Sensitivity — where it appears
Named by 16 essays across 9 fields — each of them below, with the objects they name alongside it.
A lens destroys the invariant
The cross-ratio is the one thing a projection preserves, and nearly everything checkable about a photograph is checked with it. A lens returns it one and a third per cent out where the pinhole is exact to fifteen digits — and the height error that follows tracks a quantity nobody would guess.
Measuring a room off the page
A perspective picture of a floor has to be rectified before anything on it can be measured, and the rectification is a fit that amplifies the marking error. An oblique picture of the same floor is already rectified — the page is the plan, at one scale, and a ruler on the paper is a ruler on the ground.
How wrong a measurement from one picture can be
The formula divides by a difference of two nearly equal numbers when the object is tall, which looked like the instability and is not. Measured, a taller object is recovered better and a more distant one worse — half a per cent per pixel at six metres and eleven per cent at a hundred and twenty. The argument that was wrong is as much the finding as the curve that replaced it.
A shadow can be un-cast
A shadow looks like a lossy record — a flattened smear with the shape half thrown away. It is nothing of the kind. The map from the occluder's plane to the floor is a plane projectivity, so it has an inverse, and the outline comes back out of its own shadow exactly. What breaks it is not the light and not the shape: it is the floor not being flat.
One plane is nearly free
The near and far planes enter a depth buffer's precision through 1/near − 1/far, and one of those reciprocals is enormous. Pushing the far plane out by a factor of a thousand costs a tenth of a per cent; bringing the near plane in by the same factor costs a factor of a thousand — and an infinite far plane is the limit of the first rather than a separate case.
The seats a screen will accept
Collect the seats whose picture is within a pixel of the one intended and the result is a solid — half a cubic centimetre in front of a curved desk monitor, a litre in front of a curved television. Ten times the tolerance is a thousand times the room, which is the pavement anamorph's own law arriving on an object that has nothing else in common with it.
Square to the camera is the worst mirror
A mirror pair's baseline runs along the mirror's normal, so a mirror facing the camera puts the second eye directly behind the first — the forward-motion arrangement, with the epipole in the middle of the frame and the rays to a mark crossing at 23°. Turning it forty-four degrees opens that to 65° and cuts the worst depth error threefold, and the number to watch is not the angle but where the reflected lens sits on the print.
How well the floor has to be known
“Supply the surface” is an instruction with no number in it, and an instruction with no number is a wish. Here is the number: the error in the recovered design is very nearly proportional to the error in the assumed curvature — the constant varies by 1.4% across a twentyfold range — so about nine parts in a thousand of the curvature buys one millimetre on a design 1800 mm wide, and 40% out costs 43.5 mm.
Size that means rank
In a great many pictures the drawn height records importance rather than distance. That is a decision rather than a mistake, and it can be caught with a straightedge by carrying one figure's height across the room by the taught construction and see where it lands. A tenth of rank in the picture already misses the drawn head by 21 px, the miss is exactly linear in how much rank is there, and the whole test needs two references and no arithmetic.
The corner that answers every eye
Three mirrors at right angles compose into the point inversion, so a ray entering leaves antiparallel to itself whatever direction it arrived from — a picture surface with no correct viewpoint because every viewpoint is correct. Tilt one face half a degree and the worst returning ray is out by exactly one degree, twice the error; the best is out by a twelfth of it, which is why a tolerance quoted from one measured ray is a statement about that ray.
An angle on the ground
An angle needs no length at all — it is scale-free, so the one thing a single view cannot give is not the thing an angle wants. What it wants instead is the plane's shape, and the difference is measurable — assume the reference rectangle's aspect a quarter wrong and the angle moves 11.5°, a length running into the picture is out by exactly that quarter, and a length along the reference's own direction does not move at all.
The answer is an ellipse
A mark read with a round error does not come back as a round region on the ground. The ray is grazing, so a pixel across the picture is worth a little and a pixel up it is worth a great deal, and the recovered point's uncertainty is an ellipse pointing at the camera — 5.07 to 1 at eight metres from a camera 1.62 m up, which is the depth over the height. Propagated and sampled agree to 0.5 per cent, and a ray that is not grazing gives a disc.
Which reference to measure from
Given four candidate scale bars in one photograph, the best is not the longest and not the nearest — it is the longest in the picture. A five-point-two metre bar near the horizon is the longest thing in the scene and the worst reference in it; a four metre bar close to the camera is the best. Walk one bar outward and the term it controls falls as one over its length in pixels, with a fitted exponent of −1.08.
The floor a better camera cannot reach
Sweep the marking error from four pixels down to a hundredth and the measurement's error falls thirty-fold and then stops — at 6.0 per cent, which is exactly the six per cent the reference's assumed shape was wrong by. With the closure exact the same sweep keeps falling to 0.06 per cent. The crossing is at half a pixel, and it can be computed before the photograph is taken, which makes it a decision about equipment rather than a discovery about it.
A fold names the height
A pavement anamorph’s marks fix where the reader must stand and leave how tall they are entirely free — every height explains the marks exactly, to the last bit. Put one crease in the floor and the freedom is gone, because two degrees of fold makes a ten-centimetre error in the height leave six tenths of a millimetre, and a right angle makes it nine.
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.
Named alongside it
The objects these essays reach for when they reach for this one.
error propagationConditioningsingle-view metrologyHomographyRectificationCross-ratioDepth uncertaintyHorizonIdentifiabilityinstrument limitReference lengthResidual