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The thread: What a projection destroys

Length goes, angle goes, area goes, and the ratio in which a point divides a segment goes. Knowing precisely what is lost is what makes the one surviving quantity worth anything, so the losses are measured alongside it rather than mentioned.
plane — bend 0.0%cylinder — bend 6.0%stereographic — bend 3.5%equidistant — bend 4.5%equal-area — bend 5.1%equirect. — bend 5.7%120° across in every panelsame scene, same angle, six surfaces Surfaces that are not flat

No picture surface keeps everything

A picture has to be cast onto something, and every candidate surface destroys something different. Six of them are measured here on the same three properties, and the corner of the plot where a surface pays nothing is empty — not because nobody has thought of one, but because a theorem says there is none.

 elevationx 1.000y 1.000z 0.000cavalierx 1.000y 1.000z 1.000cabinetx 1.000y 1.000z 0.500isometricx 0.816y 0.816z 0.816dimetricx 0.943y 0.943z 0.471axis scales measured from the drawingall 5 preserve midpoints The other systems

Parallel projection is not primitive perspective

Isometric and oblique drawing are not what people used before they worked perspective out. They are a different answer to a different question, and the difference is one measurable quantity — a parallel projection preserves the ratio in which a point divides a segment, and a perspective projection destroys it by 7% of the segment's drawn length at a comfortable depth, rising to 13% over the range the slider covers.

 horizonABCDon the groundin the picturelength AB1.00011.3930ratio AB:CD0.56670.6837cross-ratio1.31681.3168correct from 26 cm, at 160 mm wide34° across What survives

What a projection destroys

A projection loses length, angle, area and the ratio in which a point divides a segment. Exactly one quantity comes through untouched, and almost everything that can be checked about a picture is checked with it — including, as it turns out, some things it cannot check at all.

 fitted conic: 1.467 from being a circlecylinderthe samples, fitted Surfaces that are not flat

The cylinder, and the price of going all the way round

A cylindrical picture can hold three hundred and sixty degrees, keeps every vertical vertical, and bows every horizontal. Its cost is a stretch of sec φ in elevation, which is also the equirectangular surface's cost exactly — two surfaces that are always described as different and are identical in the one respect anybody notices.

 elevation0.0000two equal, orthographiccabinet0.5000two equal, obliquecavalier1.0000all three equal, obliquedimetric0.4714all three different, orthographicisometric0.8165all three equal, orthographic ←trimetric0.5479all three different, orthographicsmallest of the three axis scalesmeasured from each projection The other systems

What isometric actually means

The three axis scales are equal to each other. They are not equal to one. Every unit along every axis is drawn at 0.8165 of its true length, which is √(2/3), and a great deal of confusion about isometric drawing comes from the word promising something it does not deliver.

 horizonequal stepshalve thetaper byworst error, in metres of depthequal steps to the horizon11.72 mhalve the remaining gap225.13 mtaper by eye347.80 mcorrect from 26 cm, at 160 mm wide34° across Drawn confidently

Dividing depth by eye

Three methods for spacing a receding row, all of them taught, all of them wrong. Read back as distances, the best misplaces a post by 3.5 metres in a row that is supposed to be spaced 1.4. And the obvious way of checking them gives one of the three a perfect score.

 00.50011.502-0.400-0.20000.200k₁departure of the cross-ratio from the world's value (%)pinhole1.29%the pinhole's own error, on the same four points2e-16 — the control The real instrument

A lens destroys the invariant

The cross-ratio is the one thing a projection preserves, and nearly everything checkable about a photograph is checked with it. A lens returns it one and a third per cent out where the pinhole is exact to fifteen digits — and the height error that follows tracks a quantity nobody would guess.

 the water surfacethe pinholethe rays miss by 9.9 mmno single viewpoint — the rays miss by 9.9 mmdry control: 0e+0 m Through water and glass

A picture through water has no viewpoint

Continue the rays of a refracted picture into the water and fit them to a common point. They miss it by ten millimetres. The same fit with the water taken away misses by zero, which is what makes ten millimetres a measurement rather than a number.

 1234512345a point is chosen hereand must be on this linematch to its own epipolar line: 1.4e-13 px5 of 44 correspondences drawn The second eye

A point is a line over there

Knowing where a mark sits in one photograph does not say where it sits in the other. It says which line it is on, which removes one of the two unknowns and leaves one — and the five lines drawn here meet at a point to within a thirtieth of a billionth of a pixel.

 × 4.6e+6seven flatand the rest stiffsingular value ÷ the largest, log scale, smallest firstσ₈/σ₇ = 4.6e+6168 parameters · 528 residuals Many pictures at once

Seven numbers no picture can name

Shift a whole reconstruction by a metre and a half, turn it half a radian, scale it by 2.7, and every photograph of it stays where it was to a hundredth of a billionth of a pixel. Move one point by fifty millimetres and they move by two thirds of a pixel.

 a receding straight line, and the chord it is not9.59 px of sagthe control — the same line at constant depth0e+0 pxno single viewpoint — the rays miss by 6.1 ma straight line's image is a hyperbola The eye that moves

A straight line in a scroll is a hyperbola

Under a pushbroom the image of a straight world line is a Möbius function of the paper coordinate, which is a rectangular hyperbola. It is straight exactly when the line holds its depth — so a curve in a handscroll is a depth signal rather than a stylistic one, and the sag is computable in pixels.

 midpoint — 2.77 mm gapfrom the left eyefrom the right eyegap 2.77 mm at 7.22 mexact marks: 2.2e-16 m What a pair is for

Two rays that do not meet

Triangulation is described everywhere as the intersection of two rays, and two rays in space do not intersect. Read the same two marks to a whole pixel and they miss by 2.77 mm at seven metres, which is a real length and is the part a residual will not report.

 0100200204060distance of the object, metresits drawn height, pxa pinholean oblique systemdrawn height of a fixed object against its distance20.0× against 1.000 Systems that kept the measure

A picture with no size–distance signal

In a system with no diminution the drawn size of an object falls at exactly zero pixels per metre, so nothing in the picture says how far away anything is. Depth has to be carried by something else, and what carries it is height on the page — linearly, and without a horizon.

 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 What a machine computes

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.

 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 What each system gave up

A centre and a measure are exclusive

Eight drawing systems, measured on five questions, with the pinhole as a row rather than the header. Exactly one has a centre of projection and it is exactly the one with no true measure — and loosening the measure test by a hair lets it in, which is what says the boundary is real.

 02468020406080angle off the optical axis (degrees)area scale, relative to the centreplanecylinderstereographicequidistantequal-areaequirect.measured by differencing on the spherethe plane leaves the frame Surfaces that are not flat

Every fisheye is a different rule

The word "fisheye" names a shape of lens and not a projection. There are several, they disagree with each other by tens of per cent at the frame edge, and each is the right answer to a different question — one is a protractor, one is a counting instrument, one preserves shape. Which one a lens implements is a fact about that lens, and it is rarely printed on the barrel.

 centre of the ellipseimage of the centrecorrect from 22 cm, at 160 mm wideoffset 21.1 px What survives

The circle whose centre moves

The image of a circle is an ellipse, and the image of the circle's centre is not the centre of that ellipse. The gap is 3 or 4% of the ellipse's width in an ordinary view — large enough to see, small enough to be dismissed as a slip by anyone not expecting it, and the reason drawn wheels look wrong.

 a room 2.8 m across, eye 1.6 m up1 mthe same plan, 137× bigger137 midenticalpicturesthe picture — both scenes, drawn twice, one on top of the otherlargest disagreement 1e-13 px over 8 verticesone length has to come from outside the picture Measuring from one picture

The one thing a single view cannot give

Make the world a hundred and thirty-seven times larger and move the eye a hundred and thirty-seven times further away, and the picture does not change by a measurable amount. Every ratio in a scene is recoverable from one photograph and no size is, and that is not a caveat about the method — it is the shape of the method.

 2 — every orthographic projectionisometric · axonometric 0.816 · 0.816 · 0.8162.0000dimetric · axonometric 0.943 · 0.943 · 0.4712.0000trimetric · axonometric 0.876 · 0.966 · 0.5482.0000cavalier · oblique 1.000 · 1.000 · 1.0003.0000cabinet · oblique 1.000 · 1.000 · 0.5002.2500elevation · orthographic 1.000 · 1.000 · 0.0002.0000military · planometric 1.000 · 1.000 · 1.0003.0000isometric's three equal scales are forced to √(2/3) = 0.816497not chosen — the identity leaves no other value The other systems

Which axis scales are possible

An orthographic projection's three foreshortening ratios always satisfy one identity — their squares sum to two. Isometric's famous 0.8165 is forced by it rather than chosen, and cavalier's 1, 1, 1 sums to three, which is the arithmetic saying cavalier is not the ORTHOGRAPHIC projection of anything. A later rung shows what the departure is a measurement of.

 elevationx 1.000y 1.000z 0.000cavalierx 1.000y 1.000z 1.000cabinetx 1.000y 1.000z 0.500isometricx 0.816y 0.816z 0.816dimetricx 0.943y 0.943z 0.471axis scales measured from the drawingall 5 preserve midpoints The eye that moves

A map along, and a picture across

The midpoint of a segment lying along a scroll's length images to the midpoint of its image, exactly. The midpoint of a receding one lands 22% of the way off. One projection, two answers — and the direction that keeps measure keeps something a perspective picture never offers.

 no picture at 300°the plane is unbounded at 180°plane — bend 0.0%cylinder — bend 6.0%stereographic — bend 3.5%equidistant — bend 4.5%equal-area — bend 5.1%equirect. — bend 5.7%300° across in every panelsame scene, same angle, six surfaces Surfaces that are not flat

What a 360-degree photograph actually is

The format every spherical camera writes preserves nothing — not straightness, not shape, not area — and it is the right choice anyway, for a reason that has nothing to do with looking at it. An equirectangular file is a lookup table of directions, and the picture only exists at the moment something re-projects a piece of it.

 head0.668profile — the outline that names a faceeye0.669frontal — an eye in profile is a wedgeshoulders0.669frontal — the width that says two armslegs0.668profile — a stride is a side viewpond0.326plan — a rectangle of water is a rectangle1.000 — what its own aspect keepsshare of each part that reaches the picture, from the best single direction|d · n| for each part3.000 of 5, swept and in closed form Systems that kept the measure

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.

 the ratio along a receding line15.6%0% parallelthe depth range in the last tenth69.0%10% lineara square metre, near against far53.5% larger0% parallelthe floor of the far room6.2 points0 parallelwhat it costs, and what the other systems have insteadmeasured by the same computationthe price of a station point What each system gave up

What perspective gave up

The field ends by turning its own battery on the system it has been comparing everything against. Four quantities a pinhole destroys that the other systems keep, each measured by the same computation as the systems it is set against, and each the price of the one thing perspective has and they do not.

 -0.400-0.200000.2000.4000.6000.8001position across the drawn surfacehow far along the real surface, minus how far along the drawn one(√k−1)/(√k+1) = 0.5195at the page's midpoint, 40.9%depth ratio 10 : 1peak 0.5195 at s = 0.760 What a machine computes

A texture does not interpolate on the page

Walking across a drawn surface at a constant rate walks across the real one at a rate that changes, and the worst gap is a closed form in the depth ratio alone — 0.52 at ten to one, more than half the whole range. It is exactly the error a person makes dividing depth by eye, made by a machine, and the fix is the fourth coordinate the pipeline kept.

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