Every essay — page 4
What survives Constructing a view Where to stand Surfaces that are not flat The other systems Light and mirrors Mirrors that are not cameras Measuring from one picture Through water and glass The real instrument The second eye What a pair is for Many pictures at once The eye that moves Systems that kept the measure What each system gave up What a machine computes The rectangle behind the lens The second projection Drawn confidently
Systems that kept the measure
A removed roof, a ground plane tilted into the picture, a figure assembled from the aspect that identifies each part. Every one of them gives up the single station point and buys something exact with it — a floor a reader can measure with one ruler, a depth scale that does not bunch, a picture in which nothing is edge-on. What they have instead of one centre is two, or several, and a picture drawn from two eyes turns out to be a picture drawn from one of a room that has been sheared.
What survives being copied
A workshop copying a drawing from a drawing is a random walk — the spread across lineages grows as the square root of the generation, with a fitted exponent of 0.5001. A workshop copying the method is not, and its exponent is 0.012, which is no growth at all, and after forty generations two lineages started from different originals end up 5 × 10⁻¹⁵ apart. Copying the marks loses the picture; copying the recipe loses the original and keeps the recipe.
The workshop that throws drawings away
Adding a rule that discards a drawing which looks wrong turns the mark copyist's spread from a growing random walk into a stationary process — the fitted exponent falls from 0.542 to 0.022, indistinguishable from the method copyist's 0.033 — and the two mechanisms then separate only by where they settle, 0.7002 against 0.8000, seven and a half spreads apart.
A grid on the wall is a scale without a projection
An Egyptian canon rules a wall into squares and counts a figure's height against the ruling — no horizon, no centre, and a length recovered to 1.4e-14% of error where the same reading taken off a pinhole misses by 58%. Applied instead to a pinhole picture, the furthest of six equal figures reads at 19% of its true height.
Higher on the page, and where that stops being true
A pinhole's image height above the horizon falls monotonically with depth over the whole 2–400 m range sampled here, and 57% of that whole range lands in the ground's last drawn tenth — the accumulation the oldest depth convention is quietly built from. Above eye level the ordering inverts, and at eye level exactly, five different depths draw one height, a spread of 0.0e+0 px.
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°.
Two stations in one picture
A parallel floor under a perspective room found one map absorbing two centres into one sheet. Split a two-rule picture down the middle instead and each half hands back its own horizon — 7.80 px apart at a rule-mix of 0.02 — and no eye's position has anything to do with the gap, because an ordinary pinhole picture's recovered horizon does not depend on where the eye stood at all.
The room a divergent picture is a photograph of
A divergent construction depicts a rectangle on a plane leaning toward the camera, and the splay alone sets how far — 36.3° at a splay of 1.32. Stand a second such construction on the first one's far edge, as a wall, and the two recovered planes meet at 6.9°, not at the right angle a real room's corner would need.
What a removed wall costs that a removed roof does not
Fitting a single centre to a building with its near wall deleted lands at 3.0e-15 m — the arithmetic floor — because deleting a wall does not touch the projection, only which surfaces are drawn. Fitting the identical routine to the same building with its roof removed does not return a number at all: handed a bundle of genuinely parallel rays, it refuses outright.
Surfaces that are not flat
A cylinder, a sphere, a fisheye. Every one of them is computed here rather than described, and every one is measured on the three things a picture surface can do to the world: bend its straight lines, turn its right angles, and change its scale. No surface escapes all three.
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.
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.
Stereographic keeps every angle, and only stereographic
One surface in the family preserves shape exactly — every right angle stays a right angle and both its arms are magnified equally, to the last bit the arithmetic has. It also sends every circle in the world to a circle in the picture, which the site's existing conic fit can be pointed at and asked to confirm without being told what it is looking at.
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.
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.
The arcs a curvilinear drawing uses
The taught way to draw a very wide view by hand is to run every straight edge of the world as a circular arc. That recipe has been repeated for sixty years without a surface attached to it, and it turns out to name one exactly — fitting a general conic to the image of a straight line returns a circle to nine decimal places under stereographic projection and returns nothing like a circle under any of the other standard picture surfaces.
One parameter between two surfaces
Wide architectural views are usually made on a projection with a number attached to it — a family running from the flat plane at one end toward the cylinder at the other, with everybody using the value one. That value has never been given a geometric defence. Measured across the family with the same battery of tests as every other surface, the worst angular error over the field has a minimum, and the minimum is at 1.04.
Conformal is not undistorted
The most distorted-looking picture in ordinary circulation is the little planet — a 360 photograph re-projected from below, with the ground curled into a ball. Its worst angular error over 160 degrees of the sphere is 4.4e-8 degrees, which is arithmetic noise. Every crossing in the original crosses at exactly the same angle in the result, and what has gone is area, over a factor of 255.
Six flat pictures of everything
There is one way to photograph the whole sphere and keep every straight line straight, and it is to stop using one surface. Six flat pictures at ninety degrees cover everything, each of them a perfect pinhole, and the price is paid entirely at the seams — where a straight line does not bend but kinks, by an angle that reaches 45 degrees and is exactly zero for the lines lying in the seam's own plane.
The lines a surface leaves alone
Only the plane draws every straight line straight, which is easy to measure and which gets read as though a curved surface bent everything. It does not. Every curved picture surface here keeps a one-parameter family of world lines exactly straight, and the family is a curve on the sphere of lines rather than a region of it — so a grid of samples finds none of it, which is what the first version of this measurement reported.
Undoing a picture made on a curve
Three rounds of work here have measured what a curved receiving surface costs going forward — 5.67 mm on a dish, 9.07 mm on a ridge, 529.4 mm on a vault. None of them asked whether the design can be got back. It can, exactly, and the price is stated precisely: you have to supply the surface. Told the floor, the recovery returns a design to 1.1e-12 mm; told nothing, and fitting the four marks every rectification tool fits, it is 111 mm 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.
The floors that unroll
A ridged floor curves visibly and can be laid flat without stretching anything — 7.4e-9 of strain across the patch. A dished floor curves less and cannot be laid flat by any means whatever. The difference is one number, Gaussian curvature, and it is the number Gauss proved no bending can change: 0 for the ridge, 0.0144 per square metre for the dish, and no cleverness in the flattening touches it.
The third column is area
This field has measured what each picture surface does to straight lines and to shape. Both are questions for somebody looking at the picture. Somebody counting in it wants a third column, and the same projections have been returning it all along without anybody asking: the equal-area fisheye holds a square degree at one printed area to 8e-8 across 80° off axis, while a flat plane inflates it 191-fold.
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.
The pivot that is not the eye
A camera bolted to a tripod turns about the screw under its baseplate, and the light crosses somewhere else. Every ray of the panorama that results passes the pivot by e sin γ — the offset times the sine of how far off its own frame's axis the ray points — so the picture has a radius where a projection would have a point.