single viewpoint — the rays miss by 0.54 mtwo centres, 1.30 m apart
Fig. 20 A figure on that list, printing the strip, with the miss that justifies it named in the strip itself.
The rays of a refracted picture, continued into the waterEvery ray leaves the pinhole, bends at the surface and carries on. Fitted to a common point they miss it by 9.9 mm — the circle is that miss drawn at the figure's own scale. With the water removed the same fit misses by 0e+0 m.the water surfacethe pinholethe rays miss by 9.9 mmno single viewpoint — the rays miss by 9.9 mmdry control: 0e+0 m
Fig. 21 And the entry that opened the list: a bundle continued into water whose rays miss their own centre by millimetres.

The list now has four reasons on it — bent light, a moving eye, a curved reflector, and two centres — and the fourth is the only one that can be taken away without changing a drawn mark. That is what makes it the interesting entry and it is why the absorbed figure prints the opposite strip.

An exemption a figure could take silently would be a hole rather than an exemption. One that has to be named beside the number justifying it is a decision on the record, and the number is the same kind of number in all four cases because the same solver produces it.

The same question about the other three reasons

It is worth running the essay’s own distinction over the other entries on the no-viewpoint list, because two of them answer differently and the difference is informative.

A refracted picture. Told the scene, the rays refuse. Told nothing, is there a scene whose ordinary projection this is? Not in general — the displacement a slab adds depends on how far away the point is, in a way no single projective map reproduces, so the refracted picture is not a projection of anything from anywhere. That is a stronger statement than the two-centre one and this collection has made it.

A case front 25 mm thick, with and without itEvery point has moved — by up to 4.5 px — and every vanishing point has not, to 4e-6 px. So the camera recovered from this picture's own vanishing points is the camera that took it, to 4e-10 relative, out of a picture in which nothing is where it was.no single viewpoint — the rays miss by 4.5 px, depth-dependentf recovered from it: 396.88 px
Fig. 22 The reason. A slab’s displacement is a function of depth, so undoing it would need a map that knows depth, and a projection does not.

A handscroll. Same answer, same reason: the eye’s position varies continuously along the roll, so absorbing it would need a different map per column and there is no single map of space that does it.

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. 23 The signature of that: a straight line images as a hyperbola, which no projective map of space produces from a straight line.

A curved mirror. Same again — the reflected bundle is spread over a region rather than issuing from a point, and no rearrangement of the scene collects it.

So the two-centre picture is the only one of the four that can be absorbed, and the reason is that it is made of finitely many honest projections rather than of one dishonest one. That is worth having as the sharp version of this essay’s finding: it is not that a picture without a centre can generally be re-read as one with a centre, it is that a picture assembled from projections can.

The transferable form

“Can this be detected?” is not a question about the thing. It is a question about what else is known, and the honest answer names the extra information rather than reporting a yes or a no.

The two-centre picture is undetectable given the picture, detectable given the room, and undetectable again given only that the room is some room. Three answers to what looks like one question, and the difference between them is which side information was on the table.

These essays have now met that shape three times over: a drawn rectangle’s proportion is undetermined until the centre of the picture is supplied; a ball’s outline gives a direction and no size until a length is supplied; and a picture’s number of eyes is free until a shape is supplied. In each case the free thing was a named object and the closing information was one number or one known solid, and in each case reporting a single figure without naming the assumption would have been the error.

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. two eyesno single viewpoint — the rays miss by 0.54 mtwo centres, 1.30 m apart A picture with two eyes in it Several traditions draw the floor from one place and the people on it from another. This site has said no single camera produces both. What such a picture actually is has a measurement attached: give the rays their world points and ask for the one place they all pass through, and at a stride of separation the best answer misses them by six tenths of a metre. 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.

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.

centre of projectionCorrespondenceDegeneracyDemonstrationDrawing systemerror propagationFree parameterleast squaresPicture planescale ambiguity