The thread: What a projection destroys — page 5
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.
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.
The second eyeSteering swings a camera too little to see ahead
A vehicle yaws about its rear axle, so a camera mounted ahead of the axle is swung sideways whenever the vehicle steers — a real baseline, where a turn alone gave the blind point ahead nothing. Half a degree of yaw swings a windscreen camera 1.3 centimetres and a bumper camera 3.1. Straight ahead at eight metres a pixel then costs 96 and 40 per cent of the depth. The cost falls as one over the arm times the yaw, and a usable 10 per cent needs 7.2 metre-degrees: two degrees of steering every frame on a bumper. Ordinary lane-keeping gives a fraction of that.
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.
The other systemsNo view draws a curved plate true
The auxiliary view is descriptive geometry's answer to a foreshortened plane — turn until the plane is parallel to the paper and it draws at true shape. A bent plate has no such direction and a dished one has none twice over: the best view of the first is out by 1 − cos w and the best of the second by 1 − cos²w, worse by exactly 1 + cos w, because its normals need two parameters rather than one.
The second eyeA drift across the lane closes the blind point, slowly
A car that wanders twenty centimetres either side of its lane every four seconds moves its camera twenty-eight centimetres sideways in a second, far more than steering ever swings it. Looking back over that second, the road twenty metres ahead is known to 23 per cent a pixel; over four seconds, to 10.5. The error falls as one over the wander and grows with its period and the car's speed, so a car that keeps its lane well, drifts slowly or drives fast leaves the point it is heading for nearly as blind as before.
The other systemsThe drawing and the development
A bent plate gets two flat pictures on the same sheet and each is exact in what the other loses. The parallel drawing keeps the generators at one scale and stretches the arc over a factor; the development keeps every length on the surface and keeps nothing of the shape in space. Neither is the plate and the pair of them is.
The second eyeA stereo pair and a lane wander add as two readings of one depth
A pair of cameras 0.3 metres apart across a car ranges the road twenty metres ahead to 15.8 per cent a pixel from one moment's two pictures — and to 4.6 per cent from the last second's sixty, because every picture is another reading. Add the car's ordinary drift across its lane and the second's pictures give 4.5: exactly what the pair's look and a single camera's wander would give as two independent readings. The two baselines neither waste each other nor multiply, while the pair's is the larger; only a narrow pair on a car that drifts half a metre does better than the rule, by a quarter.
What survivesA line is a space of its own
Most of what is said about projective geometry in pictures is said about the plane. One dimension down there is a smaller object with a complete theory: a point of a line is one ratio, a map of a line is three numbers, three pairs fix it, and the cross-ratio is not merely an invariant but the only one — which is a claim that can be made to fail.
The other systemsA drawn fold has a phantom
A Necker cube has two readings and so does a drawn fold, and the fold's second reading is not the supplement of the first. The drawing fixes each plate edge's component in the picture and leaves its component along the ray free up to a sign; a reflection identifies two of the four sign pairs, so there are exactly two plates — and a hundred-degree fold reads as eighty-seven as well.
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.
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 second eyeA car ahead costs a pair its speed, and a car that may drift costs a single camera everything
A stereo pair's last second of pictures ranged the still road twenty metres ahead to 4.5 per cent a pixel. A car twenty metres ahead is not still, and each old picture saw it somewhere else. Read together with its speed, a car keeping pace is ranged to 4.5 per cent all the same — it stood twenty metres from every picture, and reading its speed spends exactly what that was worth. A car that may also drift across the lane costs the pair a fifth more and takes a single wandering camera from 38 per cent to 185. A braking car caps the useful look at about a second.
Drawn confidentlyThe rims carry a part's taper, as finely as they are drawn
A turned part's two rim ellipses are enough to read its taper: the one axis that makes both of them circles lands on the true vanishing point, and looked at down that axis the rims give a one-degree taper back to the sixth decimal. Drawn by hand to half a pixel they give it to ±0.4°, a part pointing away from the eye to ±0.15° and one lying across the view to ±3°. The rims' sizes alone cannot do it, since a turn shrinks the far rim exactly as a taper does. And rims drawn with ellipse guides carry no taper at all — the guide's step reads as one.
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.
Surfaces that are not flatWhere a surface spends its pixels
A picture surface is a budget before it is anything else, and the six named ones distribute the same marks over the same directions quite differently. The flat plane lays 25.0 times as many on a square degree at the edge of a 70° field as on one at the centre; the equal-area fisheye is flat to 8.3e-6 per cent.
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 second eyeA growing car times the gap without its width
A pair of cameras ranges the car in front from its disparity, and reads how soon the gap will close only as well as it reads the car's speed: to 39 per cent a pixel from a second's pictures, for a car 20 metres ahead closing at 5 metres a second. One camera watching the same car grow reads the same four seconds to 8.8 per cent, without knowing how wide the car is. The car's own width is a baseline six times the pair's. The distance needs the width, and the pair, reading both, measures the width as well.
Drawn confidentlyA turned part drawn from the side keeps its taper in its sides
A turned part's two rims read its taper well when the part points away from the eye and lose it as the part turns across the view: drawn to half a pixel, a one-degree taper lying across the view reads ±3.1°. Its two sides do the opposite. Fitted together with the rims as one tapered part, the same drawing reads the taper to ±0.03° — a hundred times better — and a part seen from the side, the view it is usually drawn in, shows a taper of six hundredths of a degree.
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.
The second eyeThe road under a car ahead gives its growth a distance, while the road stays flat
One camera that knows its own height reads the row where the car in front meets the road, and that row is the camera's height seen from the car: a baseline of 1.3 metres standing on the road, read in every picture. With the car's growth it ranges a car 20 metres ahead to 0.80 per cent a pixel, six times better than a stereo pair's disparity, and reads the car's width to 0.74. A camera height a few centimetres out scales the distance and leaves the time untouched. A road that starts rising by one degree between the camera and the car moves the distance by up to 24 per cent, and neither horizon puts it right. The lane's own edges bend where the road does and read the rise back.