The thread: Determined up to something — page 9
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°.
The second eyeThe road under a car ahead gives its growth a distance, while the road stays flat
One camera that knows its own height reads the row where the car in front meets the road, and that row is the camera's height seen from the car: a baseline of 1.3 metres standing on the road, read in every picture. With the car's growth it ranges a car 20 metres ahead to 0.80 per cent a pixel, six times better than a stereo pair's disparity, and reads the car's width to 0.74. A camera height a few centimetres out scales the distance and leaves the time untouched. A road that starts rising by one degree between the camera and the car moves the distance by up to 24 per cent, and neither horizon puts it right. The lane's own edges bend where the road does and read the rise back.
The eye that movesThe 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.
Through water and glassA crooked stick hides in a quarter pixel, and its residual finds it only below a tenth
Eight sticks stand in still water, one of them bowed three millimetres and the rest one, and a refraction fit holds every stick to a straight line. Read exactly, the bad stick carries the largest share of the fit's residual in about seven sets of ten. Read to a quarter pixel, as a careful photograph is, it carries it in one set of five, barely more than chance: a millimetre of bow moves the index more than the reading error does and leaves its rays missing by a twelfth of a quarter pixel. The residual stays blind until the bad stick is bowed five or ten millimetres, and dropping the stick it ranks worst does nothing measurable for the index.
Systems that kept the measureA 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.
What survivesAn error with two terms
Two results from machineries with nothing in common have now found the same shape. A panorama's parallax separates into a term that halves every time the frame count doubles and a term with no frame count in it at all; a silhouette's error into an excess that falls as one over the square of the view count and the area of a concavity that is the same number at four views and at a hundred and twenty-eight. Fitting both terms turns the distinction into a measurement, and pointed at seven of this collection's own laws it reads every one of them the way its own essay does.
Light and mirrorsThe 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.
Constructing a viewStepped, 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.
Where to standA flight that has ends
A real staircase has three families of faces and the rule proposed last round predicted three pictures. It supports four, because a picture is fixed by a direction and the flight's two sides point opposite ways — one kind of face that no single direction reaches both of. The rule counts words and the object counts orientations.
The real instrumentTwo 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.
What each system gave upScatter 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 survivesWhat 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.
Light and mirrorsA 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.
Constructing a viewWhat 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.
What survivesThe ladder of assumptions is a ladder of conditioning
Push the four corners of a board by one pixel and read three quantities through the one recovered map. A cross-ratio does not move at all — it is read in the picture and never went through the map. A ratio of parallel lengths moves by a tenth of a per cent at twenty degrees of obliquity and by 1.6 per cent at seventy-eight. An angle moves by sixteen thousandths of a degree and by nine tenths. The stratification ladder is usually taught as a hierarchy of what is assumed; it is also a hierarchy of what a pixel costs.
Light and mirrorsThe 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.
The other systemsTwo distances to infinity
A parallel projection is the limit of a perspective one, and the limit arrives twice. Which faces get painted settles within three object radii, because a face is either round the back or it is not; where each mark lands falls like one over the distance and is still out at thirty-four. Far enough away has two answers an order of magnitude apart.
Light and mirrorsThe 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.
What survivesFive tangents name the same conic
Fitting a conic to five lines is the same six-coefficient nullspace problem as fitting one to five points, with the roles exchanged, and it lands on the same curve to 2.6 × 10⁻¹¹ pixels. What duality does not promise is that the two are equally good evidence — and measured at equal marking precision they are, to within seventeen per cent, because a drawn tangent is made of points.
What survivesThe quadrilateral that finds the middle
The harmonic conjugate is usually built from four points. Built instead from four lines — the dual construction, the same number of straightedge steps — it lands on the same fourth point to 1.2 × 10⁻¹⁵ of the range's own length, gives a cross-ratio of exactly −1, and refuses the midpoint, whose conjugate is at infinity.
Light and mirrorsHow many shadows determine the object
One lamp's shadow says only that a convex section lies inside a wedge 10.5 times its own area; four already cut that down to 1.17 times, and 128 close a convex section's boundary to 0.10 mm everywhere. The identical sweep on a section with a bite taken out of it stalls at 70.3 mm, 675 times worse, because an outline is the boundary of the smallest convex body with that shadow and no direction ever sees inside a concavity.
Light and mirrorsThe lamp and the floor cannot both be recovered
Every member of a one-parameter family of lamp-and-floor pairs draws the identical photograph to a fraction of a thousandth of a pixel — one member swings the lamp 90 cm and tilts the floor 4.3° and the picture does not move at all. A tape measure or a post of known height each pin the true member uniquely, and so does a wall known in advance to be square to the floor, whose angle to the recovered floor peaks at exactly 90° at μ = 1.000 and nowhere else; a receiver merely known to be parallel drifts thirteen times more weakly.
What survivesThe picture contains what is behind the camera
A pinhole maps a direction, and a line has one direction, so a point behind the eye lands on exactly the same mark as its reflection in front — here to 6.4e-14 pixels. The sign the division throws away is why cheirality is a fact supplied from outside the picture rather than measured in it.
Drawn confidentlyThe rule is exact for a floor that lengthens
The constant-ratio rule for spacing receding boards is an exact perspective — to the last digit, on the panel's own horizon — of a floor whose boards grow by the inverse of the ratio, 0.74 braccia deep at the front and 1.31 at the back on an eight-braccio pavement. The orthogonals agree with that floor. What says the tiles were meant to be square is a diagonal, which bends 8.1 pixels off straight where the reader's fitting test finds a sixth of one.