Visual Plane
every picture states the point it is correct from
Concept

Sampling — where it appears

Taking a finite set of readings of something continuous, and the question of what falls between them. The gap decides what a measurement cannot see, so a sampled reading is honest only when the departure that could hide between samples is quoted alongside it.

Named by 10 essays across 6 fields — each of them below, with the objects they name alongside it.

00.2500.5000.7501-10-50510across the edge, in pixels of the picturehow much of the background reaches the sensorwhere a pinhole puts the edgethe pupilone depth per samplethe same edge, two ways of blurring it70% apart

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.

pipeline · Zbuffer 21 of 36 tiles still countablethe floor starts 18 m away

Counting is a measurement

A tiled floor gives its area with no reference length at all — count the tiles and multiply. The count is an integer, so it is exact wherever it can be made, which is a completely different error law from the rectifier's smooth decay. And the distance at which it fails is set by the tile's depth edge, which foreshortens as one over the depth squared, so 18.7 m for a 62 cm tile, where the across edge alone would have allowed 217.

metrology · Area 2 transversalsnoneunfalsifiable3 transversalsnoneunfalsifiable4 transversals15 transversals26 transversals38 transversals510 transversals713 transversals1017 transversals14independent statements a pavement makesthree unknowns, one equation per mark

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.

construction · Attribution 8 lamps · 20 mm counts as read8.5% of 9.0 m²

A floor is read along curves

Whatever a shadow says about the floor it landed on, it says only where the shadow is — and a shadow is a curve while a floor is a surface. Shadow curve length grows exactly linearly in the number of lamps, by a fitted exponent of 0.999, and the fraction of floor within two centimetres of one grows more slowly at 0.94, because the curves begin to overlap. At thirty-two lamps, seventy-one per cent of a nine square metre patch has still never had a shadow on it.

light · Twolamps -4-3-2-1000.50011.502how much more effort, log₁₀ of the controlhow much of the error is left, log₁₀ of the first samplepanorama-seam · −0.92panorama-vertical · floorsilhouette-hull · −2.00silhouette-notch · flooranamorph-sagitta · −2.30anamorph-sagitta-intervals · −2.00mirrorball-size · −1.007 laws, 12 samples each2 with a floor · 5 without

An 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.

foundations · Buyable 00.2500.5000.7501-1-0.50000.500how far apart the lamps are, log₁₀ metreshow often the drawing is read as two lampshalf the drawings1 px of clicking, 12 drawings each, separated across the viewbelow the crossing the second lamp is not in the picture

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.

light · Twolamps 05101520123how many decades of the control were sweptlargest floor the data cannot rule out (% of the first sample)everything in here is still possiblea floor-free law at 1% noise21.5% at 0.3 decades · 0.00% at 3

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.

foundations · Buyable -1.50-1-0.50000.50011.5022.50how far back the eye stands, in object radii (log₁₀)how far a mark lands from where the direction puts it, in metres (log₁₀)one design samplethe faces settlethe marks settlea seating rake, from a point and from a direction3 radii against 34

Two 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.

parallel · Limit -3-2-1010123how many measurements are averaged, log₁₀the error in the recovered height, log₁₀ mmthe spread — -0.52the bias — flat, and microns1 px on the clicked topspread 0.177 mm at m = 1024, bias 20.0 µm

The bias out of reach

A single-view height recovered from the cross-ratio has an error with two terms. The spread of an average of m readings falls as one over the root of m, by a fitted exponent of −0.52 over three decades. The bias does not fall at all — and it is eight microns, against a single-shot spread of six millimetres, so the two cross at 535,000 measurements. There is a floor here and no experiment anybody will run can see it.

foundations · Crossratio under the lampa scatter, in planground conditioning 2.1e-1

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.

light · Pointcount

Named alongside it

The objects these essays reach for when they reach for this one.

instrument limitConditioningIdentifiabilityAsymptoticsDemonstrationPower lawResidualError termToleranceCross-ratioHorizonleast squares

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