The thread: One picture, one length — page 3
A shadow across a second object
A straight edge held in front of a lamp defines one plane, and the shadow's boundary is wherever that plane meets something. That turns a picture of a shadow into a measurement: a camera ray and a known plane meet in one point, and the object the shadow is falling on comes back out.
Measuring from one pictureWhere 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 camerasA 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.
The other systemsThe drawing that gives the solid back
One parallel view of a general point determines nothing: two equations, three unknowns, and the kernel is free. What closes it is not a second view but the correspondence — knowing which drawn edge runs along which world axis — and with it the whole solid comes back out of one drawing, exactly.
Constructing a viewThe arc every eye stands on
Four drawn corners known to be a rectangle fix the horizon of their plane and nothing else. The eye that drew them has to see the two vanishing points at a right angle, so it lies on the circle those points are a diameter of — and every point of that arc reconstructs a genuine rectangle, with right angles to five parts in ten million million of a degree, and a different proportion.
Measuring from one pictureA wrong model shows at its own level
Give a single-view measurement the wrong length and the error stays in the length. Give it the wrong model — a slope where it assumed level ground, a horizon five pixels out, a focal length off the barrel — and the error lands on the level that model belongs to and on every level above it, never below. That is a diagnostic. Two equal things at different depths catch a wrong plane; two equal things at right angles catch a wrong shape; marks read to 0.4 px see a slope of 0.38°. A wrong length, alone, nothing in one picture can see.
Mirrors that are not camerasTwo 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.
Mirrors that are not camerasThe 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.
Constructing a viewThe proportion is the assumption
Read the proportions of a rectangle out of a photograph of it and the answer is a function of where the centre of the picture is assumed to be. Sweeping that assumption across the horizon takes one drawn quadrilateral from one part in fourteen to slightly wider than square, every reconstruction a genuine rectangle, and only a fiftieth of the sweep within five per cent of the truth.
Measuring from one pictureA plumb wall tells a slope from a horizon, faintly
A sloping ground and a misplaced horizon fail the ground's depth test alike. A wall standing plumb beside the ground separates them in kind: a slope leaves the wall's test at 1e-16, and a horizon moves it. But it moves it a tenth as far as it moves the ground's — 0.20 per cent for a horizon five pixels out, against 2.11 — so a pair of windows a metre wide cannot see the difference, and a whole facade six metres wide can tell a one-degree slope from its matching horizon 95 times in a hundred. A wall running straight away from the camera sees nothing at all.
Mirrors that are not camerasThe wall convicts a pair set too wide
A window set wider than its partner's reflection keeps every joining line on the mirror's point, so the one test a symmetric photograph was known to carry cannot see it. The wall the other pairs fix can: it says where the partner should be seen, and a window set ten centimetres too wide misses by 6.0 pixels. With marks read to 0.4 px it convicts a 3-centimetre offset eight or nine times in ten and a 5-centimetre one every time. What no single photograph can see is a move along the camera's own ray to the window.
Mirrors that are not camerasA false pair is named by its neighbours, not by itself
Read against the rest of a symmetric facade, a pair whose right half has moved six centimetres is explained exactly by two stories: the right half moved, or the left half moved the mirror way from a pair standing six centimetres over. Predicted from left to right and from right to left, the pair gives one residual twice. What names the half is a mark that shares a line with it — the window's upper corner names a move across ninety-three times in a hundred at two centimetres — and what the readings recover is the move as the camera sees it, blind along its ray.
Measuring from one pictureA facade's corners see the horizon its widths could not
Read by its level widths, a six-metre facade saw a misplaced horizon only when it was 11.5 pixels out. Read by its two corners — the same wall's plumb edges — it sees 4.2, as well as the ground does, and four posts beside it bring that to 2.4. The corners' vanishing point is nearly six thousand pixels below the picture, but what the lines measure is its reciprocal, and in pixels of horizon their precision does not change from a camera pitched two degrees down to one pitched eight.
Measuring from one pictureA leaning post is worth its lean, not its length
A facade's two plumb corners read a misplaced horizon at 4.2 pixels; a street's lamp posts and poles are more numerous and never quite plumb. Each lean is random from post to post, so it averages away with the count and puts no floor under the reading — but it is large beside a long line's reading error, so it sets the price of every post: four steel columns leaning a twentieth of a degree match the two corners, and it takes sixty-four timber poles at a quarter. Counted as the corners' equals, eight such poles make the corners' horizon 2.4 times worse; weighted for their lean, they leave it where it was.
Where to standThe stretch decides the band
A design band chosen by geometry — the rays that meet the object — includes rays that graze along it, and a grazing ray lands two design points twenty times further apart than the design says. Cap the stretch at four and a bare floor keeps 55 per cent of its design, a corner 80, and the top of a descending flight all of it.
Measuring from one pictureA 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.
Measuring from one pictureA 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.
Measuring from one pictureA 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.
What survivesThe 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.
Constructing a viewThe stair that turns has a vanishing point that moves
A spiral stair's treads are one rectangle turned by a constant angle, so every front edge has its own vanishing point and the twelve lie on one horizon to 2.8e-14 pixels. Four consecutive of them read a cross-ratio of 1.353720, and the formula behind it holds no focal length, no principal point and no eye — so a photograph gives the builder's 12° turn back.
Light and mirrorsThe lamp's size over its distance, and nothing else
One straight-edge shadow gives a lamp's angular size and nothing about its actual size or distance — scale a 36 cm lamp and its 3.0 m distance together and the band it casts differs by 0.0e+0. A second card at a different height breaks the tie, recovering 36.00 cm at 3.000 m from bands of 18.0 cm and 144.0 cm alone, at a condition number of 37.1.
Constructing a viewFigures on a street that slopes
Equal-height figures have their heads on one line, and the taught rule says the line is the horizon. On a street rising at 8.33 per cent the heads are still collinear to 2.8e-14 pixels and the line is 58.9 pixels above the horizon — the street plane's own vanishing line. The taught rule loses 1.46 m of a 1.62 m figure at the far figure, and runs out of figure altogether at 19.4 m.