Every essay — page 14
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.
A photograph grades its posts across the sight, and not along it
Once two plumb corners have fixed a picture's vertical vanishing point, every other post in it can be read for its own lean: the angle between its line and the plumb line through its foot. Posts 2.5 metres tall at 7.5 metres are read to about a twelfth of a degree, enough to tell a quarter-degree lean from a twentieth ninety-one times in a hundred. But the reading sees only the lean across the line of sight. A post leaning towards the camera is caught five times in a hundred near the centre of the frame and a third of the time at its edges — and the posts read against one another lose the corners' share of the error, though not the street's own shared lean.
A second photograph reads the lean the first cannot, if it is taken round the corner
One photograph with two plumb corners grades a street's posts for their lean across its line of sight and is blind to a lean towards or away from it. A second photograph, read the same way, sees each post's lean across its own sight — so the pair reads both components wherever their two sight lines through a post cross at a wide angle. At right angles they read the lean along the first sight to a tenth of a degree, and catch 65 in 100 of the posts the first picture missed, against 26. The picture the question proposed, from across the street, is nearly the worst place to stand: its sight lines run back along the first, it sees the same component reversed, and it catches 29.
A third photograph closes the hole the second leaves, if it brings a building
A second photograph of a row of posts reads every post's lean except the one whose two sight lines run nearest parallel, and that post sits at the end of the row away from the second camera. A third photograph from the other side puts its own worst post at the other end, so between them no post is left: sixty degrees round either side, the worst post is read to 0.10° where either picture alone left it at 0.23 or 0.20. But the third picture does not buy what was hoped of it. Without buildings of their own, three pictures catch fewer of the hidden leaning posts than two at a right angle with one, and the spare equation each post gains cannot read the further pictures' verticals: a lean the whole row shares, or one that grows along it, looks exactly like a vertical misread.
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.
The corner that answers every eye
Three mirrors at right angles compose into the point inversion, so a ray entering leaves antiparallel to itself whatever direction it arrived from — a picture surface with no correct viewpoint because every viewpoint is correct. Tilt one face half a degree and the worst returning ray is out by exactly one degree, twice the error; the best is out by a twelfth of it, which is why a tolerance quoted from one measured ray is a statement about that ray.
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.
Two mirrors are three cameras
A photograph with two mirrors in it holds three views of the scene from three places, at baselines of 2.90, 3.10 and 2.26 metres. Two of the three pairwise geometries are mirror pairs and are skew; the third is a rotation by twice the angle between the mirrors, and it is not new evidence — five numbers read off the print rebuild it to 9.4 × 10⁻¹⁴ pixels, where a general three-view arrangement needs eighteen.
A symmetric object is its own stereo pair
A building with a plane of symmetry photographed once gives fourteen correspondences whose joining lines meet at one point to 1.9 × 10⁻¹² pixels, a skew-symmetric matrix, and the object's whole shape to fifteen digits — with no mirror anywhere and no second exposure. What it does not give is the size, and the instrument that decides whether any of it applies is the same meeting point, which opens to 12.7 pixels when the symmetry is half a per cent out.
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.
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.
A mirror pays for its field out of its aperture
Any map from the eye's angles to the world's is a mirror, by one line of arithmetic, so a mirror can be designed by what it does. The field is then a sum along the glass of one over the size factor, which caps it at the aperture over the smallest factor allowed, and a simple map reaches that cap. A surface free to take any profile still cannot read both directions at one constant factor unless it is flat.
Two wide pairs are most of a symmetry
A facade with a symmetric frame and an asymmetric middle is still its own stereo pair. Two pairs whose halves stand wide apart find the symmetry plane to 0.40° with marks read to 0.4 px; twelve pairs find it to 0.26°; two narrow pairs only to 1.35°. The middle that pairs with nothing comes back too, to 0.8 per cent, carried along its rays to the wall the pairs have fixed. The risk is not too little symmetry but a false pair — and a window set wider than its partner is one the standard test cannot see.
The mirror with an eye is the one that is sharp
A convex mirror focuses a distant point at two depths, and the gap between them is exactly the gap between its two size factors, so a mirror's blur can be read off its map. Hold the two factors equal at every point and the blur vanishes. That mirror turns out to be a familiar shape, and it is also the only one with a single centre of projection.