The thread: One machine used twice — page 4
A projector in the viewer's eye
A projector paints a wall along its own rays, so from the projector's own position the wall's shape is invisible — exactly, on a cylinder, on a dome and on a plane alike. Move a hand's breadth away and the residual is pixels. The one place a projector can stand and ignore the shape of what it is throwing onto is the eye of the person watching.
Mirrors that are not camerasThe 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.
What a machine computesA point at infinity is an ordinary vertex
Give a vertex a zero in its fourth slot and it stops being a point and becomes a direction — and the projection matrix draws it anyway, through the same multiply and the same divide, on that direction's vanishing point to seven trillionths of a pixel. Slide the eye ten metres and it does not move. Two of them bound a ground that reaches the horizon, where a ground drawn to ten kilometres stops a fifth of a pixel short.
The rectangle behind the lensTurning and travelling blur different worlds
A subject 8 m away crosses the frame at 4 m/s, and the camera keeps it sharp over a thirtieth of a second. Turn to follow it and every still thing blurs by the same 13.5 px, whatever its depth. Travel beside it and the still world blurs as one over its depth — 49 px at 2 m, 1.5 px at 64 m — while everything moving with the subject is sharp at every depth.
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.
Light and mirrorsTwo lamps and one map
A flat object lit by two lamps casts two shadows, and one is the other scaled about a point — ratio 1.2509 here, carrying every point of the first outline onto the second to 1e-15 m. No rotation and no shear is available to it, because a projection between two parallel planes has its axis at infinity. And the ratio is exactly 1 when the two lamps are at the same height, which makes a pair of shadows a measurement of the lamps.
Where to standAn anamorph has one eye
From the design point exactly — a camera's single eye — the floor marks give the design back to sixteen decimal places. A head carries two eyes 63 mm apart, and neither of them is the design point. The difference between the disparity the floor gives and the disparity an upright board would give runs to 47 arcminutes, against a stereoacuity of a few tens of arcseconds. This is why pavement paintings are photographed.
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.
The rectangle behind the lensA dolly zoom is a step and a zoom, and they meet at one depth
Step 1.5 m toward a subject 5 m away while shortening the lens to hold its size. Every mark moves along the line from the centre of the picture, to a ten-trillionth of a degree — outward if nearer than the subject, inward toward a limit if further, and not at all on the subject's own plane. The step and the zoom each move everything one way; the dolly zoom is where they cancel.
What a machine computesA shadow map's texels land by two distances and two cosines
A renderer finds its shadows by taking a second picture from the lamp and storing a depth in every texel. Each texel reaches the screen through the surface it falls on, and how many pixels it covers there is a closed form — two focal lengths, two distances and two cosines. From a lamp beside the eye every texel lands at 0.79 px; from a lamp 40 m ahead facing back, the same map lands texels of 8.69 px on the floor 5 m out.
Through water and glassThe wedge recovered with the camera
Admit the glass into the model and the fit finally has something left over. A two-degree wedge is invisible to a reading confined inside eighteen degrees of the axis — the fit calls the whole displacement a rotation of the camera and is right to — and by fifty-five degrees it comes back to 0.07 of a degree. What the picture does not separate is the ordinary glasses — assuming an index of 1.50 for a true 1.52 costs one per cent in the angle and 0.07 pixels of residual.
What survivesThree constructions, one map
A shadow cast on a floor, an anamorph painted on one, and a reflection in a mirror are usually treated as three different subjects, each with its own derivation and its own figure. Decomposed into their fixed points and lines, three of them are the same kind of map — a central collineation with a line of fixed points — and the fourth, a rectification, is not. That difference separates changing where a picture is seen from, from changing one picture into another.
Light and mirrorsA wire with a corner in its shadow
A bent wire has no corner anywhere on it, and its shadow has one. The lamps that do it are not a coincidence — they are a surface in the room, two-dimensional, made of the wire's own tangent lines, and a lamp being carried across the room passes through it.
Mirrors that are not camerasTwo mirrors are three cameras
A photograph with two mirrors in it holds three views of the scene from three places, at baselines of 2.90, 3.10 and 2.26 metres. Two of the three pairwise geometries are mirror pairs and are skew; the third is a rotation by twice the angle between the mirrors, and it is not new evidence — five numbers read off the print rebuild it to 9.4 × 10⁻¹⁴ pixels, where a general three-view arrangement needs eighteen.
What survivesThe conic a circle becomes
A circle photographed is an ellipse, or a parabola, or a hyperbola, and which one is decided by a single incidence: whether the circle reaches the plane through the eye parallel to the picture. Not the lens, not the tilt, not how far away it is. The discriminant of the image agrees with that one test at every point of a sweep, and at the crossing it is zero to 1e-13.
Light and mirrorsThe edge of a shadow is drawn on the object
The outline of a cast shadow is the image of a curve, and the curve is on the caster. It is not painted there: it slides when the lamp moves, it is not the outline the camera sees, and the two coincide only in the arrangement where no shadow is visible at all.
Surfaces that are not flatThe pivot that is not the eye
A camera bolted to a tripod turns about the screw under its baseplate, and the light crosses somewhere else. Every ray of the panorama that results passes the pivot by e sin γ — the offset times the sine of how far off its own frame's axis the ray points — so the picture has a radius where a projection would have a point.
Light and mirrorsA hole is not preserved
The shadow of a connected object is connected — always, at every lamp position, and for a reason with no geometry in it. A hole survives in neither direction: a ring's shadow closes up at a computable tilt, and an object with no hole in it casts a shadow that has one.
Light and mirrorsA shadow across a second object
A straight edge held in front of a lamp defines one plane, and the shadow's boundary is wherever that plane meets something. That turns a picture of a shadow into a measurement: a camera ray and a known plane meet in one point, and the object the shadow is falling on comes back out.
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.
Light and mirrorsThe ball stands at a focus
A ball on a floor casts an ellipse, and the point where the ball touches the floor is a focus of it — not near a focus, on one, to two parts in ten thousand million million. That is Dandelin's theorem arriving somewhere nobody puts it, and it is the only thing about the lamp that the shadow gives away.
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.