The thread: Determined up to something — page 2
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.
Measuring from one pictureOne picture of a ball
The outline of a ball in a photograph fixes the cone of rays that graze it, so the cone's axis is the direction of the ball's centre and its half-angle is the arcsine of radius over distance — both exactly, and neither of them separately. A ball a hundred and thirty-seven times larger, a hundred and thirty-seven times further away, draws the identical conic. And the drawn ellipse's own centre is not the image of the ball's.
Mirrors that are not camerasA fitted radius is wrong before it is uncertain
A sphere and a paraboloid of the same vertex radius agree to second order, so a fit over a small aperture cannot separate them. What it does instead is return a confident radius that is wrong by a stated percentage, with a residual far below any measurement floor — 0.03% of bias behind a residual of three ten-thousandths of a degree. The residual only clears a two-hundredth of a degree at six times the aperture, by which point the bias is thirty-six times larger.
The second projectionThe screen that names the seat
A flat screen shows a homography of the intended picture from every seat in the room, and an observer's own framing is free to be a homography too — so a flat screen's picture is consistent with every seat there is. A curved one is not, and the seat comes back out of the picture in all three directions, in units of the screen's own radius.
The second eyeSeven marks, three answers
Seven correspondences leave a two-dimensional nullspace, and the requirement that a fundamental matrix be singular is a cubic in the mix — one or three real roots. Here it has three, and all three satisfy every one of the seven marks to 8.9 × 10⁻⁹ pixels. The eighth mark, withheld, separates them by more than an order of magnitude.
What a pair is forWhole pixels cut space into shells
A disparity read to whole pixels can report only the depths fB/k, so a stereo pair does not measure distance on a scale — it chooses among 113 shells between half a metre and twelve, 6.7 cm apart at two metres and 1.39 m apart at ten. A level floor comes back as 35 standing plates. And a finer step and a better reading are different purchases: at a quarter pixel with a quarter pixel of matcher error the pair prints 449 depths and can tell 149 apart.
What each system gave upFour surfaces, and no one camera that draws them
Read under one assumed camera, the floor, footstool, table and book of a constructed divergent picture imply tilts of 36.7°, 65.1°, 54.2° and 78.2°, where one camera photographing four parallel surfaces gives each of them 35.0°. But the spread between the tilts is 46.8° under a 260 px lens and 6.7° under a 5,000 px one, so it measures the lens as much as the picture. The measure that owes nothing to a lens is on the page: the nearest drawing one camera could make moves the far corners by 17.6 px.
The real instrumentA barrel model folds at a radius it sets itself
The polynomial every calibration fits to a wide lens stops increasing at a radius fixed by its own first coefficient — 47.49° of field at k₁ = −0.28 — and past it two directions land on one picture radius. The routine that undistorts pictures with it does not refuse there. It hands back wrong directions from 46.75°, by as much as 106.5°, and refuses only at 65.5°: a fifth of the field returned silently wrong.
Many pictures at onceThe eighth held number bends the scene
Four surveyed points, each 10 mm out in a different direction. Hold seven of their coordinates during an adjustment and the courtyard's shape moves by a trillionth of a millimetre; hold eight and it moves by 0.59 mm, because seven numbers choose a frame and the eighth makes a claim the pictures disagree with.
What survivesFour lines have a cross-ratio
The cross-ratio is introduced as a property of four points on a line. Its dual — four lines through a point carry the same number, and any transversal reads it back — is not a second theorem. It is why the first one is true: four rays from an eye are a pencil, every picture of them is a section of that pencil, and a quantity belonging to the pencil cannot depend on which section was taken.
Where to standWhere the anamorph still works
A picture correct from one point raises an obvious question that nobody answers with a number — how far may the eye move. The answer here is exact rather than approximate: a wrong viewpoint composes the intended picture with a central collineation whose axis is the line the picture stands on, so the error is zero along that line and grows linearly upward, and a step sideways costs precisely as much as the same step upward.
Measuring from one pictureTwo pictures of a ball
Two outlines give two tangent cones, and where their axes cross is the ball's centre — with the radius following from either half-angle. There is no correspondence problem, because a ball's outline in one picture and its outline in another are guaranteed to be the same ball; and there is a degeneracy, which is the two eyes and the ball in a line, and the solver refuses it rather than returning something plausible.
The second projectionThe seats a screen will accept
Collect the seats whose picture is within a pixel of the one intended and the result is a solid — half a cubic centimetre in front of a curved desk monitor, a litre in front of a curved television. Ten times the tolerance is a thousand times the room, which is the pavement anamorph's own law arriving on an object that has nothing else in common with it.
Drawn confidentlyThe rule that draws another room
The taught rule for spacing receding boards — each gap a fixed fraction of the last — is not a projection of anything, and it produces a pavement that is a correct perspective to within a fifth of a pixel. Of a room whose horizon is a hundred and seventy pixels from the one the panel drew. The error is not incoherence; it is a disagreement between two halves of one drawing.
The second eyeThe surface two pictures cannot separate
There is a quadric through both camera centres on which two genuinely different motions draw identical pictures. Built explicitly, forty-two marks satisfy both epipolar geometries to 2 × 10⁻¹³ pixels, and the two scenes they reconstruct place the same mark at 15.6 metres and 35.1. The design matrix's nullspace has two dimensions rather than one, which is the seven-point situation arrived at from the other side.
Mirrors that are not camerasOne shutter, two views
A photograph with a mirror in it is a stereo pair, and a peculiarly well-behaved one. Its fundamental matrix is skew-symmetric, so both epipoles are the same point; that point is where the camera would see its own lens; and every line joining a mark to its reflection passes through it, to 1.4 × 10⁻¹² px. The baseline is twice the distance to the glass, which is the one number a single view cannot supply and a tape measure can.
Many pictures at onceA narrow view keeps a second answer, inside out
Six pictures of a courtyard through a 3° field, every mark exact. Started from the scene turned inside out, the adjustment settles there — near points far, far points near — and misfits the marks by under a pixel. The misfit grows in proportion to the field and to the sweep of the cameras, and vanishes only where perspective does.
What each system gave upA camera count needs a tolerance
Asked how few cameras could have drawn a splayed picture, a constructed stack answers four, three, two or one depending on how many pixels of redrawing a reader will allow — one camera at 17.6 px, two at 11.1, three at 4.6. The count is real: a picture built with two groups hidden in it gives back exactly those two, anywhere between nothing and 47.0 px. What it is not is a property of the picture alone, and the floor under it belongs to the hand — a one-camera drawing made by a hand that scatters its far corners by σ splits below about 0.57σ, where the 0.57 belongs to that hand and runs from 0.17 to 1.34 across a hundred and twenty of them.
Where to standThe marks name the place, not the height
Run the camera-recovery round trip on an anamorph — hand it the floor marks and ask for the eye back — and it returns the spot on the floor to eleven decimal places with nothing assumed at all. It does not return the height. What the marks fix is the product of the eye's height with the design's aspect ratio, and no amount of looking at the floor separates the two.
The other systemsThree views do not fix the solid
A stepped block on a six-cell grid draws a front, a top and a side view. So does a solid with a third of its material, and so does one with more than the block has — 192 cells against 64, every filled square in all three views identical. The drawing office's triple bounds a part between two solids and does not determine it, and the gap runs to a factor of n.
Light and mirrorsThe floor that is not a plane
A shadow on a flat floor is a homology, so four marks determine the whole map and the rest of the outline comes back exactly. Dish the floor and the same four marks mispredict the rest by 5.67 mm; ridge it and 9.07 mm; put a step in it — two planes, each of them exactly a homology — and 74.95 mm. The receiver's shape is what breaks the projective description, and it breaks it worst where the surface is flattest.
Systems that kept the measureA 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.
Measuring from one pictureThe dish no outline reaches
An outline is a pair of numbers per direction and nothing more, so a shape built from outlines has two errors that behave differently. The part outside the convex hull falls as one over the square of the view count. The part inside a concavity is the same area at four views and at a hundred and twenty-eight, because no pair of supporting lines ever reaches into a bite.
The second projectionOne picture and three people
A curved screen can be pre-warped for one seat, and the search over which seat to choose returns the middle one to a quarter of a per cent — there is nothing to be clever about. What the correction buys the sofa as a whole is six per cent, and the worst seat grows at fourteen pixels for every metre of audience, with no width at which it is zero except one person.