The thread: The viewer is in the geometry — page 4
Two circles, one picture
A photographed circle leaves its own pose ambiguous, and not a little: two congruent circles in planes 23.6° apart draw the same conic to 1.1e-16 on normalised coefficients, both of them in front of the camera. On top of that the distance is free, so the family is two discrete poses each with one continuous parameter — and a plane one degree from either draws a conic 2.5e-4 away, which is what makes the agreement a measurement.
Where to standWhat an eye can paint
A flight of steps has eighteen faces and no eye reaches more than fifteen. Pointed at a cluster of blocks, a seating rake and a corridor with a doorway in it, the same measurement finds 8 of 21, 7 of 13 and 6 of 7 — and the plane, which offers its whole self to every eye, is the control that makes the law a law rather than a fact about stairs.
The real instrumentA refocused panorama is six lenses
Refocus between the frames of a room panorama and each frame is its own camera: its pupil carried forward by its own extension, its picture made at its own principal distance. The best fixed pivot then leaves 12.9 arcminutes along the worst seam, and a head that slides the camera back as the lens extends leaves 2.82 — the lens's walk alone. The pivot's share is first order and worth a tracking head indoors. But a stitcher that reads all six frames with one focal length misregisters by up to 270 arcminutes, which is the larger mistake by ninety times.
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.
Where to standFacing the reader is not being reachable
A face turns toward the eye or it does not, and that is a dot product any reader can compute. Whether the eye’s rays actually land on it is a different question with a different answer — on a seating rake, three faces of ten that face the reader receive nothing, and they are 27 per cent of the facing area. On a corner the same test loses nothing at all, which is what makes the gap occlusion rather than arithmetic.
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.
The real instrumentOne focus stopped down is as sharp as six in a deep room
Refocusing a room panorama frame by frame costs a seam of 12.9 arcminutes on a fixed head. One focus stopped down costs none of that, and at f/40 holds half a metre to four metres to 5.27 arcminutes — sharper than the fixed head's seam, blunter than a tracking head's 2.82. But the comparison flatters refocusing: the frame on the table also holds the wall behind it, so it cannot be held sharper than 5.27 however it is focused. Counted properly, one focus on a plain head comes within 2.4 per cent of the geared one, and pays for it in light.
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.
Where to standThe eye that reaches the most
A higher eye buys the faces occlusion was hiding and loses design off the far end of the object, so "the best eye" is not a question with an answer until somebody says which of the two they are paying for. On three objects the answer is as high as possible; on a corridor with a doorway in it the two quantities cross and the best height is two and a bit metres.
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.
The real instrumentA level picture shows its rise on its horizon, and hides its slide
A shift lens moves the principal point off the middle of the frame, and the textbook focal length from two vanishing points pays for assuming it did not. On a level camera the payment is optional. The principal point lies on the horizon, so a rise is read straight off the picture and costs nothing; a slide along the horizon is read by nothing at all, because two vanishing points fix only a semicircle of possible cameras. What a sideways crop costs then depends on how the building is turned: a tenth of the frame trimmed off the side is 23 per cent of the focal length on a facade seen eleven degrees off square and under one per cent at forty-five.
Mirrors that are not camerasA mirror's top edge waits for an eye above it
Between two upright mirrors a path of light from one eye to another, unfolded, climbs in a straight line from one eye's height to the other's, so every bounce lies between the two. While both eyes are inside the glass's height its top edge takes nothing at all. Above it, the images go in a fixed order — not the one with the most bounces, not the longest, but the one whose last bounce falls furthest along its path, in forty arrangements out of forty. The barber's corridor that sinks out of the glass is a mirror leaning a fraction of a degree, and its length goes as one over the square root of the lean.
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.
Systems that kept the measureA walk's order is on the floor, not on the page
A narrative panel whose copies are moments of one walk holds its order after all — in where the copies stand, because a walker moves a step between moments. Joined by the shortest route on the page, six copies seen from a standing eye come back in the order walked 68 times in 100. Undo the floor first, with the horizon the copies themselves supply and the angle of view guessed anywhere from half to twice the truth, and the same rule returns it every time the walk allows — leaving one bit, the direction, which a figure facing its way supplies.
The real instrumentA second rectangle fixes the column a level picture hides
Two vanishing points of a building put a level camera on a semicircle and leave it free to slide along it; a second rectangular object turned against the first puts it on a second semicircle, and the two cross once, at the camera. On a facade seen twelve degrees off square and slid a hundred pixels, a second object twenty degrees further already reads the focal length to 1.3 per cent where the building alone, even with its principal point known, manages 2.9 — and where the frame's middle column is wrong by a third. The reading does not care where the column was slid, and it is best when the second object stands at forty-five degrees to the camera, whose semicircle is flat over the true column.
Mirrors that are not camerasA corridor of mirrors counts their angle, not their leans
Between two facing mirrors that both lean, the barber's corridor of self-images ends where it leaves the glass, and the count depends on the two leans only through their sum — the angle between the mirrors. Forty-two pairs of leans fall on one curve, within an image. Counted from near either mirror the number is the same, so counting twice cannot say which mirror leans, and two mirrors tilted together run the corridor on for ever. A photograph can: each round trip lowers the image by the sum of the leans, and where the row of images starts shares the sum between the two — provided the camera knows which way is level.
The second projectionTwo walkers share a layout drawn for the worse of them
A projector turned to its wall can re-lay its corrected picture for one walking viewer and give them up to thirty per cent more detail per degree on the projector's side. For two, it draws the layout that serves the worse-served best. Walking the same way a metre apart, the one nearer the picture's centre keeps 99 per cent of its own gain and the other 77, falling to 44 at four metres apart. Crossing, they keep two per cent. A viewer on the far side vetoes the other only while it is the worse-served — and at different depths the shared layout takes from the walker it serves better, leaving them below the even picture.
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.
Constructing a viewThe slip that leaves no trace
A distance point put twenty-four pixels wrong moves the pavement by two and a half and leaves the reader's projective test reading exactly four thirds. The same slip on Alberti's section moves the drawing by the same amount and is caught, so the difference is not the size of the error — it is that one of them lands back on the set of correct drawings.
Systems that kept the measureEach copy's facing reads the walk past the place it doubles back
A walk that doubles back defeats every reading of a narrative panel from where its copies stand, because a later stop lands nearer an earlier one than a step. The way each copy faces is a short arrow on the floor, and a route that has to agree with the arrows as well as the distances follows the walk past the turn: on walks turning 70° between stops, facings read to 15° return the order 94 times in 100 where positions alone reach 59 — and say which end came first. But only once the arrows are undone onto the floor with the horizon the copies supply. Read straight off the page, an arrow pointing away from a standing eye is drawn nearly flat, and the same facings return 82.
The real instrumentA circle on the ground names the column and the focal length at once
The picture of a circle lying on the ground meets the horizon in two complex points, and for a level camera they are exactly the principal point's column plus or minus i times the focal length. So one circle does what two rectangles turned apart do, with no turn to choose: a manhole cover 1 m across, 8 m off, marked at 24 points to half a pixel, reads the column to 14 px and the focal length to 1.5 per cent, as two rectangles twenty degrees apart do — wherever it lies across the picture. It pays for distance as the square: the same cover at 24 m is out by 128 px. And the building beside it is worth its horizon and nothing more.
Mirrors that are not camerasTwo mirrors nearly facing bend the corridor two ways, and each bend runs out at its own edge
Between two mirrors that lean and are also turned a little about the vertical — the barber's shop, seen past the shoulder — the corridor of self-images steps down by the sum of the leans and sideways by the sum of the turns on every round trip. Its count is not a reading of the angle between the mirrors: each bend runs out against its own edge of the glass, and the corridor ends at whichever does first. A quarter of a degree between the planes gives 22 images when it is all lean, 21 when it is all turn and 25 when it points between. A photograph reads both bends from how the images step, with no reference at all, and needs gravity only to share the lean between the mirrors and the room only to share the turn.
The second projectionA rule that weighs two walkers spares the one serving the worse would starve
A projector following two walkers can share one layout between them by serving whichever is worse-served. Weighed instead — the most detail in all, the most of both walkers' shares of their own best, or the largest of the smaller share — the shared layout gives up a little of the worse walker's gain and changes who pays for it. Walking two and five metres from the wall, serving the worse gives the near walker 11.7 per cent over the even picture and takes the far one below the even picture, out of resolvable detail in a fifth of the frames; keeping the smaller share largest gives the near walker 6.8 per cent and leaves both at least 61 per cent of their own best, never below what an eye resolves. The most detail in all is the worst rule of the four: crossing, it serves the worse walker 9 per cent below the even picture.
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.