Concept

instrument limit — where it appears

The distance or precision past which a picture no longer contains the distinction a measurement is asking about. It is a property of the arrangement rather than of the instrument, so the repair is usually to move the camera rather than to buy a better one.

Named by 66 essays across 17 fields — each of them below, with the objects they name alongside it.

123-2-10baseline (m, log scale)range past which 1 px of disparity error is unbounded (m, log scale)65 mm → 58.5 mrange = fB/δ — 58.5 m at a 65 mm baseline900 px focal length, 1 px reading

The range a pair cannot see past

A stereo rig has a distance beyond which it cannot say "no further than", and the distance is fixed before anything is built. It is the focal length times the baseline divided by the reading precision, and for a human pair of eyes it is fifty-eight and a half metres.

depth · Disparity
00.2500.5000.7501-10123distance from the eye — log₁₀ metresfraction of the buffer's codes used uphalf the codes by 0.20 ma linear map, for comparisonnear 0.1 m, far 1000 mharmonic mean 0.20 m against arithmetic 500 m

The precision a depth buffer has left

Depth is stored as an affine function of one over the distance, so half of a buffer's codes are spent before the harmonic mean of the near and far planes — twenty centimetres out of a kilometre. The resolution goes as the square of the distance, and the fix that works is not more bits.

pipeline · Zbuffer
010203040-2502550disparity on the screen — millimetreswhere the point is depicted — metres from the eyesd = 63 mm — the eyes' separationon the glassscreen at 2.0 m · eyes 63 mm apartthe ceiling is the head, and it does not move when the screen does

Two pictures on one screen

A stereoscopic display puts a point where two sightlines cross, so the depicted depth is b·D/(b−d) and the disparity that reaches infinity is exactly the separation of the reader's eyes — 63 mm, at any screen distance whatever. The depth budget is set by the width of a head and by nothing about the scene.

screen · Stereodisplay
a centretrue measurediminutionbounded depthstraight linesperspectivehandscrollisometricdimetrictrimetriccavaliercabinetelevationmilitaryfilled means the system keeps itmeasured from each system's own projectionno row is filled in both of the first two

Each system answers its own question

A comparison in which every system wins its own column proves nothing if the columns were chosen after the systems. The test that makes it a result is whether any system wins something it was not designed for — and two of them do.

choices · Exclusion
3 m · 47 px6 m · 24 px12 m · 12 pxnear ÷ far = 4.000 against a depth ratio of 4.000exposure 33.3 ms · 6 m/s across the frameeach streak is straight; the set of them is not one kernel

A frame is an interval

An exposure is not an instant, so a frame is an integral of projections and every moving point draws a streak. The streak is straight, because the image of a straight path is straight — and its length goes as one over the depth, so two objects at 3 m and 6 m blur by lengths in the ratio 2.000. No single kernel describes the frame.

sensor · Exposure
k₁ = -0.28 · the round trip closes to 8.0e-13 pxpale: what the eye receives · dark: what the renderer drawsthe inner 86% of a 72° frame, where the inverse is exact

The render is distorted on purpose

A headset renders a bent picture so its lens can straighten it, which is a lens's distortion polynomial run backwards, and the one case in which distortion is introduced deliberately. The round trip closes to a thousandth of a millionth of a pixel, and the price is that one rendered pixel becomes 0.493 delivered pixels at the edge of the field and one at the centre.

screen · Distortion
a room point at a known place115 cm at 2.40 m403 cm at 8.40 mthe ball scaled, the room left where it isoutline identical · reflection 8.90° apart

A mirror ball does not know its size

The outline of a mirror ball in a photograph gives the ratio of its radius to its distance and stops there — a ball three and a half times bigger, three and a half times further away, draws an outline identical to the last bit. What breaks the tie is a point of the room, and only a near one: the sensitivity falls as one over the room's distance, so a mirror ball photographed against a landscape has no recoverable size at all.

mirrors · Mirrorball
00.2000.4000.6000.800204060how far off the axis the ray leaves the pinhole (°)how far the dome bends it (°)10 cm dome20 cm domeboth at offset/radius = 0.060identical to 1e-16°

The dome knows its offset in units of itself

A dome port centred on the entrance pupil bends nothing at all, exactly. One that is not bends rays by an amount that depends on the decentring over the radius and on nothing else, so a ten-centimetre dome six millimetres off centre and a twenty-centimetre dome twelve millimetres off centre are the same instrument, bit for bit. The picture carries the ratio, which means it never carries the radius.

refraction · Port
481632641282560.3131030100how far away the scene is (m, log scale)pixels of parallax, and degrees of pose error (log scale)parallax, pxpose error, degreesparallax goes as distance to the -0.977 distances

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.

manyviews · Degenerate
a point at 1.5 mthe pupilthe sensorwhere it focusesf/2.8, focused at 3 m5.8 px across

The centre has an area

Every theorem of perspective follows from a projection through a point, and no instrument has one. Give the pupil a radius and each world point images as a disc — whose centre is the pinhole's mark to 5.7 × 10⁻¹⁴ millimetres, at every aperture, every distance and every field angle. The geometry survives exactly; only the sharpness is spent.

sensor · Aperture
a dished floor · truly 0.0600.0600a flat floor · truly 00a floor with a 60 mm step · truly 00.4820one lamp, one ring occluderthe step reports 0.482

The curvature a shadow reports

A flat floor makes a shadow a homology, so four marks predict the rest exactly and a curved floor mispredicts them by millimetres. Turn that round and the mispredict is a measurement of the floor. It recovers a dished floor's own curvature to a per cent — and returns 0.482 for a floor made of two planes, whose curvature is zero everywhere, with a residual of ten microns.

metrology · Shapefit
cusp — R/2 = 0.800 mvertexaperture 50 cm of a mirror of radius 1.6 mR from the cusp: 1.5996 m (0.023%)

The caustic is the mirror's own ruler

Four of the five instruments in this row return a ratio and refuse a size. The bright curve a mirror throws is the exception, and the reason is that it is a length lying on the table rather than an angle in a picture. Its tip is the paraxial focus, so a ruler laid from the mirror's vertex to it returns the radius of curvature — 1.5996 m against 1.6, and exactly right on a paraboloid, where there is no aberration to bias the envelope.

mirrors · Caustic
00.2000.4002468how far off the perpendicular the sightline is (°)how far the pane moves the point (mm)13.2 mm, n = 1.3510.0 mm, n = 1.528.0 mm, n = 1.756.5 mm, n = 2.1t(1 − 1/n) = 3.421 mm for all four1.8 µm apart over 8°

A pane gives a product before it gives two numbers

A flat pane of glass displaces every point it is seen through, and the displacement at small angles is the thickness times one minus the reciprocal of the index. So the two numbers arrive multiplied together. Four panes from 6.5 to 13.2 millimetres thick, with indices from 1.35 to 2.1, agree to under two microns over an eight-degree fan and separate by more than a millimetre over sixty — and a fit over the narrow fan returns whichever pair it started near.

refraction · Slab
0306090-3-2-1out-of-plane spread, as a fraction of the scene's own extent (powers of ten)error in the recovered translation direction, in degreeswith 0.3 px of reading errorexact marksthe geometry is a step and the measurement is a slope9 reliefs

How flat is flat enough

With exact marks the transition has no width at all — 84° of pose error at exactly coplanar and 0.000° at eight parts in ten thousand of relief. Put three tenths of a pixel of reading error in and the same sweep becomes a slope three decades wide, crossing into usefulness when the out-of-plane parallax reaches about ten times the marking error.

twoviews · Planar
00.50011.50-0.25000.2500.5000.750the blur a reader is prepared to call sharp, in pixels (powers of ten)near and far limits of the band, in metres (powers of ten)the focus distance2 pxone lens, one focus setting, five criteria5 bands

The sharp band is a decision

One 50 mm lens at f/2.8 focused at three metres has a sharp band half a metre deep or an unbounded one, and nothing about the optics changes between them — only how large a blur disc a reader is prepared to ignore. Every quantity usually quoted about depth of field is that acceptance restated, including the rule that a third of the band lies in front, which is true at one distance and nowhere else.

sensor · Aperture
the entrance pupilthe stopthe glassthe chief rays, from four object distances3.6e-15 mm apart

The hole a scene actually sees

The stop is not the centre of projection. Model a 50 mm lens with its stop 18 mm behind the glass and the chief rays from every object distance cross the axis at one point 28.1 mm on the other side of the lens — 10.1 mm from the stop and 1.56 times its size — to 3.6 × 10⁻¹⁵ mm. That point is the entrance pupil, and it is where a picture is a projection from.

lens · Pupil
0501000.2000.4000.600half the mirror's aperture (m)how wrong the fitted radius is (%)how wrong the answer ishow wrong the fit says it ismeasurement floor, 0.02°a paraboloid fitted to a sphere of radius 1.6 mhidden below 0.3 m of aperture · 0.030% of bias there

A 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.

mirrors · Paraboloid
-0.10000.1002.5057.5010depth reported from the stepped disparity (m)height reported for each image row (m)the floor as it is35 platesspacing 3.9 cm → 1.95 mslope 1.997

Whole 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.

depth · Disparity
00.50011.50020406080field angle off the axis (degrees)picture radius, in focal lengthsfolds at 47.49°43.91°50.54°the radial factor reaches zeropinholefolds at 47.49°43.91° and 50.54° share one radius

A 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.

lens · Distortion
true ellipse — axis ratio 0.5774four arcs — 5.72% short along the major axisthe four tangent points are exactworst departure 5.72% of the semi-major axis

The ellipse the drawing office draws

Every isometric ellipse template is cut to a construction of four circular arcs, and every account of it calls the result an ellipse. It is not one and cannot be: a circular arc has constant curvature and a conic's varies, so the two can agree at four points and nowhere between. The four-centre curve reaches 2√2/3 of the true semi-major axis — 5.72% short — and its minor axis is 3.53% too long.

parallel · Isocircle
00.2000.40005101520field angle, in degrees from the axislight lost, in stops0°10°17°23°one foreshortening, one tilt, two of distance0.48 stops at the corner

The corner sees an ellipse

A circular pupil viewed from off the axis is foreshortened by the cosine, so the blur patch a corner receives is an ellipse of axis ratio 0.920 at the edge of a full-frame picture with a 50 mm lens — and the light through it falls as the fourth power of the same cosine, 0.480 stops. Both are geometry, both happen to a perfect lens, and no design removes either.

sensor · Aperture
slit leaning 10° forwardslit leaning 10° back4 m · 37 px9 m · 83 px15 m · 138 px23 m · 211 px35 m · 321 px52 m · 477 px9.169 px of disparity per metredepth to 7e-15 m

A 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.

scroll · Pushbroom
hull 3.1098 · object 2.9193 · floor 3.07498 outlinesthe notch survives all of them

The 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.

metrology · Visualhull
the seat3 chords against the arccurved monitor, 3 flat pieces6.80 mm of sag · 5.50 px at the seat

A 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.

pipeline · Tiling
00.50011.50210203040how many marks the fit was givenerror in the recovered translation direction, in degrees (powers of ten)flatwith depththe same reading error in both5 counts

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.

twoviews · Critical
at one depth — a stadium, to 1.8e-5receding — tapered, 4% outone point, one exposure, the whole pupil1.8e-5 against 4%

The 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.

sensor · Aperture
248163264128256100100010000depth of the point from the track (m, log scale)separation of its two drawings (px, log scale)straight trackoutside a 500 m bendoutside a 200 m bendoutside a 100 m bendinside a 200 m bendstraight: 9.169 px per metreslits ±10° · 26 px per metre of roll

A 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.

scroll · Pushbroom
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
paraxial pupilthe stopa curved front elementcrossings 15.0 · 14.5 · 13.5 · 11.6 mm

The 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.

lens · Pupil
a centretrue measurediminutionbounded depthstraight linesperspectivehandscrollisometricdimetrictrimetriccavaliercabinetelevationmilitarycrossed-slitsfilled means the system keeps it10 rowsthe last is filled in neither of the first two

The 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.

choices · Exclusion
020040060018202224along-roll scale measured beside the point (px of paper per metre of ground)separation of the point's own two drawings (px)5 m out10 m out20 m out40 m out80 m out120 m out6 points, one bendradius read back to 0e+0 m

A scroll can be asked its own radius

The two marks a bend leaves separate exactly. The along-roll scale alone fixes the angle in the disparity, so one point and a neighbour at its depth give back the radius and the depth in closed form — 200 m and 40 m returned to a part in 10⁹, with no search. The two answers are not equally held: a scale read one per cent too large under-reads the depth by one per cent and over-reads the radius by tan(φ − α)/α, which is 50 for a point ten metres from a five-hundred-metre bend. And a painter who evens the scale out by eye reports a gentler bend, never a bend that was never there.

scroll · Pushbroom
025507510010203040how wrong the assumed curvature is, as a % of the true oneworst error in the recovered design (mm)0.91% buys 1 mma a ridged floor, k = 0.0644.0 mm at 41% out

How well the floor has to be known

“Supply the surface” is an instruction with no number in it, and an instruction with no number is a wish. Here is the number: the error in the recovered design is very nearly proportional to the error in the assumed curvature — the constant varies by 1.4% across a twentyfold range — so about nine parts in a thousand of the curvature buys one millimetre on a design 1800 mm wide, and 40% out costs 43.5 mm.

curved · Curvedrectify
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
00.50011.5020123where a pinhole would put the point, in focal lengths from the centrewhere the model puts itwhere the polynomial foldsthe division model's horizonboth at k = -0.42fold 42° · horizon 1.54

A model that inverts has a horizon instead of a fold

The polynomial every calibration fits turns around at a finite radius and stops being a map from direction to picture. The division model, chosen because it inverts in closed form, never turns around — it rises for ever toward a horizon at one over the root of its own coefficient, so the whole hemisphere of directions lands inside a finite disc. Fitted to the four fisheye laws it follows every one of them more closely than the polynomial at every field from forty degrees to eighty — a hundred times more closely for the equidistant law at forty, and the stereographic law exactly.

lens · Distortion
78.510.51421421051050∞78.510.51421421051050∞where the horizontal slit sits (m)where the vertical slit sits (m)8 with a centre, 17 with a measurenone with both

The exclusion is two conditions, not ten rows

Ten systems have been measured and none has both a centre and a true measure. Ten is not a proof. Swept across the whole two-slit family — eighty-one members, both parameters run out to infinity — a centre appears at exactly the eight members whose slits sit at one finite distance, a measure at exactly the seventeen with a slit at infinity, and at none of them both. The case the two conditions appear to leave open is closed by the rays themselves: one divide is a slit, and eight page points use 6.02 m of it.

choices · Exclusion
05010015020011.502log₁₀ of the lamp's distance (m)pixels the recovered foot sits below the horizonthe sun: on the horizonthe recovered height stays exactwhat falls off is the evidence that the light is finite

A light far enough away

The evidence in a photograph that its light is in the room rather than at infinity is one number — how far below the horizon the shadow lines put the light's foot. It falls off as one over the distance, from 211 px at 4 m to 10.8 px at 266 m, while the recovered height stays exact to 5e-13 of itself. What fails first is not the arithmetic; it is the evidence, and one pixel of error costs 0.21 mm of height at the near end and 0.07 m at the far one.

light · Lightrecovery
where the eyes are aimedin plan, framed to the shells drawnzero at 0.92 m, 40° aside

Vergence moves the shells and does not respace them

Turn two eyes inward and the depths a whole-pixel reading can report stop being planes and become a family of near-circles through both eyes — the twenty-pixel shell standing at 0.74 m forty degrees aside where a parallel pair puts it at 3.82. The spacing between consecutive shells is the same to 0.07 per cent across the whole field, so vergence relabels the rays and does not sharpen them, and the resolution argument for turning the eyes in does not exist.

depth · Disparity
the object0 of them no light reaches7 of 7 seen

Two mirrors show fewer images than they make

Two mirrors at 55° generate seventy-one images of a point and an eye between them can reach six. The count the field teaches — three hundred and sixty over the angle, less one — is out by as much as sixty-six against the orbit and never by a whole image against what a viewer standing on the bisector actually sees. It is a correct rule about the eye, quoted as a rule about the mirrors.

mirrors · Twomirrors
left pictureright picturethe post it is matched toone image row, drawn as two strips19.8 m instead of 9.0

A third eye that lands on the next post

Match one post of a railing to its neighbour and the pair reports it at 19.8 metres instead of 9.0, with every test two photographs can run at the arithmetic floor. A third picture usually exposes that by hundreds of pixels — but at five azimuths in seventy-eight degrees the wrong point lands within three pixels of another post, and the third view confirms the mistake. Narrow the railing to twenty centimetres and those places cover 28 per cent of the arc.

depth · Outlier
0.313100510how far away the subject is, in metreshow far the pupil moves forward, in millimetres0.50 mthe whole walk with field anglea 50 mm lens, focused as one pieceequal at 0.50 m

Focusing moves the pivot past its best place

Focusing a fifty-millimetre lens to one metre carries its entrance pupil 2.63 millimetres forward of the camera body, and to half a metre 5.56 — which is more than the whole 5.53 that the pupil walks with field angle, so past a subject at 502 millimetres the focus decides where the pupil is. A panorama head aligned at infinity and used at a metre leaves 7.46 arcminutes along its seam; aligned at four metres it leaves 2.21, better than pivoting at the pupil at all.

lens · Pupil
share of the 70° field counted as cloud — the truth is 10.31%equal-area fisheye10.31% (-0.0%)equidistant fisheye9.37% (-9.1%)stereographic7.56% (-26.6%)flat plane2.40% (-76.8%)401² of picture, four caps of cloud0.01% on the equal-area rule, -77% on the flat plane

Counting cloud by counting pixels

A sky camera looks up and something counts the white pixels. On an equal-area fisheye that answer is right to 0.01%, which is the grid's own error. On an equidistant one it is 9% low, on stereographic 27% low, and on an ordinary flat lens 77% low — against a cover that is known exactly, because the clouds here are caps whose solid angles add. Weighting each pixel by the surface's area scale repairs every one of them to better than a fifth of a per cent.

curved · Equalarea
0.2000.4000.6001.8022.20range, mheight above the plane of the eyes, mhorizontalverticalat 2 m, 40 cm off the plane79° between them

The second disparity cuts cells

A point off the plane of the eyes has a vertical disparity as well as a horizontal one, and quantising both, on an 86,400-point lattice of a room, gives 7,663 labels where one coordinate gives 179 — a count that belongs to the lattice rather than the room, as the essay after this one found. The gain is entirely vergence's — two eyes looking straight ahead have no vertical disparity at all, exactly — and it is largest where the first reading is already finest: 60.8 in the near metre and 3.7 in the far band.

depth · Disparity
0102030401234the marking error, in pixelsthe error of the answer, in per centupper curve: the closure wrong · lower: the closure exactfloor 6.0%

The floor a better camera cannot reach

Sweep the marking error from four pixels down to a hundredth and the measurement's error falls thirty-fold and then stops — at 6.0 per cent, which is exactly the six per cent the reference's assumed shape was wrong by. With the closure exact the same sweep keeps falling to 0.06 per cent. The crossing is at half a pixel, and it can be computed before the photograph is taken, which makes it a decision about equipment rather than a discovery about it.

metrology · Uncertainty
010203051015how far each camera is turned in, degreesper centwhat the vertical disparity addslabels merged by rectifyingof those, from the warpfocal length to merge noneeach pair rectified at its own 900 px4.0% for eyes at 1.2 m

Rectifying a pair spends what its epipolar lines lean

Counted from the two disparities alone, rectifying a verged pair looks as if it throws away at least 99.7 per cent of what the pair can tell apart. That cannot be true of a warp that loses no ray, and it is not: once the place in the picture is counted, the vertical disparity adds 4.1 per cent for eyes verged at 1.2 m, and rectifying at the same focal length gives back all but 4.0 of it.

depth · Disparity
0204060204060frames in the panoramadistance from the pivot (mm)π/β = 9.5across the seamup the framepivot 60 mm · frame 38° tallfloor 19.53 mm

The parallax you cannot shoot away

A stitched panorama's parallax has two halves and they do not behave alike. The one across the seam falls as the sine of half a frame spacing, so more frames buy it off; the one up the frame is the sine of half the frame's own height, and no quantity of shooting touches it. They cross at π over β, which has no pivot error in it at all.

curved · Pivot
25102050131030how far away the object really is, in metreshow far away it appears, in metresa flat mirrortwo metres of radiusboth axes logarithmic · the eye 0.8 m from the glassceiling 1.30 m

Closer than they appear, by a factor with a number in it

A wing mirror of one metre radius held eighty centimetres from the eye reports 1.30 metres for an object at sixty-four, because the image of anything distant sits half a radius behind the glass. The size such an object subtends reads as a distance 2.60 times the true one, and the factor is exactly one plus twice the eye's distance over the radius — so the warning is a number, and it is larger for the mirror that is further away.

mirrors · Curvedmirror
9 m out82.6 px apart along7.47 px apart in rows18 m out165.2 px apart along7.47 px apart in rows36 m out330.5 px apart along7.47 px apart in rows72 m out661.0 px apart along7.47 px apart in rowsrows parted by 7.47 px at every post · slits ±10° · drawn at 1.5×no single viewpoint — the rays miss by the height climbed between two momentsa 5% climb, eye upright

A scroll of a climbing road measures its grade

Every reading of the two-slit scroll has leaned on its two drawings of a point sharing a row, because the eye is at one height at both moments. On a road that climbs they do not — and what parts them is the height climbed between the two moments over the reach, which on a straight climb is 2·f·g·sin φ for every point at every depth and height. The scroll does not lose its rows to a hill. It gains a third mark, a gradient meter that a level bend cannot counterfeit.

scroll · Climb
the pupil, and the pivotsubject at 34°the filmcorrect from 15 cm, at 160 mm wideagrees with the cylinder to 1.1e-16

The camera that is a cylinder

A swing-lens camera turns its lens about its own entrance pupil and sweeps a slit across film bent into a circle concentric with it. Compute where the light lands, unroll the film, undo the pinhole's inversion, and the result is not similar to the cylindrical picture surface — it is the same map, to the arithmetic floor. What it pays instead is detail, and a shear on anything that moves.

curved · Pivot
9 m outleans 1.58°0.00 px apart in rows18 m outleans 3.16°0.00 px apart in rows36 m outleans 6.31°0.00 px apart in rows72 m outleans 12.46°0.00 px apart in rowsrows parted by 0 px · posts lean with depth · slits ±10° · drawn at 1.5×no single viewpoint — the rays miss by the height climbed between two momentsa 5% climb, eye pitched with it

An eye that pitches with the road keeps its rows

An upright eye climbing a road parts each point's two drawings by the same few rows, and that offset reads the grade. Fix the eye to the vehicle instead, so it pitches with the road, and the offset vanishes exactly — for every point, at every depth and height. The grade has not gone. It has moved into the posts, which now lean by an amount that grows with their depth, and into one drawing, which can now read the grade on its own.

scroll · Climb
second eye1% error5% error20% errordisc area ∝ error10 m past30 m pastprobes 1 m up · marks read to 1 px · 60 trialscameras side by side

A plane's coefficient reaches as far as its parallax

After a known plane's map, every raised point's displacement is its height over its depth, read as a coefficient on the epipole — exactly, for any point either picture sees. The worry was that the number would be local, good only near the floor whose marks fixed the map. Read to a pixel, it is not a distance on the floor that runs out. It is a length in the picture: the point's error is about 260 per cent over its parallax in pixels, wherever the point stands.

twoviews · Planar
first viewersecond viewer6 paths, each used both ways6 seen each way

Two people between mirrors see each other equally often

Put two people between a pair of mirrors and each sees some number of images of the other. The two numbers are always equal, and so is the apparent distance of each image against its partner — a path of light walked backwards is the same path. What is not shared is where each image appears, and what the shared count depends on turns out to be neither person's position but two quantities made of both: the difference of their angles about the mirrors' meeting line, and the sum.

mirrors · Twomirrors
024020406080how deep the point is, cmthird ray's miss of the pair's point (px)one pixelraised 25 mmraised 50 mmraised 100 mmraised 200 mma one-metre stick, 30° into waterzero at the surface, every offset

A third eye sees the water the pair cannot

A level stereo pair looking at a stick in water gets two rays that meet exactly and a stick 285 mm too short, with nothing to warn it. A third eye raised ten centimetres above the pair sees a disagreement of 2.6 pixels at the stick's tip. Asked to agree with the other two as if there were no water, it hides nearly all of that and makes the stick shorter still. Told only that there is a flat surface, the same three eyes find its refractive index and put the metre back.

refraction · Refraction
3 m3.6°6 m2.4°12 m1.8°24 m1.6°far1.5°no single viewpoint — the rays miss by the camera's travel during the readoutturning 0.4 rad/s, travelling 2 m/s

A frame's shear knows travel only over depth

Read a frame row by row while the camera turns and travels, and every vertical post leans — the near ones more. The lean is the turn plus the travel over the post's depth, and that sum is all the frame holds: twice the travel past posts twice as far draws the same frame to eighteen decimal places. Two posts cannot separate turn from travel. A facade can, because a turn leans the edges of the frame more than its middle, but the two signals are 99.8 per cent alike, and reading them apart takes a pixel on every row.

sensor · Rolling
0510the six seams of the room, each at the nearer of its two frames' subjectsworst disagreement along the seam (arcminutes)0.5 m0.7 m1.5 m4.0 m1.2 m0.5 mfixed head, best pivothead that tracks the focussix frames, each focused on its own subjectworst 12.9′ against 2.82′

A refocused panorama is six lenses

Refocus between the frames of a room panorama and each frame is its own camera: its pupil carried forward by its own extension, its picture made at its own principal distance. The best fixed pivot then leaves 12.9 arcminutes along the worst seam, and a head that slides the camera back as the lens extends leaves 2.82 — the lens's walk alone. The pivot's share is first order and worth a tracking head indoors. But a stitcher that reads all six frames with one focal length misregisters by up to 270 arcminutes, which is the larger mistake by ninety times.

lens · Pupil
2481632641280.5125102050100f-number (log scale)worst blur between 0.5 m and 4 m (arcminutes, log scale)refocused, fixed head: 12.9′ seamrefocused, tracking head: 2.82′defocus at the endsdiffractionboth togetherone focus at 0.89 m, 550 nm lightleast at f/40

One focus stopped down is as sharp as six in a deep room

Refocusing a room panorama frame by frame costs a seam of 12.9 arcminutes on a fixed head. One focus stopped down costs none of that, and at f/40 holds half a metre to four metres to 5.27 arcminutes — sharper than the fixed head's seam, blunter than a tracking head's 2.82. But the comparison flatters refocusing: the frame on the table also holds the wall behind it, so it cannot be held sharper than 5.27 however it is focused. Counted properly, one focus on a plain head comes within 2.4 per cent of the geared one, and pays for it in light.

lens · Pupil
-5-2.5002.505-10-50510where the post stands along the road, from the change of grade (m)rows between the post's two drawingsriding the pointwheelbase 2.7 mwheelbase 4.5 mposts 12 m out, level to 6% at 0two chords, one difference

A vehicle's pitch lags the road by its wheelbase

A camera fixed to a vehicle does not pitch with the road under it; it pitches with the chord between its wheels. Over a step from level to six per cent, a two-slit scroll's rows — silent on any steady grade — depart by up to 2.18 rows over 6.5 metres of road for a 2.7-metre wheelbase, against 4.26 over 4.0 for a camera that pitched at the point. The excursion's width is the eye's own chord plus about the wheelbase, and one line of posts reads the wheelbase back to ±8 centimetres. A vertical curve does not silence the rows either; it shrinks them as one over its length.

scroll · Climb
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
0204060204060how oblique the view of the plane is (°)what one pixel costs, against its cost at 20°a ratio of lengths — affinean angle — metric1 px on four corners, one homographya cross-ratio moves by 0e+0

The 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.

foundations · Stratification
-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
00.2000.4000.6000.50011.502how many points were measured along each line, log₁₀how wrong the recovered focal length is (%)a pinhole — no floora lens, k₁ = -0.05the floor, 0.72%two vanishing points, three lines eachpinhole floor 3e-16 · lens floor 0.72%

A floor with a referent

Recover a focal length from two vanishing points and measure more points along each line. Through a pinhole the error falls from 0.34 per cent to 0.05 and the instrument finds no floor at all. Through a lens of k₁ = −0.05 it falls, turns, and rises to 0.70 per cent — because the noise the extra points removed had been partly masking the lens's bend. The floor is 0.72 per cent of the focal length, and doubling the distortion coefficient doubles it to 1.44. It is not noise and not conditioning; it is the model, priced.

foundations · Recovery
horizon1234the fifth pointcorrect from 23 cm, at 160 mm widecross-ratio 1.627695 at position 1 of 22

Four points on a conic look the same from anywhere on it

Four marks on a photographed circle subtend the same cross-ratio at every point of the curve — 1.627695, unmoved over twenty-two positions of the fifth point, to 1.1e-13 degrees of projective spread. A fifth point 6.1 pixels off the conic reads anything from 1.00 to 2.52, so the invariant belongs to the curve rather than to the four marks.

foundations · Chasles
a plan, looking down: the object on the left, the frame, the hook on the rightthe hookthe frame, edge-on160 cmplan at 0.25 px per mm · a hook 160 cm behind a 56 cm framecorrect from 46 cm at 160 mm wide

The hook is the centre, and the eye is not

Dürer's string frame projects from a ring of iron driven into a wall, so its centre of projection has a position anybody can measure with a tape. A hook 120 centimetres behind a frame 56 centimetres wide makes a drawing correct from 34.3 centimetres shown 160 millimetres across, with its principal point 161 px off the middle of the sheet. The gridded veil projects from a head instead, and 30 millimetres of head moves its marks by 13.27 millimetres.

construction · Instrumentdrawing
the photographv1v2v3orthocentre = principal pointcorrect from 4 cm, at 30 mm wideself-polar to 3.1e-13 px · orthocentre 1.4e-13 px

The triangle a camera cannot move

Three mutually perpendicular directions give three vanishing points, and that triangle is self-polar with respect to the image of the absolute conic — to 3.1e-13 px, with no length and no angle anywhere in the statement. Turn one direction two degrees out of square and the polars miss their sides by 65.8 px. The statistic this collection has been printing as evidence for the same claim, meanwhile, is an identity that cannot fail.

foundations · Poleandpolar

Named alongside it

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

ConditioningDemonstrationcentre of projectionEntrance pupilFocal lengthResidualDepth uncertaintyDisparityParallaxPushbroomReconstructionBaseline

All concepts