The thread: Not a projection at all — page 2
The screen is a picture surface too
A curved television is one of the six named picture surfaces sitting in a living room, and from its own axis it delivers azimuth in proportion to the picture, exactly. What it is shown is a rectilinear picture from a sofa, and the difference is not a matter of degree — a flat screen from any seat shows a homography of the intended picture, so it is a correct picture of a transformed scene, and a curved one shows a map that is not a homography from any seat at all.
What each system gave upA page is bounded by a divide, not a centre
A pinhole draws the whole of an infinite ground in a bounded patch of page — each doubling of distance half the one before — while a handscroll spends the same page on every doubling and an isometric drawing spends three quarters of its page on the last one. It is tempting to credit the centre. A crossed-slits camera, whose rays miss any common point by 0.46 m, is bounded too: what does it is dividing by depth in both directions of the page.
Systems that kept the measureCounting 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.
The rectangle behind the lensA pupil sees around an edge
Two backgrounds identical everywhere a pinhole can see, differing only in the strip an occluder hides from it, produce identical pinhole pictures and pupil pictures 42 per cent apart. So no function of the sharp image — no kernel, no depth-dependent kernel, nothing — produces the picture a real lens makes, and the reach behind the edge is R(Z₂/Z₁ − 1), which is 120 mm here.
Through water and glassOne surface, two images
A water surface reflects what is above it and refracts what is below it in the same photograph, from the same plane. The reflected half has a centre of projection to 1 × 10⁻¹⁴ m and every theorem about central projection applies to it; the refracted half misses its own best-fitting point by 28.6 mm and none of it does. And the landscape manual's rule for drawing a reflection turns out to be the epipole placed at infinity, which is why it costs nothing at zero tilt and 11.5 px at twenty-two degrees.
The eye that movesA pond in a scroll is not an ellipse
A pinhole draws a round pond as an exact ellipse whose widest row is 3.58 px off the row of the pond's centre — the drawn-circle error every perspective textbook warns about. A handscroll draws the same pond widest exactly on its centre's row, and draws it as a quartic that no conic fits: the best ellipse misses it by 3.08 px. Each keeps what the other loses, and off to one side the pinhole's pond leans 9.67 px while the scroll's does not lean at all.
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.
Systems that kept the measureTwo grounds, and what the second one costs
The miniature convention wants its floor drawn from overhead and its figures drawn from in front, and the two optical axes it asks for are exactly ninety degrees apart. Read as a picture with two centres rather than as a picture with none, the arrangement stops being a contradiction and becomes a quantity: the rays of the composite miss their own best point by more than a metre, and the absorbed reading is a floor that leans.
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.
Through water and glassA ball of water has no eye either
A flat interface is not a projection through a centre and misses by ten millimetres. A sphere of water misses by more than that on a ball the size of a plum — 1.3 mm on a fifty-millimetre radius, and the axis crossings spread over 18.6 mm at seven tenths of the aperture. But at two per cent of the radius the same fit returns 24 nanometres, so a ball does have a centre — one at zero aperture and none by the time it is gathering any light.
The eye that movesA scroll through two slits ranges in a straight line
Draw a scroll twice, through a slit leaning 10° forward along the track and one leaning 10° back, and every point appears in both drawings on the same row, separated by 9.169 px for every metre of its depth — at four metres and at fifty-two. Depth is proportional to that separation rather than reciprocal to it, so a pixel of error costs 10.9 cm at every distance, averaging leaves no bias, and there is no range past which the depth runs off to infinity. The price is paid in roll: a 100 m scroll ranges nothing past 283.6 m.
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.
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.
What a machine computesA curved screen is eight flat ones
A projection matrix is a plane and nothing else, so a curved display cannot be rendered — it has to be driven as several planes and assembled. The gap between chord and arc is the whole error, it goes as the square of the angle each piece spans, and the piece count therefore goes as the inverse root of the tolerance — three for eight pixels, eight for one, fifteen for a quarter.
The rectangle behind the lensThe disc and the streak
A frame integrates over the pupil and over the exposure at once. Hold the point's depth and the patch is exactly the streak of its centres with one disc slid along it, to 1.8 × 10⁻⁵ of its own width. Let it recede over the same exposure and the disc's radius falls by 3.7 along the streak, and the patch departs from any single kernel by 16 pixels.
Through water and glassA point under water has two depths
The apparent depth of a submerged point is not one number even along one line of sight. The rays it sends to an eye pass through two focal lines, and at 60° from the vertical a point 1.50 m down has an image 0.740 m down and another 0.320 m down. Two eyes side by side read the first, a head moving up and down reads the second, and a pair of eyes tilted between them reads neither — their two rays miss each other by as much as 6.39 mm.
The eye that movesA scroll round a bend loses its straight-line depth
Draw a scroll through two slits leaning ±10° from a track that bends, and the separation that was 9.169 px for every metre of depth stops being proportional. Outside a 100 m bend it is 653.8 px at 256 m where a straight track gives 2347, and it never passes 907.6 px however deep the point; inside a 200 m bend it runs nearly three times ahead of depth and no slit reaches past 165.3 m. The two drawings still share their rows, and the scale along the roll becomes a function of depth.
Where to standThe ceiling that is not a plane
Paint the same design for the same eye onto a floor and onto a barrel vault, then fit the best possible homography to each set of marks. On the floor it misses by femtometres, because the map is a collineation and four marks determine every other. On the vault it misses by half a metre, and no choice of four marks helps — which is where every projective construction made for a floor stops applying.
What a machine computesOne depth per sample is not enough
A depth buffer keeps a single distance at each sample, so a post-process blur can only ask how far away the thing at this pixel is. Across an occluding edge that answer is two depths and an occlusion, and the gather it produces differs from the pupil's own integral by 70 per cent of full scale over a band eleven pixels wide.
The real instrumentThe entrance pupil walks with the angle
The place a picture is a projection from is not a point in a wide-angle design. Chief rays traced through a strongly curved front element cross the axis 15.07 mm behind its front vertex when they are nearly on the axis, and 4.23 mm nearer the front at 80° of field. So no pivot makes a wide panorama seam clean: at one metre, pivoting at the paraxial pupil leaves 4.39 arcminutes of misregistration along a seam, and the best pivot still leaves 1.41.
Through water and glassA straight stick in water is a kink and a curve
The bent stick is described as one kink at the surface. Traced point by point, the image two level eyes see leaves the surface 14.96° off a stick leaning 30° — the same whichever way it leans — and then keeps turning, by 1.11° when it leans away from the eye and 7.62° when it leans toward it. A photograph from the eye shows the kink and almost none of the curve, 1.63 px over 166 px, and when the stick leans straight toward or away from the eye it shows neither: the picture is one straight line.
What each system gave upThe tenth row has neither
A crossed-slits camera divides by depth in both page directions and its rays miss any common point by 0.354 m. Put on the comparison table it prices 38.8% on length, 69.9% on area and 58.2% on angle against a pinhole's 39.5%, 70.8% and 60.9%, and its midpoint drift is 15.6% — the pinhole's own figure. It keeps a true scale in no direction at all, and it bows a straight run of ground by 1.11 px, which no row with a centre does. Giving up the point buys nothing and costs a third thing besides.
Through water and glassA wedge of glass turns the camera behind it
A pane with parallel faces moves every point and no direction, so the camera recovered through a window is the camera that took the picture. Tilt one face 2° and every direction turns, by 1.04° on the axis and 1.67° forty degrees off it. The best rotation of the frame, 1.14°, still leaves 0.94 px, and no homography does much better, so the picture is no longer a projection from the camera's centre. The camera recovered from three vanishing points through the same glass is turned 3.37° — three times as far — because vanishing points lie where the glass bends most.