Every essay — page 11
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
Measuring from one picture
A photograph read backwards for the scene that made it: a height from a cross-ratio, a façade flattened by a homography, a plan of the ground. Every measurement is a ratio, because a single view has no size — and that is shown rather than said.
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.
One picture of a ball
The outline of a ball in a photograph fixes the cone of rays that graze it, so the cone's axis is the direction of the ball's centre and its half-angle is the arcsine of radius over distance — both exactly, and neither of them separately. A ball a hundred and thirty-seven times larger, a hundred and thirty-seven times further away, draws the identical conic. And the drawn ellipse's own centre is not the image of the ball's.
Two pictures of a ball
Two outlines give two tangent cones, and where their axes cross is the ball's centre — with the radius following from either half-angle. There is no correspondence problem, because a ball's outline in one picture and its outline in another are guaranteed to be the same ball; and there is a degeneracy, which is the two eyes and the ball in a line, and the solver refuses it rather than returning something plausible.
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.
An area, out of one photograph
A patch of ground comes back at 5.205 m² from one photograph, to 1.8 × 10⁻¹⁴, through a homography built from four marks and their four known positions. What is worth knowing is how it degrades — the patch's extent across the picture is read with an error growing as the depth, its extent into the picture with an error growing as the depth squared, and at twenty-eight metres the two are sixteen times apart — which is the depth divided by the camera's height.
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.
An angle on the ground
An angle needs no length at all — it is scale-free, so the one thing a single view cannot give is not the thing an angle wants. What it wants instead is the plane's shape, and the difference is measurable — assume the reference rectangle's aspect a quarter wrong and the angle moves 11.5°, a length running into the picture is out by exactly that quarter, and a length along the reference's own direction does not move at all.
The answer is an ellipse
A mark read with a round error does not come back as a round region on the ground. The ray is grazing, so a pixel across the picture is worth a little and a pixel up it is worth a great deal, and the recovered point's uncertainty is an ellipse pointing at the camera — 5.07 to 1 at eight metres from a camera 1.62 m up, which is the depth over the height. Propagated and sampled agree to 0.5 per cent, and a ray that is not grazing gives a disc.
Which reference to measure from
Given four candidate scale bars in one photograph, the best is not the longest and not the nearest — it is the longest in the picture. A five-point-two metre bar near the horizon is the longest thing in the scene and the worst reference in it; a four metre bar close to the camera is the best. Walk one bar outward and the term it controls falls as one over its length in pixels, with a fitted exponent of −1.08.
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.
Five facts that close the same gap
The gloss that one length has to come from outside the photograph names a single option, and there are at least five — a length on the ground, the camera's own height, a repeated object of unstated size, a standing object of known height, and the focal length with the horizon. All five give an answer on the same picture with the same noise; their spreads run from 0.36 to 3.60 per cent, a factor of ten, and which one is available depends on the photograph rather than on the arithmetic.
Where each closure enters the stratification
The five facts that turn a photograph's ratios into metres do not all do the same job. Read on four quantities through the map each supplies — a cross-ratio, a ratio along a line, an angle, a length — three of them return all four exactly and two return only the first three. Set the free scale ten per cent wrong and the whole ten per cent appears in the length and nothing appears in the other three, to thirteen digits.
Mirrors that are not cameras
A flat mirror is a second camera and a curved one is not a camera at all: continue the lines of sight behind a mirror ball and they pass through no common point, so a photograph of it is a projection of nothing from anywhere. What they have instead is a caustic — and one shape, the paraboloid, that does focus exactly, for one bundle of rays and no other.
A curved mirror has no eye
A flat mirror is a second camera — reflect the eye in the plane and every line of sight passes through the reflected point, to 2.8e-12 mm. Curve the mirror and the point is gone. Over 20 cm of a mirror ball two metres across, the lines of sight miss their own best-fitting point by 2.8 mm, and by 52.9 mm on a ball half a metre across. What replaces the eye is not a worse eye; it is nothing.
Where the focus went
If the reflected rays do not meet at a point, they meet each other in pairs, and the curve they are all tangent to is what the mirror has instead of a focus. On a sphere of 1.6 m vertex radius it is 184.1 mm long and has a cusp; near the axis it sits at R/2, which is why a small enough spherical mirror passes for a good one. This is the bright shape in the bottom of a coffee cup, computed rather than admired.
The one shape that focuses
A paraboloid takes every ray parallel to its axis exactly through one point — 4.8e-9 mm of envelope across a 1.24 m aperture, which is arithmetic. A sphere with the same vertex curvature spreads the same bundle over 184.1 mm. So there is exactly one curved mirror that is a projection through a centre, it is a projection for exactly one bundle of rays, and every telescope in the world is built out of that sentence.
A mirror ball is an equal-area fisheye
Photograph a mirror ball from far enough away and its rule is ρ = R·sin(θ/2), which is the equal-area fisheye — not an approximation to it, the rule. Measured, the departure falls from 4.27% of the picture's radius at 3 radii to 0.01% at 2000, while the next-best named rule stays 21% out at every distance. And the ball reflects 100.0% of the directions there are, which no designed surface does.
The cone that reads the floor
A conical mirror standing on a design turns it into a picture, and the map it performs is exact, one-to-one, and not a projectivity — a homography fitted to four of the marks returns those four to 7e-13 mm and puts the rest 2480 mm away, on a design 369 mm wide. The reason is that the cone turns the annulus inside out: the middle of the picture comes from the far edge of the floor and the rim from the near one.
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.
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.
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.
One shutter, two views
A photograph with a mirror in it is a stereo pair, and a peculiarly well-behaved one. Its fundamental matrix is skew-symmetric, so both epipoles are the same point; that point is where the camera would see its own lens; and every line joining a mark to its reflection passes through it, to 1.4 × 10⁻¹² px. The baseline is twice the distance to the glass, which is the one number a single view cannot supply and a tape measure can.
Two matches are enough
A general fundamental matrix has seven free numbers and needs eight correspondences. A mirror pair's has two, and two correspondences fix it — with a straightedge, on a print, by drawing the line from each mark to its reflection and marking where the two cross. Given the same sixteen marks read to four tenths of a pixel, the constrained fit lands 4.8 times closer to the truth than the eight-point algorithm.
Square to the camera is the worst mirror
A mirror pair's baseline runs along the mirror's normal, so a mirror facing the camera puts the second eye directly behind the first — the forward-motion arrangement, with the epipole in the middle of the frame and the rays to a mark crossing at 23°. Turning it forty-four degrees opens that to 65° and cuts the worst depth error threefold, and the number to watch is not the angle but where the reflected lens sits on the print.
Two mirrors make one turn
Reflect a point in two mirrors meeting at 36° and the images arrive nine at a time, every one of them on a single circle about the line where the mirrors meet, to 4 × 10⁻¹⁶ m. The rule taught for the count — three hundred and sixty over the angle, less one — is right at six of nine angles tried and wrong at the rest, because it is a rule about angles that divide a half turn and it is quoted for angles that divide a whole one.