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
Light and mirrors
A shadow is a projection from the lamp; a reflection in a flat mirror is the view from a camera on the far side of it. Neither needs a new method, and both are computed with the projections already here — which is exactly why the mirrors that bend get a field of their own.
The residual has a shape
A flat-floor map mispredicts a shadow by millimetres on any floor that is not flat, and the number everybody quotes is the worst one. Tune a dish, a ridge and a step until all three mispredict by exactly 25.0 millimetres and the scalar can no longer tell them apart — by construction. The signed residual around the ring still can. The second harmonic of it reads 0.09%, 2.79% and 19.37%, a factor of two hundred across three floors the headline number calls identical.
The floor is a choice of coordinates
Four rungs of this field have measured what a curved floor costs a shadow reading, in millimetres. It costs nothing. A shadow mark sits on the ray from the lamp through the occluder's edge, and where along that ray the floor caught it is a fact about the floor alone — so un-casting in rays returns the occluder to three parts in ten thousand of a millionth of a millimetre on a plane, a dish, a ridge and a step alike, while the same shadow read in the plan is wrong by up to 216 millimetres.
The lamp comes out in rays and not in plan
One drawing of three posts and their shadows yields two points, and a curved floor treats them completely differently. The lines through each post's top and its shadow's tip meet at the lamp's image to a ten-thousandth of a pixel at every curvature, because a top and a tip are two points of one real ray. The lines through each foot and the same tips meet 113 pixels from the lamp's foot — and the lamp placed from an exact point and a wrong one lands 1.3 metres away.
The corners a floor cannot add
A wire with no corner anywhere on it casts a shadow with one, wherever its tangent runs along the ray. That condition contains the lamp and the wire and no surface at all — so the same helix over a plane, a dish and a ridge draws shadows that differ by metres and each has exactly one corner. A floor with a crease draws five more, and the two kinds are separable by a hundredfold: a real corner is where the shadow stops dead, and a crease is where it turns at full speed.
A floor is read along curves
Whatever a shadow says about the floor it landed on, it says only where the shadow is — and a shadow is a curve while a floor is a surface. Shadow curve length grows exactly linearly in the number of lamps, by a fitted exponent of 0.999, and the fraction of floor within two centimetres of one grows more slowly at 0.94, because the curves begin to overlap. At thirty-two lamps, seventy-one per cent of a nine square metre patch has still never had a shadow on it.
How many lamps a drawing has
The shadow field recovers a lamp by intersecting drawn lines. Two lamps make that a partition rather than an intersection — and two centres fit any bundle better than one, on a one-lamp drawing as readily as on a two-lamp one, so a count is a decision that needs a noise level before it exists. A criterion built on a penalty instead of a noise level returns four.
The distance at which two lamps part
Two lamps five centimetres apart are one lamp, and the drawing is right to say so. The separation at which they become two is proportional to how carelessly the picture was clicked — 3.6 centimetres at half a pixel, 7.2 at one, 28 at four — with no floor anywhere, so nothing but care stands between a reader and any separation at all.
A floor cannot fake a second lamp
Cast the same two lamps onto four floors at four curvatures and ask how well one centre explains the drawing. Every one of the thirteen answers is 138.3277 pixels — the same to fifteen digits, because a floor decides where along a ray the shadow's tip landed, and a line through a point and another point that has slid along it is the same line.
The drawing does not run out of lines
Every post supplies a line to every lamp, so a drawing of five posts offers ten lines to two lamps and twenty to four — the unknowns and the constraints grow together and two posts fix any number of lights. What runs out is the partition, whose margin falls from 251 pixels to six as the share of lines assigned correctly falls from all to just over half.
A lamp behind the camera
A light behind the photographer has no image — the projection refuses it — and the shadows it casts are in front of them, drawn as ordinary shadows. The construction that recovers a lamp from those shadows works anyway, meeting to a ten-thousandth of a pixel at the point the reversed divide puts it, and the taught reading of where the answer lies gets the case exactly backwards.
Counting shadows is not counting lamps
Two lamps close together cast one connected dark patch and the drawn lines say two. One lamp behind two cards casts two patches and the lines say one, exactly. And the arrangement where the patch count is right — two lamps far apart — has no fully dark region at all, so the same floor answers one, two or zero depending on which darkness is being counted.
The arrangement the count cannot see
Five posts laid out five different ways give the same leverage to a sixth and the same separation limit to a quarter, and the sixth arrangement — posts strung out along their own shadows, built to be exactly degenerate — is no worse than the rest. The degeneracy belongs to the family the count does not use, and its conditioning there is exactly zero.
A soft shadow on a curved floor is not the lamp's image
On a flat floor the soft edge a lamp with a size casts is exactly the lamp's own image, cast through the occluder's edge as though through a pinhole. On a floor with a step the same construction lands 1.047 m off the line that fits a flat one, and on a dished floor the image's own shape departs by 0.0453 of the lamp's width even where its overall span barely moves.
The 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.
How many lamps make one lamp
An array of point lamps spread across the width a real lamp would occupy leaves a staircase rather than a ramp — 4 lamps step by 25.0 per cent of the whole, 64 by 1.6 per cent — and the worst departure from the true ramp falls as the -1.007 power of the count. A single point lamp is not a coarse version of that staircase; it is wrong by 0.50, the most a fraction can be wrong by.
What a point in shadow can see of the sky
A place directly under a slab is not sealed off from the sky at all — it sees 83.4 per cent of it, 5.242 of the 6.283 steradians a place in the open has, against 5.240 from the closed form for a rectangle's own solid angle. The same counted-directions call that measures a lamp's penumbra measures this fraction too, and the two agree to 2.50e-4 with the source simply swapped from a lamp to the whole sky.
The lamp a low shadow cannot locate
Five posts, their shadows, and the line from each foot through its own shadow's tip meet at the lamp standing over them — at 52 degrees up that meet moves 0.11 m for half a pixel of marking error; at 11.5 degrees, 0.64 m. The two eigenvalues of the same pencil of lines part by a factor of 8849084 across the sweep, and half a pixel becomes more than a metre of lamp below 14.4 degrees.
Where a shadow splits in two
A gantry's shadow is two pieces at a lamp height of 1.36 m, and the crossing to one piece happens at a tangency running the whole length of the beam rather than at a point — the same plane that meets a ball at an aspect of 1.00 to 1 meets the beam at 1736 to 1, and a grid finds the true crossing height to a fitted exponent of 1.00 as it is refined. A ring tipped 70° keeps its hole for a completely unrelated reason, closing only at 71.34°, which is the warning that a shadow's topology changes at a tangency names two different accidents rather than one.
How many shadows determine the object
One lamp's shadow says only that a convex section lies inside a wedge 10.5 times its own area; four already cut that down to 1.17 times, and 128 close a convex section's boundary to 0.10 mm everywhere. The identical sweep on a section with a bite taken out of it stalls at 70.3 mm, 675 times worse, because an outline is the boundary of the smallest convex body with that shadow and no direction ever sees inside a concavity.
A dent breaks the terminator
A ball's lit boundary is one closed curve, 313.2 cm around; press a dimple into its top and, at 45° of elevation, the walk that traces it finds 181 crossings where a point still faces the light but is occluded by the body's own rim — a second curve, 23.1 cm long, that a convex surface can never produce. The two curves meet where the dent's own deepest point goes dark, at 50.952°, with the ray's clearance reaching zero at a stationary point rather than merely a small one.
A shadow edge read as a profile
A lamp, a stick and a camera recover a stepped object's profile to 4.9e-15 m rms when the marks are exact, and to 10.4 mm once they are read to two tenths of a pixel — the same linear law a fitted exponent of 1.001 confirms. What actually sets that number is the angle between the sweeping light plane and the camera's own ray — the amplification is least, 17.0 times a pixel, broadside at 6°, and grows without bound toward -36.1°, where the plane contains the camera's own eye and the recovery keeps none of its marks at all.
The lamp and the floor cannot both be recovered
Every member of a one-parameter family of lamp-and-floor pairs draws the identical photograph to a fraction of a thousandth of a pixel — one member swings the lamp 90 cm and tilts the floor 4.3° and the picture does not move at all. A tape measure or a post of known height each pin the true member uniquely, and so does a wall known in advance to be square to the floor, whose angle to the recovered floor peaks at exactly 90° at μ = 1.000 and nowhere else; a receiver merely known to be parallel drifts thirteen times more weakly.
Many pictures at once
Raise the count and the answer changes character. A sequence determines its own camera track and the scene together — up to seven numbers that no quantity of pictures supplies, counted here in the Jacobian and then walked along to show that they are free rather than merely small.
The track and the scene together
Six photographs go in and one hundred and sixty-eight numbers come out — every camera's position and orientation and every point's place in space, solved for at once. Nothing in the solve was ever told where a camera or a point was.
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.