Identifiability — where it appears
Named by 61 essays across 18 fields — each of them below, with the objects they name alongside it.
Assembled from several views
An Egyptian relief takes each part of a figure from the direction that identifies it — head in profile, eye and shoulders frontal, a pond in plan. What that buys is exactly measurable: any single viewing direction keeps at most √k of k perpendicular aspects, so the best compromise view retains 58% of each.
Far enough away, a pair is one eye
Hold the baseline and walk the scene away, and the parallax a single homography cannot explain falls as the distance to the power −0.968 — one over the distance, which says the ratio of baseline to depth is the whole of it. The recovered translation direction follows it down, from 3.3° at four metres to 74.5° at two hundred and fifty-six.
The 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.
A 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.
One 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.
An ambiguity is not an uncertainty
Eight marks to forty cuts a solid scene's pose error from 19.8° to 0.7° and leaves a flat one at 48°. The two failures look identical from inside — a confident answer, a residual at the floor — and they respond to opposite remedies, so telling them apart is worth more than either measurement.
One 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 same person, twice on one panel
A panel showing one figure at four moments is geometrically the least strange thing in this field — one camera, one floor, and every pair of copies meeting the horizon to 8.5 × 10⁻¹⁴ px. What the picture withholds is the order, and four copies admit twenty-four readings, and a reading convention supplies 4.6 bits from outside the marks. Enlarge one figure by six per cent and the horizon test that passed the panel catches it at 36 px.
Size that means rank
In a great many pictures the drawn height records importance rather than distance. That is a decision rather than a mistake, and it can be caught with a straightedge by carrying one figure's height across the room by the taught construction and see where it lands. A tenth of rank in the picture already misses the drawn head by 21 px, the miss is exactly linear in how much rank is there, and the whole test needs two references and no arithmetic.
A picture in bands
A register picture stacks its scene in horizontal bands, each with its own ground line and every figure drawn at one height. The feet line and the heads line of any pair are then parallel to the arithmetic floor — 0° against 15.4° in a photograph of the same figures — so the picture has no horizon anywhere in it, and what a reader recovers is an ordering with no metre attached.
The pond with its trees laid flat
An Egyptian garden pond is drawn in plan with its trees rotated outward about the bank they stand on. A rotation is an isometry, so every length in that drawing is exactly the length it is in the garden — zero error, not a small one. What is spent is the angle between any two faces, which reads 180° across every hinge and is 90° in the garden, and the four walls admit sixteen assemblies, so a reader supplies four bits to fold it back up.
The wedge recovered with the camera
Admit the glass into the model and the fit finally has something left over. A two-degree wedge is invisible to a reading confined inside eighteen degrees of the axis — the fit calls the whole displacement a rotation of the camera and is right to — and by fifty-five degrees it comes back to 0.07 of a degree. What the picture does not separate is the ordinary glasses — assuming an index of 1.50 for a true 1.52 costs one per cent in the angle and 0.07 pixels of residual.
Bands with no overlap name a height for the eye
Read as a view, a stack of bands is one drawing of a family of eyes and row spacings. Keeping every figure clear of the row in front sets the least elevation: 40° for a procession and 74° for a crowd. There is a real parallel view that draws it exactly, an oblique one onto the wall itself, but it draws its people as paper. What the arrangement buys, on one wall, goes as the square of the number of bands.
A wedge moves the centre, not the lens
A wedge of glass in front of a lens deflects every ray by a little more the further off the axis it goes, which looks like the shape a radial distortion coefficient describes. Fitted together, the two are nearly independent — correlated at 0.16 at most — and a calibration that knows nothing of the glass does not invent a lens: it reports a coefficient of about 0.002, moves its principal point by 4.5 to 14 pixels, and leaves a swirling residual that no radial model takes. The blame goes to the camera's centre, and the residual says so from twenty-two degrees off the axis.
No solid casts an aspective figure
Fitting the best single rigid view to an aspective figure — head and legs in profile, eye and shoulders turned square — misses its own marks by 2.6% of the drawn height, and no yaw does better than 3.0% in a full sweep. A genuine single-view drawing of the same body fits to 7.6e-13 pixels, and the five rotations recovered from the marks alone match the convention's own list to 0.0e+0°.
A calibration through glass reports a prism
Give a calibration that knows nothing of a 2° wedge in front of the lens the two decentring terms every standard model carries, and it takes three-quarters of the swirl the wedge leaves, moves the principal point further rather than less, and reports a lens decentred by about 0.013 — a lens that does not exist. Give it thin-prism terms instead and it takes more — nine-tenths at twenty-five degrees off the axis, almost all of it nearer — and reports what the glass is: a prism. Either way the residual that announced the glass goes quiet, out to 41° and 46° off the axis instead of 22°.
A closer pose sharpens one coefficient and not the pair
Photograph a calibration target tilted four ways at several distances, and the two radial coefficients of the lens stay locked together near a correlation of −0.97. A target further off adds nothing at all. Bringing it closer sharpens the first coefficient by about a half, then stops, and pushes the pair closer together. What moves the correlation furthest is being told each pose's distance.
Facing the reader is not being reachable
A face turns toward the eye or it does not, and that is a dot product any reader can compute. Whether the eye’s rays actually land on it is a different question with a different answer — on a seating rake, three faces of ten that face the reader receive nothing, and they are 27 per cent of the facing area. On a corner the same test loses nothing at all, which is what makes the gap occlusion rather than arithmetic.
Turning the glass tells the window from the lens
A calibration done through a two-degree wedge of glass reports a lens decentred by about 1e-2 that does not exist, and hides the glass inside it. Calibrate twice with the glass turned between, and the glass's part turns while the lens's stays: a half turn finds a real lens decentring of 2.2e-3 to 5.2e-4 with readings to 0.4 px, where one calibration was 1.0e-2 out. Three calibrations spread evenly round the turn do better, because the loop the glass traces is not a circle. What never separates is the radial coefficient, which has no direction for a turn to carry.
The eye that reaches the most
A higher eye buys the faces occlusion was hiding and loses design off the far end of the object, so "the best eye" is not a question with an answer until somebody says which of the two they are paying for. On three objects the answer is as high as possible; on a corridor with a doorway in it the two quantities cross and the best height is two and a bit metres.
One more habit takes two hands, and only part of a walk
A picture whose strips disagree was not painted by one steady hand, but the disagreement does not say whether a stepping hand or a second painter put it there. Refit the picture with a second habit and the answer is in how much of the disagreement that one number takes: nine-tenths and more for two hands, a share that shrinks with every strip added for a walk. Read against a line set on separate paintings, eight strips tell the two apart eighty-nine times in a hundred — and the split the second habit chooses is a map of who painted what, wrong mostly where an unknown starting edge lets one habit counterfeit the other.
A level picture shows its rise on its horizon, and hides its slide
A shift lens moves the principal point off the middle of the frame, and the textbook focal length from two vanishing points pays for assuming it did not. On a level camera the payment is optional. The principal point lies on the horizon, so a rise is read straight off the picture and costs nothing; a slide along the horizon is read by nothing at all, because two vanishing points fix only a semicircle of possible cameras. What a sideways crop costs then depends on how the building is turned: a tenth of the frame trimmed off the side is 23 per cent of the focal length on a facade seen eleven degrees off square and under one per cent at forty-five.
A followed post's own lean is read as the bend
A rolling camera on a bend reads the bend from how its roadside posts lean, and a post that leans by itself adds its own tilt to every frame, unchanged. That tilt is indistinguishable from the bend's curvature: a tenth of a degree on a post twelve metres out turns a 50-metre bend into 58. No number of frames removes it, and fitting each post's own lean along with the bend loses the bend — a sideways camera's reading widens from ±7 metres to ±29. A followed post has to be plumb to six hundredths of a degree, finer than a well-set steel column; timber posts at a quarter of a degree help only in their dozens.
A panel's creeps keep the order it was painted in, while the hand is steady
A painter who tires lets each pavement's gaps creep a little more than the last one's. Fit each pavement's creep from its transversals and rank them, and the ranking is the order the panel was painted in: with the creep growing a point a braccio from one pavement to the next and the hand at a pixel, the whole order of four pavements comes back fifty-seven times in a hundred, against one in twenty-four by chance. The step is the weak link — four short gaps fix its creep to two points — and a creep set by each pavement's length rather than by time reads as a confident order that is not the painting's at all.
Four marks before anything is said
A reader fitting a correct perspective to a row of transversals has three numbers to choose, so three transversals fit whatever they are and the fourth is the first that can disagree. Below that count a pavement is unfalsifiable, and a great many painted pavements are below it.
A walk's order is on the floor, not on the page
A narrative panel whose copies are moments of one walk holds its order after all — in where the copies stand, because a walker moves a step between moments. Joined by the shortest route on the page, six copies seen from a standing eye come back in the order walked 68 times in 100. Undo the floor first, with the horizon the copies themselves supply and the angle of view guessed anywhere from half to twice the truth, and the same rule returns it every time the walk allows — leaving one bit, the direction, which a figure facing its way supplies.
Groups in a row are one painter twice; groups in turn say little
A second habit splits a divergent picture into two groups, and nothing in the split says whether they are two painters or one painter before and after a change. The order of painting can, because a hand that changed once leaves its groups in a row and two hands taking turns leave them alternating. On eight strips read to a tenth of a pixel, 93 panels in 100 by one hand who changed habit are read with their groups in a row and none by two hands in turn. But the change and the fatigue the order is read from are both drifts in time, so a change at one moment costs the rate of tiring three-quarters of its precision again — and a panel read as alternating has acquitted nobody.
A second rectangle fixes the column a level picture hides
Two vanishing points of a building put a level camera on a semicircle and leave it free to slide along it; a second rectangular object turned against the first puts it on a second semicircle, and the two cross once, at the camera. On a facade seen twelve degrees off square and slid a hundred pixels, a second object twenty degrees further already reads the focal length to 1.3 per cent where the building alone, even with its principal point known, manages 2.9 — and where the frame's middle column is wrong by a third. The reading does not care where the column was slid, and it is best when the second object stands at forty-five degrees to the camera, whose semicircle is flat over the true column.
A photograph grades its posts across the sight, and not along it
Once two plumb corners have fixed a picture's vertical vanishing point, every other post in it can be read for its own lean: the angle between its line and the plumb line through its foot. Posts 2.5 metres tall at 7.5 metres are read to about a twelfth of a degree, enough to tell a quarter-degree lean from a twentieth ninety-one times in a hundred. But the reading sees only the lean across the line of sight. A post leaning towards the camera is caught five times in a hundred near the centre of the frame and a third of the time at its edges — and the posts read against one another lose the corners' share of the error, though not the street's own shared lean.
A corridor of mirrors counts their angle, not their leans
Between two facing mirrors that both lean, the barber's corridor of self-images ends where it leaves the glass, and the count depends on the two leans only through their sum — the angle between the mirrors. Forty-two pairs of leans fall on one curve, within an image. Counted from near either mirror the number is the same, so counting twice cannot say which mirror leans, and two mirrors tilted together run the corridor on for ever. A photograph can: each round trip lowers the image by the sum of the leans, and where the row of images starts shares the sum between the two — provided the camera knows which way is level.
A post read both ways sets its own lean aside
A rolling camera reads a bend from how its roadside posts lean, and a post's own tilt is read as bend: a quarter of a degree leaves a 50-metre bend read to ±12 metres. Followed again by a car coming the other way, the same post shows the same tilt and the opposite shutter lean. Half the sum of the two passes is the tilt and half the difference is the bend, and a timber pole read both ways gives the bend to ±1.7 metres — better than a plumb steel column read one way, at ±2.3.
The slip that leaves no trace
A distance point put twenty-four pixels wrong moves the pavement by two and a half and leaves the reader's projective test reading exactly four thirds. The same slip on Alberti's section moves the drawing by the same amount and is caught, so the difference is not the size of the error — it is that one of them lands back on the set of correct drawings.
Two painters on their own clocks leave no record of their turns
If each of two painters tires with their own work rather than with the day, the habits a panel's strips report record each painter's own sequence and nothing of how the two sequences were interleaved: painted in turn or one after the other, the panel is the same to the last digit. What does carry the difference is the shape of the fatigue — one trend through the panel, or two that start fresh — and that shape names one hand or two painters 59 times in 60 at a tenth of a pixel. But only once the comparison is calibrated: taken at face value, the model that searches seventeen times as many orders wins by searching, and calls two painters one hand four times in five at a quarter of a pixel.
Each copy's facing reads the walk past the place it doubles back
A walk that doubles back defeats every reading of a narrative panel from where its copies stand, because a later stop lands nearer an earlier one than a step. The way each copy faces is a short arrow on the floor, and a route that has to agree with the arrows as well as the distances follows the walk past the turn: on walks turning 70° between stops, facings read to 15° return the order 94 times in 100 where positions alone reach 59 — and say which end came first. But only once the arrows are undone onto the floor with the horizon the copies supply. Read straight off the page, an arrow pointing away from a standing eye is drawn nearly flat, and the same facings return 82.
A circle on the ground names the column and the focal length at once
The picture of a circle lying on the ground meets the horizon in two complex points, and for a level camera they are exactly the principal point's column plus or minus i times the focal length. So one circle does what two rectangles turned apart do, with no turn to choose: a manhole cover 1 m across, 8 m off, marked at 24 points to half a pixel, reads the column to 14 px and the focal length to 1.5 per cent, as two rectangles twenty degrees apart do — wherever it lies across the picture. It pays for distance as the square: the same cover at 24 m is out by 128 px. And the building beside it is worth its horizon and nothing more.
A second photograph reads the lean the first cannot, if it is taken round the corner
One photograph with two plumb corners grades a street's posts for their lean across its line of sight and is blind to a lean towards or away from it. A second photograph, read the same way, sees each post's lean across its own sight — so the pair reads both components wherever their two sight lines through a post cross at a wide angle. At right angles they read the lean along the first sight to a tenth of a degree, and catch 65 in 100 of the posts the first picture missed, against 26. The picture the question proposed, from across the street, is nearly the worst place to stand: its sight lines run back along the first, it sees the same component reversed, and it catches 29.
Two mirrors nearly facing bend the corridor two ways, and each bend runs out at its own edge
Between two mirrors that lean and are also turned a little about the vertical — the barber's shop, seen past the shoulder — the corridor of self-images steps down by the sum of the leans and sideways by the sum of the turns on every round trip. Its count is not a reading of the angle between the mirrors: each bend runs out against its own edge of the glass, and the corridor ends at whichever does first. A quarter of a degree between the planes gives 22 images when it is all lean, 21 when it is all turn and 25 when it points between. A photograph reads both bends from how the images step, with no reference at all, and needs gravity only to share the lean between the mirrors and the room only to share the turn.
A body that bounces leaves a second swelling in the rows, and one pass reads both
A vehicle's body pitches and bounces on its springs at two frequencies, and a scroll camera's rows answer both: the pitch by turning the camera, the bounce by lifting it through the height climbed between the slits' two moments. One oscillator fitted to a body pitching at 1.4 Hz and bouncing at 1.0 lands between the two, at 1.22 Hz, with the camera over the body's centre, and on the pitch, at 1.41, with it 1.5 metres ahead — too lightly damped either way. Two modes fitted to the same single 800-metre pass read both, the pitch to ±0.02 Hz and the bounce to ±0.03, say plainly whether a bounce is there at all, and expose a bias the earlier readings carried even on a body that does not bounce: a camera ahead of the centre reads a 1.4 Hz pitch as 1.5 unless the fit knows its lever arm.
Frames read up and down part a post's lean in one drive
A rolling camera reads a bend from how its roadside posts lean, and a post's own tilt is read as bend unless a second car drives the road the other way. A sensor that reads alternate frames from the bottom row up reverses the shutter's lean without reversing the car. Thirty-two such frames of one drive read the 50-metre bend from a timber pole to ±1.8 metres, against ±1.7 for two passes of sixteen — and a post swaying in the wind, which costs two passes three metres, costs one drive a quarter of one.
How many lamps a drawing has
The shadow field recovers a lamp by intersecting drawn lines. Two lamps make that a partition rather than an intersection — and two centres fit any bundle better than one, on a one-lamp drawing as readily as on a two-lamp one, so a count is a decision that needs a noise level before it exists. A criterion built on a penalty instead of a noise level returns four.
A fold names the height
A pavement anamorph’s marks fix where the reader must stand and leave how tall they are entirely free — every height explains the marks exactly, to the last bit. Put one crease in the floor and the freedom is gone, because two degrees of fold makes a ten-centimetre error in the height leave six tenths of a millimetre, and a right angle makes it nine.
A painter's rate of tiring signs every strip but the first
Two painters who tire on their own work at different rates leave the difference in a panel's rows: at a tenth of a pixel a gap of a point a strip is read 97 times in 100, though the gap itself is read only to within three-quarters of a point. Carried to a second panel by the same two, where they share one habit and nothing else tells their strips apart, the rates give a strip to its painter — every strip but each painter's first, which carries no rate at all, and every strip but those whose creeps coincide, which they do whenever one rate is a small whole-number multiple of the other. Painters at one and 1.7 points a strip are told apart on every later strip; at one and two, on 88 in 100.
A circle reads a tilted camera's column, and its tilt has to come from the verticals
The picture of a circle on the ground meets the horizon at c ± i·f for a level camera. For a camera tilted down it meets it at c ± i·f/cos(tilt): the column is still exact, and the imaginary part is the eye's distance from the horizon line rather than from the picture. Every eye on an arc of that radius about the horizon fits the circle — and fits the building's right angle too, which is one of the circle's own. A handheld tilt of five degrees costs a level reading 0.4 per cent; looking 45° down at a manhole cover it costs 41. The building's upright edges, meeting far below, cut the arc once and give the camera back, and from then on the circle is the better half: looking steeply down it names the column to 1.7 px where three vanishing points manage 11.
A third photograph closes the hole the second leaves, if it brings a building
A second photograph of a row of posts reads every post's lean except the one whose two sight lines run nearest parallel, and that post sits at the end of the row away from the second camera. A third photograph from the other side puts its own worst post at the other end, so between them no post is left: sixty degrees round either side, the worst post is read to 0.10° where either picture alone left it at 0.23 or 0.20. But the third picture does not buy what was hoped of it. Without buildings of their own, three pictures catch fewer of the hidden leaning posts than two at a right angle with one, and the spare equation each post gains cannot read the further pictures' verticals: a lean the whole row shares, or one that grows along it, looks exactly like a vertical misread.
A mirror corridor shows the room once, and the room's lines are a reference by where they are
Between two facing barber's mirrors the corridor of self-images goes on for a dozen round trips, but the shop around it does not: the floor never appears in a level photograph, and the walls appear in the first reflection only, because every later path has to start and end on glass and only what stands between the two panes can be reached. That one sight of the wall behind the sitter is enough. A rail and a door jamb on it, fitted with the self-images, read each mirror's own lean and turn to about a twentieth of a degree with the camera's heading, pitch and roll unknown — better than a camera believed level and a tenth of a degree out. The reference is not the lines' angles, which can be 0.2° off at almost no cost, but their places: a rail 5 mm higher than believed reads as a lean of 0.14°.
One pair of frames reads the speed and the bend from the roadside
A camera that reads one frame from the top down and the next from the bottom up turns every roadside pole's rolling-shutter lean round and leaves its own tilt where it was. Half the difference of each pole's two leans is the shutter's alone, and across a row of poles it lies on one straight line in one over the range: its slope is the car's speed and its crossing is the bend. So one pair of frames, a fifteenth of a second, reads both — the speed to 3 per cent and the bend to ±5.6 m from sixteen timber poles, whatever they lean. With the speed known, sixteen poles read the bend to ±2.4 m, a little better than one plumb post followed for sixteen frames. The nearest pole decides most of it.
The rows read where the camera rides on the body, and a camera behind the centre hides the pitch
A scroll camera's rows over a rough road say how far ahead of the body's centre the camera stands, as well as how the body pitches and bounces. One 800-metre pass reads a camera 1.5 metres ahead to ±0.15 metres, costs the pitch's frequency nothing to do it, and removes the bias a fit that puts the camera at the centre carries — a 1.4 hertz pitch read as 1.54 from 2.5 metres ahead, and as 1.80 from a metre behind a body that does not bounce. Behind the centre the camera's rise works against its turn, and a metre back on a body that does not bounce it very nearly cancels the pitch where it rings.
A turned part's outline reads its swelling from every side, and a straight taper hides it
A turned part that swells by 15% of its radius two fifths of the way along, drawn to half a pixel, gives up the swelling's height to ±0.0024 of its radius and its crest's place to ±0.003 of its length from one side-view drawing — and to about the same from every other view, square to the picture or lying across it. A swelling of one hundredth of the radius, under a pixel on the page, is read as 0.0098 ± 0.0022. Read as a straight taper, the earlier essays' reading, the same drawings put the swelling into the taper and read a 1° part as 2.2°.
A walker who stops is ordered by the way it turns
Three copies of a walker who pauses on one spot stand a few centimetres apart, and no reading of where they stand can order them. The way they face can — but not by the rule that followed a walk that keeps moving, which reads a three-copy pause turning 90° only a quarter of the time. A reader that ignores the direction of legs shorter than a step, and charges those legs instead for the turn between the copies' facings, reads it 92 times in 100 and costs a moving walk almost nothing. The middle copy of the pause is the hard one, and it is held only where the walker turned well clear of the error in reading each facing.
The distance at which two lamps part
Two lamps five centimetres apart are one lamp, and the drawing is right to say so. The separation at which they become two is proportional to how carelessly the picture was clicked — 3.6 centimetres at half a pixel, 7.2 at one, 28 at four — with no floor anywhere, so nothing but care stands between a reader and any separation at all.
Stepped, or measured from the zero
The same hand at the same precision, laying the same braccia off two ways — dividers walked from the last mark accumulate and grow as the square root of the count, while marks set from a common origin do not accumulate at all. The difference is not which method was used — it is where the zero is, and only the second is recorded in the drawing.
Two circles leave the eye on a circle
A picture of the ground holds five numbers, and a camera has six. Any number of circles at any distances draw identically from every eye on one circle in the side view, so a second circle cannot give the pitch. It gives the circle. A tape measure cuts it, and which length to tape depends on the pitch.
Scatter breaks the later ties and never the first
Two painters' rates of tiring sign their strips, except where the two creeps meet. If a hand's slips also grow with the strips it has painted, a tied pair differs in its scatter unless both are first strips. Five rows a strip settle a later tie about two times in three, a gap of two strips four times in five, and a run of panels nearly always. Two first strips stay a coin toss, whatever is measured.
What a null result is worth in decades
The first draft of this expected a short sweep to invent a floor, on the reasoning that least squares always spends a free parameter. It does not — on exact data the fitted floor of a floor-free law comes back at three parts in a quadrillion. The failure is the other one and it is worse because it looks like a result. Over a third of a decade at one per cent noise, floors of a fifth of the first sample are still consistent with the data, and the fit reports none while telling the truth.
A floor cannot fake a second lamp
Cast the same two lamps onto four floors at four curvatures and ask how well one centre explains the drawing. Every one of the thirteen answers is 138.3277 pixels — the same to fifteen digits, because a floor decides where along a ray the shadow's tip landed, and a line through a point and another point that has slid along it is the same line.
What a panel says about its maker
The reading assembled over this row, run against every procedure sixty times and scored — with the failures reported as carefully as the successes, because three of the five rows are refusals. A drawing names the class of error in it, not the recipe that produced it, and one procedure it never names at all.
The drawing does not run out of lines
Every post supplies a line to every lamp, so a drawing of five posts offers ten lines to two lamps and twenty to four — the unknowns and the constraints grow together and two posts fix any number of lights. What runs out is the partition, whose margin falls from 251 pixels to six as the share of lines assigned correctly falls from all to just over half.
Which rule a fisheye obeys, from straightness alone
Four candidate rules for a fisheye lens part by 54.0 per cent at 45° off axis, and a plumb-line fit shown no scene, no camera and no calibration target can still name which one took a photograph — reliably from about 45° of half-field. Below that the four are indistinguishable in the marks, and naming one collapses to guessing.
Counting shadows is not counting lamps
Two lamps close together cast one connected dark patch and the drawn lines say two. One lamp behind two cards casts two patches and the lines say one, exactly. And the arrangement where the patch count is right — two lamps far apart — has no fully dark region at all, so the same floor answers one, two or zero depending on which darkness is being counted.
The arrangement the count cannot see
Five posts laid out five different ways give the same leverage to a sixth and the same separation limit to a quarter, and the sixth arrangement — posts strung out along their own shadows, built to be exactly degenerate — is no worse than the rest. The degeneracy belongs to the family the count does not use, and its conditioning there is exactly zero.
The arcs the five-point construction actually draws
The taught five-point construction draws circular arcs between five vanishing points and instructs a draughtsman to graduate the radius evenly. Read that way, the arcs miss a straight line's true image by up to 3.75 pixels on a 300-pixel disc. Read at the stereographic scale instead, the same arcs are exact to 4.3e-13 pixels — the construction was always drawing one projection, and the taught scale was never it.
Named alongside it
The objects these essays reach for when they reach for this one.
Residualleast squaresModel errordegrees of freedomHorizonConditioningDrawing conventionVanishing pointForeshorteningPrincipal pointCamera calibrationFalsifiability