The thread: Projected, not constructed — page 5
Three-point, laid out with a straightedge
Recovering a camera from a drawing is the familiar direction. The other direction — stand somewhere, measure a room, and lay the picture out — had never been taken in three-point, because the third axis needs a measuring point on a line nobody draws. With it, every corner lands where the camera puts it to three parts in ten million million of a pixel, and nothing anywhere is judged.
What a machine computesA texture reaches the horizon as a rate
A ground drawn to infinity cannot carry a texture coordinate at its far corners, because a repeating texture has no coordinate there. It can carry a rate — so many checks per metre along the direction — and a second number that is 1 at points and 0 at directions. Interpolated like every other attribute and divided once, that pair is exact half a pixel from the horizon. Give the same corner a value instead and the ground is drawn in reverse perspective.
What survivesFive marks and the sixth
Five points determine a conic exactly — five coefficients up to scale, five equations, nothing left over — so a fit through five marks on a photograph is not a fit at all. The sixth mark, withheld, lands on the curve to 1.9e-13 px. And the moment a sixth mark is used, the arithmetic changes character completely: it becomes a least-squares problem, and the residual starts telling you something the five could never say.
The other systemsThe shadow rules that hold here
Drop the foot, run a line from the top at forty-five degrees, take the intersection: the drawing manual's shadow construction is exact in a parallel drawing, to arithmetic noise, at every point of the picture and with one set square. It is where the rule came from, and carrying it into a perspective picture is what broke it.
Constructing a viewA drawing has three horizons
The triangle three vanishing points form is not a scaffold. Each of its sides is the vanishing line of one of the three coordinate planes — the floor's horizon and two walls' — and its orthocentre is the centre of the picture. So the horizon a perspective drawing is built on is one side of a triangle, and it is special only because the ground is where things stand.
Surfaces that are not flatThe camera that is a cylinder
A swing-lens camera turns its lens about its own entrance pupil and sweeps a slit across film bent into a circle concentric with it. Compute where the light lands, unroll the film, undo the pinhole's inversion, and the result is not similar to the cylindrical picture surface — it is the same map, to the arithmetic floor. What it pays instead is detail, and a shear on anything that moves.
The eye that movesA band lifted by columns bends its lines
A curved seam in a several-station landscape opens a gap because the far band is one picture lifted by one amount. Lift it column by column instead and the seam closes exactly — and the gap goes into the band's lines. Shift each column and every road bends by twice the old gap, horizon and all. Draw each column from its own eye and a road bends by 8.1 px near the seam and 3.0 at the band's far edge, dying only as one over its depth; the horizon stays straight; and the next join inherits 37 per cent of the gap the first one lost.
The second projectionKeystone correction spends the panel unevenly
Turn a projector fifteen degrees and keystone correction throws away 18 per cent of its panel — the number a specification quotes. It is the smaller half of the price. What remains is spread unevenly: the near edge of the corrected picture gets a panel pixel for every source pixel, the far edge three-quarters of one, so the far edge carries 58 per cent of the source's detail. A tipped projector keeps more than a turned one, and a lens shift that places the same picture loses nothing at all.
What a machine computesA sky is carried as a direction
A triangle of sky has all three vertices at infinity, so the weight that lets a texture reach the horizon is zero everywhere and there is nothing to divide by. The attribute that belongs to such a triangle is the direction itself: carried over w like any other, it is every pixel's own ray to 5e-14 degrees. Carry the vertices' azimuth and elevation instead and a 60° triangle is 7.7° out, a 4096-texel sky needs triangles under ten degrees wide, and a triangle across the seam where azimuth wraps is painted with the opposite sky.
What survivesThe polar with a straightedge
Two secants through a point cut a conic at four places; the complete quadrangle they make has two more diagonal points; the line through those is the polar. Not one length, angle or midpoint is used, so the whole construction survives the projection that made the picture — and three unrelated pairs of secants land on the same line to 4.3e-13, while moving the point moves it by fifteen orders of magnitude more.
The other systemsThe ball a drawing does not draw round
An orthographic drawing of a sphere is a circle wherever the sphere is, and its centre is the image of the sphere's centre, exactly. A cavalier oblique drawing of the same sphere is an ellipse of aspect exactly √2 — and the drawing office reaches for a circle template. One formula covers both and the camera as well, and only the camera moves the centre.
Surfaces that are not flatDrawn for the cylinder, shown on the cylinder
Four essays in this collection have named the arrangement where a picture's surface and a screen's surface are the same surface, and none has run it. It is exact — a tenth of a millionth of an arcminute — and the exactness is not the finding. Three separate things have to be true at once, and a curved television's own maker prints a sitting distance where the first of them is out by five degrees.
What a machine computesGround and sky meet at the horizon without a crack
A ground of rate triangles and a sky of direction triangles share their vertices along the horizon, and rasterised as a graphics processor does it — positions snapped to a fraction of a pixel, every centre given to one triangle by the top-left rule — they lose no pixel and claim none twice, at any roll and any snapping. What the horizon does have is a sliver: a centre that falls within half a snapping step of it goes to whichever side the snapped edge puts it, and one that falls exactly on it, rolled one way, is given to the ground at a weight of zero. A ground stopped at a far plane leaves the crack the shared vertex never does: f·h/D rows.
The second projectionAn even spend of the panel is an uneven picture
A keystone correction that spends the panel evenly exists: render the picture straight into the panel pixels that reach the corrected rectangle, one sample each. At fifteen degrees that is 82.2 per cent of a frame's samples, and on the wall it delivers exactly what a full frame does — 0.99 of the source's detail at the near edge, 0.76 at the far — because the panel, not the source, was setting the detail everywhere. It removes no unevenness. On fewer samples it makes the picture worse where it was worst; the even picture is a uniform source of 57.6 per cent of a frame.
The eye that movesA painter's closing can be read from one band
A band of a several-station landscape closed along a curved seam carries the record of how. A station per column bends a road by one over its depth, a column-wise shift bends every road alike, and one lift bends none. Four roads read to a pixel name the closing nearly every time, down to a station curve of about six centimetres, and the reading survives every depth being misjudged by the same factor.
What survivesThe ball at the edge of the frame
A ball photographed near the edge of a wide picture is not drawn as a circle. It is an ellipse, longer along the radius from the centre of the picture than across it, and the centre of that ellipse is not the image of the centre of the ball. Both are properties of the flat sheet the picture is on, and an exact pinhole produces both.
Constructing a viewThe quadrilateral no rectangle casts
The relation that reads a camera out of a drawn rectangle has a minus sign in it, and the minus sign is a refusal: two vanishing points on the same side of the assumed centre give the square root of a positive number, and no camera makes that quadrilateral out of a rectangle. Watching the refusal arrive shows what it is worth — one corner has to travel most of the picture's width before it fires.
Systems that kept the measureThe room a divergent picture is a photograph of
A divergent construction depicts a rectangle on a plane leaning toward the camera, and the splay alone sets how far — 36.3° at a splay of 1.32. Stand a second such construction on the first one's far edge, as a wall, and the two recovered planes meet at 6.9°, not at the right angle a real room's corner would need.
What a machine computesA ground and a sky share an elevation, not a distance
Read from its own attribute — the ground's rate and weight, the sky's direction — each side of a rasterised horizon reports the elevation of its pixels' rays to 2.2 millionths of a radian, in 32-bit arithmetic as in 64, so a haze painted by elevation crosses the edge without a seam. A fog by distance cannot, and should not: the ground a row below a walker's horizon is 1.2 kilometres away, and at thirty kilometres of visibility the fog steps by 0.85 in one row. A camera's pixel, averaging its area, records a step of 0.69.
The second projectionThe even picture is fair to a centred room, and only to it
Judged by detail per degree of each seat's view, the corrected picture that is even on the wall is already the fairest possible for an audience centred on it: a layout drawn for that audience gains its worst seat nothing. For an audience on the far side it is also the best, because the far edge is where the panel is coarsest and the even picture already spends all of it there. Only an audience on the projector's side can be served better — thirty per cent better at a fifteen-degree turn — by putting the samples on the near edge, where the panel has them to give.
What survivesEvery quadric has one outline
The curve where a solid turns away from the eye is the section of the solid by one plane — the eye's polar plane — and the outline is three matrix products with no sampling in them. It works for a ball, a dish and a hyperboloid, and it fails for a cone, whose dual outline collapses to a single point and forgets which two lines pass through it.
Constructing a viewThree procedures, one panel
Alberti's lateral section, the distance-point construction and a pinhole camera put every transversal at the same pixel — and every reading of the finished drawing therefore returns the same number for all three. The methods are distinguishable on the desk and indistinguishable on the panel, which is the fact any attribution has to start from.
What a machine computesA 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 second projectionA walking viewer gains from a following layout only on the projector's side
Re-lay a turned projector's corrected picture frame by frame for one viewer walking across the room and they gain up to thirty-one per cent of detail per degree — on the projector's side of the picture, where the panel has samples to spare, and nowhere past eighty centimetres beyond its centre. A layout drawn once for the whole walk gains nothing at all. Half a second late, the following layout keeps four-fifths of the gain and at the worst moment serves the viewer six per cent worse than the even picture; four seconds late it is worse on average. And whatever it gives the walker it takes from a seated room: up to twenty-nine per cent of the worst seat's detail.