The thread: What a projection destroys — page 3
The dish no outline reaches
An outline is a pair of numbers per direction and nothing more, so a shape built from outlines has two errors that behave differently. The part outside the convex hull falls as one over the square of the view count. The part inside a concavity is the same area at four views and at a hundred and twenty-eight, because no pair of supporting lines ever reaches into a bite.
What survivesAn angle is a cross-ratio
A projection destroys angle, which every account of perspective says and a direct measurement confirms. It destroys it only in the sense that the picture no longer carries it for free — because an angle is a cross-ratio taken against two particular points, and once those two points are located in a photograph the world's angles can be read off it directly, with no rectification, no camera, and nothing measured but which lines pass through which points.
The other systemsThe view that makes a line a point
Descriptive geometry's first drill is to look at an edge from a direction perpendicular to it, so it draws at true length, and then along it, so it draws as a point. Both are the same identity — the imaged length is the true length times the sine of the angle to the ray — and the sine holds to 1e-12 across the whole sweep of directions.
Systems that kept the measureThe same person, twice on one panel
A panel showing one figure at four moments is geometrically the least strange thing in this field — one camera, one floor, and every pair of copies meeting the horizon to 8.5 × 10⁻¹⁴ px. What the picture withholds is the order, and four copies admit twenty-four readings, and a reading convention supplies 4.6 bits from outside the marks. Enlarge one figure by six per cent and the horizon test that passed the panel catches it at 36 px.
Through water and glassA straight stick in water is a kink and a curve
The bent stick is described as one kink at the surface. Traced point by point, the image two level eyes see leaves the surface 14.96° off a stick leaning 30° — the same whichever way it leans — and then keeps turning, by 1.11° when it leans away from the eye and 7.62° when it leans toward it. A photograph from the eye shows the kink and almost none of the curve, 1.63 px over 166 px, and when the stick leans straight toward or away from the eye it shows neither: the picture is one straight line.
What each system gave upThe tenth row has neither
A crossed-slits camera divides by depth in both page directions and its rays miss any common point by 0.354 m. Put on the comparison table it prices 38.8% on length, 69.9% on area and 58.2% on angle against a pinhole's 39.5%, 70.8% and 60.9%, and its midpoint drift is 15.6% — the pinhole's own figure. It keeps a true scale in no direction at all, and it bows a straight run of ground by 1.11 px, which no row with a centre does. Giving up the point buys nothing and costs a third thing besides.
What survivesThe two points a picture hides
The stratification rung says a photograph of a plane is projective, becomes affine when the vanishing line is known, and becomes metric with one more fact. The one more fact has a size and a shape: it is two points, complex conjugates on the vanishing line, and a rectification built from those two and nothing else returns every world angle to a tenth of a picodegree and every ratio of lengths to five parts in a quadrillion — and refuses to name a length.
The other systemsNothing moves when the object does
Slide a box 3.2 m across the world and its parallel drawing is the same drawing translated 131.5 px — every edge the same length to 4e-14 px. The perspective drawing of the same move changes its edge lengths by 87.2%. One family's pictures depend on where a thing is; the other's do not, and almost everything the two families disagree about follows from that.
Mirrors that are not camerasTwo mirrors make one turn
Reflect a point in two mirrors meeting at 36° and the images arrive nine at a time, every one of them on a single circle about the line where the mirrors meet, to 4 × 10⁻¹⁶ m. The rule taught for the count — three hundred and sixty over the angle, less one — is right at six of nine angles tried and wrong at the rest, because it is a rule about angles that divide a half turn and it is quoted for angles that divide a whole one.
Systems that kept the measureSize 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.
Through water and glassA wedge of glass turns the camera behind it
A pane with parallel faces moves every point and no direction, so the camera recovered through a window is the camera that took the picture. Tilt one face 2° and every direction turns, by 1.04° on the axis and 1.67° forty degrees off it. The best rotation of the frame, 1.14°, still leaves 0.94 px, and no homography does much better, so the picture is no longer a projection from the camera's centre. The camera recovered from three vanishing points through the same glass is turned 3.37° — three times as far — because vanishing points lie where the glass bends most.
The second eyeAn epipole in the picture leaves a blind disc
Step a camera half a metre straight forward and the image of the other eye sits in the middle of both pictures. Around it lies a disc where one pixel of reading costs a tenth of the depth or more — 20 px across a surface 2 m off, 247 px at 16 m — and at its centre no depth is recovered at any range.
What a machine computesAlong a line of constant depth the page is affine
Stepping a texture by a constant amount per pixel is wrong across a receding floor and exactly right along any line of it that stays at one depth — and on every plane those lines run parallel to its own vanishing line. Turn a 120 px span 1° away from that direction and it is 0.79 px out; roll the camera a hundredth of a degree and a floor drawn to 30 m is out by 0.69 px on its worst scanline.
What each system gave upThe exclusion is two conditions, not ten rows
Ten systems have been measured and none has both a centre and a true measure. Ten is not a proof. Swept across the whole two-slit family — eighty-one members, both parameters run out to infinity — a centre appears at exactly the eight members whose slits sit at one finite distance, a measure at exactly the seventeen with a slit at infinity, and at none of them both. The case the two conditions appear to leave open is closed by the rays themselves: one divide is a slit, and eight page points use 6.02 m of it.
The other systemsThe true shape of a cut
A plane through a box makes a hexagon of 2.8996 m². The front view draws it at 1.9966 m² — the true area times the cosine, 0.6886 — and gets its corners wrong as well, the worst by 15.60°, because a foreshortening scales one direction and not the other. The area is recoverable with one number and the angles are not, which is why a section gets a view of its own.
Surfaces that are not flatThe floors that unroll
A ridged floor curves visibly and can be laid flat without stretching anything — 7.4e-9 of strain across the patch. A dished floor curves less and cannot be laid flat by any means whatever. The difference is one number, Gaussian curvature, and it is the number Gauss proved no bending can change: 0 for the ridge, 0.0144 per square metre for the dish, and no cleverness in the flattening touches it.
Systems that kept the measureA picture in bands
A register picture stacks its scene in horizontal bands, each with its own ground line and every figure drawn at one height. The feet line and the heads line of any pair are then parallel to the arithmetic floor — 0° against 15.4° in a photograph of the same figures — so the picture has no horizon anywhere in it, and what a reader recovers is an ordering with no metre attached.
The second eyeThe two pencils keep one number
Four lines through the image of the other eye in one picture, and the four epipolar lines they become in the other. The angles between them change by up to 11.4°; their cross-ratio is 3.012836 on both sides, to eight parts in a trillion. Three pairs of lines fix the map between the pencils, and the fourth is predicted to a third of a billionth of a pixel.
Drawn confidentlyA cylinder has two different ends
The two end circles of a cylinder image as two different ellipses — 14.4° apart in the direction of their major axes and 0.918 against 0.839 in aspect — and the outline's straight sides touch neither of them where its major axis ends, missing by 28° round the ellipse and 35 pixels. Walking the cylinder away removes the difference between the ends and does not remove the offset of the touch, which settles at 7.3° off the principal ray and at nothing on it.
The eye that movesA seam breaks direction, not size
An eight-metre road crossing the first join of a three-station landscape is drawn 215.4 px wide under either band's rule, identically, and a six-metre post 161.5 px tall under either — the eye's height cancels out of any size taken at one depth. What does not cancel is where those sizes sit. The road's edges are turned 23.2° from each other and the post's foot lands 64.6 px out of place, and a painter butting two bands can absorb the offset and can never absorb the turn.
What survivesWhat a flat map leaves alone
A projectivity of the plane is eight numbers in a matrix, and reading them tells a reader nothing. What it does is decided by its fixed points, and there are exactly three cases: three isolated fixed points, or a whole line of them with one point off it, or a whole line of them with the point fallen onto it. The middle case has five numbers instead of eight and every point slides along a line, and it is what most of the maps this site builds turn out to be.
Surfaces that are not flatThe third column is area
This field has measured what each picture surface does to straight lines and to shape. Both are questions for somebody looking at the picture. Somebody counting in it wants a third column, and the same projections have been returning it all along without anybody asking: the equal-area fisheye holds a square degree at one printed area to 8e-8 across 80° off axis, while a flat plane inflates it 191-fold.
Constructing a viewSeven is not a power of two
Halving a receding depth by diagonals is exact, and every book gives it. Halving reaches a half, a quarter, three eighths — and never a third, however many times it is spent, because no power of two is divisible by three. There is a construction that reaches every whole fraction, it costs three lines rather than a stack of quadrangles, and the extra ingredient is not a measurement.
Systems that kept the measureThe pond with its trees laid flat
An Egyptian garden pond is drawn in plan with its trees rotated outward about the bank they stand on. A rotation is an isometry, so every length in that drawing is exactly the length it is in the garden — zero error, not a small one. What is spent is the angle between any two faces, which reads 180° across every hinge and is 90° in the garden, and the four walls admit sixteen assemblies, so a reader supplies four bits to fold it back up.