The thread: One machine used twice — page 6
A fog band softens the horizon only by repainting the ground
Carry the sky's full haze a few rows down into a distance-fogged ground and the step at a walker's horizon is not removed but spread into a ramp, whose steepest row changes by about one and a half over the band's width of the whole rise. Two rows barely move a step of 0.63; softening it to the unaided horizon of an eye 150 metres up takes 32 rows at every visibility from one kilometre to thirty — and repaints all the ground beyond forty metres hazier than it is. The choice between an honest distance fog and a seamless horizon is real. What four rows of band do buy is the shimmer: the last row's saw-tooth falls from 0.061 a column to 0.012.
The rectangle behind the lensA followed post's own lean is read as the bend
A rolling camera on a bend reads the bend from how its roadside posts lean, and a post that leans by itself adds its own tilt to every frame, unchanged. That tilt is indistinguishable from the bend's curvature: a tenth of a degree on a post twelve metres out turns a 50-metre bend into 58. No number of frames removes it, and fitting each post's own lean along with the bend loses the bend — a sideways camera's reading widens from ±7 metres to ±29. A followed post has to be plumb to six hundredths of a degree, finer than a well-set steel column; timber posts at a quarter of a degree help only in their dozens.
Light and mirrorsThe 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.
Surfaces that are not flatA picture that can be printed
Two screens are fed the surface that is exactly right for each, so nothing about the viewer is left in the answer. What remains is whether the picture can be made flat before it goes up — and a cylinder unrolls while a sphere does not, so the dome's picture is stretched by eighteen per cent between its middle and its rim before anybody sits down.
What a machine computesJittered frames rebuild a camera's horizon, saw-tooth and all
A renderer that moves its sample a fraction of a pixel every frame and averages the frames turns each pixel's single sample into an average over the pixel's area — which is what a camera's pixel records. Along a hazy, rolled horizon that removes the quarter of the shimmer point sampling adds, 0.082 a column down to 0.061, and stops there: eight Halton frames are within a hundredth of the camera, and sixty-four are no better. The rest of the saw-tooth is the camera's own, and so is the horizon's step. Only a band of repainted ground goes below them, and jitter's own price is a flicker of 0.29 a frame until a history averages it away.
The rectangle behind the lensA post read both ways sets its own lean aside
A rolling camera reads a bend from how its roadside posts lean, and a post's own tilt is read as bend: a quarter of a degree leaves a 50-metre bend read to ±12 metres. Followed again by a car coming the other way, the same post shows the same tilt and the opposite shutter lean. Half the sum of the two passes is the tilt and half the difference is the bend, and a timber pole read both ways gives the bend to ±1.7 metres — better than a plumb steel column read one way, at ±2.3.
What survivesThe map a row of posts is
Walking one bay further down a row is a map of the drawn line to itself, and which map it is settles everything about how the spacings behave. It is parabolic — one fixed point, counted twice, and that point is the vanishing point — which is why the drawn posts crowd toward it and never arrive. Doubling a distance instead gives a hyperbolic map whose multiplier a straightedge can read.
The other systemsTwo outlines are two curves
The whole method of multiview drawing is the transfer line — a feature at a position in the front view is at the same position along that axis in the top view. On a flat-faced solid the feature is a vertex and the rule is exact. On a ball the two views draw two different great circles, meeting in exactly two points, and the transfer line joins places that are √2 radii apart.
Where to standOne flight, two pictures
From the top of a descending flight every riser faces away, so the design lands entirely on treads. From the foot of the same flight the risers take 68 per cent of the design at a median stretch of 1.4, and the treads take the rest at a median of 2.6. Give each reader the faces the other cannot use and one staircase carries two pictures, with no face asked to hold both.
What a machine computesA rolling horizon outruns the history that was meant to quiet it
Jittered frames averaged in a history draw a still hazy horizon as a camera does, four times nearer than one centred sample. Roll the camera and the history has to follow. Left in place, it is worse than no anti-aliasing at all from a degree a second. Reprojected exactly through the roll, it still reads last frame's picture between pixels, and an interpolated step is a softened step, softened again every frame: past one and a half degrees a second the history sits further from the camera than a single centred sample, and at sixteen it paints the first row of ground as sky. No blend fixes both its error and its pops. Four rows of band move with the horizon popping a fortieth as much — at the price of a quarter of the fog's range in repainted ground.
The eye that movesA body that bounces leaves a second swelling in the rows, and one pass reads both
A vehicle's body pitches and bounces on its springs at two frequencies, and a scroll camera's rows answer both: the pitch by turning the camera, the bounce by lifting it through the height climbed between the slits' two moments. One oscillator fitted to a body pitching at 1.4 Hz and bouncing at 1.0 lands between the two, at 1.22 Hz, with the camera over the body's centre, and on the pitch, at 1.41, with it 1.5 metres ahead — too lightly damped either way. Two modes fitted to the same single 800-metre pass read both, the pitch to ±0.02 Hz and the bounce to ±0.03, say plainly whether a bounce is there at all, and expose a bias the earlier readings carried even on a body that does not bounce: a camera ahead of the centre reads a 1.4 Hz pitch as 1.5 unless the fit knows its lever arm.
The rectangle behind the lensFrames read up and down part a post's lean in one drive
A rolling camera reads a bend from how its roadside posts lean, and a post's own tilt is read as bend unless a second car drives the road the other way. A sensor that reads alternate frames from the bottom row up reverses the shutter's lean without reversing the car. Thirty-two such frames of one drive read the 50-metre bend from a timber pole to ±1.8 metres, against ±1.7 for two passes of sixteen — and a post swaying in the wind, which costs two passes three metres, costs one drive a quarter of one.
Constructing a viewOne hand step each
Every classical perspective construction has exactly one step a person performs by hand, and the four constructions perform four different steps. That single difference decides everything a finished drawing can say about its maker, because the answers are identical and only the mistakes are not.
The other systemsA picture with no eye
The anamorph field measured which parts of an object one eye can paint, and every finding in it is about a point. A parallel projection has no such point, so the same questions have a different shape — and the two that change are the dimension of the answer and the map onto each face. The one that does not change is occlusion, which is routinely assumed away.
What a machine computesA clamp keeps the sky clean and leaves the rolling horizon smeared
Clamping a jittered history to the range of each pixel's neighbours is the renderer's standard guard against a moving edge smearing, and on a rolling horizon it does half the job. Two rows from the horizon, where the neighbours agree, it pulls the smear back at once: a sky row the history had darkened to 0.88 is restored to 0.99. On the rows the horizon crosses, the nine neighbours hold sky and ground together, the window runs from 0.12 to 1.00, and the smear passes through it untouched. The crossing with a plain centred sample moves from 1.5° a second of roll to 1.8; at 16° a second the history is 0.070 from a camera instead of 0.093, still nearly three times a centred sample's 0.025. And a clip tight enough to reach the smear brings the jitter back: on a still camera it pops four times as hard.
The rectangle behind the lensOne pair of frames reads the speed and the bend from the roadside
A camera that reads one frame from the top down and the next from the bottom up turns every roadside pole's rolling-shutter lean round and leaves its own tilt where it was. Half the difference of each pole's two leans is the shutter's alone, and across a row of poles it lies on one straight line in one over the range: its slope is the car's speed and its crossing is the bend. So one pair of frames, a fifteenth of a second, reads both — the speed to 3 per cent and the bend to ±5.6 m from sixteen timber poles, whatever they lean. With the speed known, sixteen poles read the bend to ±2.4 m, a little better than one plumb post followed for sixteen frames. The nearest pole decides most of it.
The eye that movesThe rows read where the camera rides on the body, and a camera behind the centre hides the pitch
A scroll camera's rows over a rough road say how far ahead of the body's centre the camera stands, as well as how the body pitches and bounces. One 800-metre pass reads a camera 1.5 metres ahead to ±0.15 metres, costs the pitch's frequency nothing to do it, and removes the bias a fit that puts the camera at the centre carries — a 1.4 hertz pitch read as 1.54 from 2.5 metres ahead, and as 1.80 from a metre behind a body that does not bounce. Behind the centre the camera's rise works against its turn, and a metre back on a body that does not bounce it very nearly cancels the pitch where it rings.
Constructing a viewStepped, or measured from the zero
The same hand at the same precision, laying the same braccia off two ways — dividers walked from the last mark accumulate and grow as the square root of the count, while marks set from a common origin do not accumulate at all. The difference is not which method was used — it is where the zero is, and only the second is recorded in the drawing.
Where to standA flight that has ends
A real staircase has three families of faces and the rule proposed last round predicted three pictures. It supports four, because a picture is fixed by a direction and the flight's two sides point opposite ways — one kind of face that no single direction reaches both of. The rule counts words and the object counts orientations.
The other systemsNothing moves along the direction
A part slid four metres toward the reader along the direction an isometric drawing projects along keeps its drawn place to a ten-thousandth of a pixel, and the same slide seen from a station point moves it forty-four. An exploded drawing is not an approximation that works because the parts do not move far — it is an identity, and it is why cutaways are drawn in parallel systems.
Light and mirrorsA 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 other systemsOne face, one scale
A design carried from a point onto a flat face varies in scale by nearly six across that one face; the same design carried along a direction varies by 1.000000000. The stretch is reported here as the two singular values of the local map rather than as one directional difference, which is the honest form and which the previous round owed.
Light and mirrorsA 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.
Light and mirrorsThe 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.