The thread: The surface is a choice — page 3
No 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.
Light and mirrorsThe floor is a choice of coordinates
Four rungs of this field have measured what a curved floor costs a shadow reading, in millimetres. It costs nothing. A shadow mark sits on the ray from the lamp through the occluder's edge, and where along that ray the floor caught it is a fact about the floor alone — so un-casting in rays returns the occluder to three parts in ten thousand of a millionth of a millimetre on a plane, a dish, a ridge and a step alike, while the same shadow read in the plan is wrong by up to 216 millimetres.
Surfaces that are not flatThree conditions, and three prices
A matched picture needs the seat, the horizontal scale and the vertical law all at once. Each is broken alone here with the other two held, and all three turn out to be linear in the mismatch — the unforgiving case, with no margin at all. The vertical law costs a third of an arcminute on a television and eighty-four on a dome, because it is the difference between an angle and its tangent and that difference is cubic in the picture's vertical field.
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.
Where to standA design that lands in two rooms
Cast a design down a corridor with a doorway in it and four per cent of the picture goes through the opening and lands on a wall three metres beyond, at 9.4 metres from the eye against the end wall's 6.4. The picture is continuous across the edge of the doorway and its scale jumps by half again, which is a corner anamorph's discontinuity with a gap in the middle of it instead of a fold.
Surfaces that are not flatThe surface a screen wants
Six picture surfaces laid on one screen and viewed from the seat its curvature names, each with its own extents fitted so the ranking is about shape rather than scale. Each screen's own surface is exactly right on it and nothing else is — and on a curved television the runner-up is a fifth of an arcminute behind, which nobody can see.
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.
Where to standThe stretch decides the band
A design band chosen by geometry — the rays that meet the object — includes rays that graze along it, and a grazing ray lands two design points twenty times further apart than the design says. Cap the stretch at four and a bare floor keeps 55 per cent of its design, a corner 80, and the top of a descending flight all of it.
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.
Light and mirrorsA floor is read along curves
Whatever a shadow says about the floor it landed on, it says only where the shadow is — and a shadow is a curve while a floor is a surface. Shadow curve length grows exactly linearly in the number of lamps, by a fitted exponent of 0.999, and the fraction of floor within two centimetres of one grows more slowly at 0.94, because the curves begin to overlap. At thirty-two lamps, seventy-one per cent of a nine square metre patch has still never had a shadow on it.
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.
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.
Surfaces that are not flatThe horizon's shape belongs to the surface
The horizon is one great circle of directions whatever draws it, and at zero tilt all six named surfaces draw it straight. Tilt the camera and they separate — and the cylinder, not the equirectangular surface, is the one whose horizon is exactly a cosine, to 9e-16 against 8.3e-3.
Surfaces that are not flatA pole is a line
An equirectangular surface sends one direction to a whole edge, so an 8° cap of sky at the pole takes 4.50 per cent of the marks against a 0.49 per cent share of the world. The worst singularity is not the pole at all — the equidistant fisheye's antipode costs ×17.9 — and only the cube map, which never holds a sphere in one chart, is bounded.
Surfaces that are not flatThe kink at a seam
A cube map is six flat pictures, so every great circle is drawn exactly straight inside a face — to 3e-15 of its chord — and breaks at the join. The break is a kink and not a bend, it is exactly zero at a seam's midpoint whatever the slant, and it is bounded by 2·atan(½) = 53.130° at the corner.
Surfaces that are not flatShot on one surface, shown on another
Two picture surfaces are two charts of the same pencil of rays, so a reprojection between them is a change of coordinates and loses no geometry at all — bit-exact at all 408 sampled directions. What it costs lies elsewhere — 70 per cent of the source has nowhere to go, and the target wants ×5.49 the marks at its edge.