Every essay — page 5
What survives Constructing a view Where to stand Surfaces that are not flat The other systems Light and mirrors Mirrors that are not cameras Measuring from one picture Through water and glass The real instrument The second eye What a pair is for Many pictures at once The eye that moves Systems that kept the measure What each system gave up What a machine computes The rectangle behind the lens The second projection Drawn confidently
Surfaces that are not flat
A cylinder, a sphere, a fisheye. Every one of them is computed here rather than described, and every one is measured on the three things a picture surface can do to the world: bend its straight lines, turn its right angles, and change its scale. No surface escapes all three.
The parallax you cannot shoot away
A stitched panorama's parallax has two halves and they do not behave alike. The one across the seam falls as the sine of half a frame spacing, so more frames buy it off; the one up the frame is the sine of half the frame's own height, and no quantity of shooting touches it. They cross at π over β, which has no pivot error in it at all.
The 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.
A rig is right on one surface
Several cameras looking outward have several centres, and no warp registers all of a scene. The residual is a disparity, so the depth the stitch is computed for has an exact optimum — the harmonic midpoint of the depth range, not the middle of it — and the arithmetic middle costs a factor of 2 z_far over the sum, which tends to two.
Drawn for the cylinder, shown on the cylinder
Four essays in this collection have named the arrangement where a picture's surface and a screen's surface are the same surface, and none has run it. It is exact — a tenth of a millionth of an arcminute — and the exactness is not the finding. Three separate things have to be true at once, and a curved television's own maker prints a sitting distance where the first of them is out by five degrees.
Three 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 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.
A 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.
Where 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.
The 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.
A 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.
The 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.
Shot 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.
Which rule a fisheye obeys, from straightness alone
Four candidate rules for a fisheye lens part by 54.0 per cent at 45° off axis, and a plumb-line fit shown no scene, no camera and no calibration target can still name which one took a photograph — reliably from about 45° of half-field. Below that the four are indistinguishable in the marks, and naming one collapses to guessing.
The eye is a picture surface too
A retinal sphere behind an off-centre nodal point takes every measurement this collection puts to a lens or a screen, and answers all of them. It is the equal-area fisheye to within 110 micrometres of retina rather than the equidistant one everybody draws it as, and a flat picture at its own correct distance leaves the identical arc on it, to 1.4e-14 degrees.
The arcs the five-point construction actually draws
The taught five-point construction draws circular arcs between five vanishing points and instructs a draughtsman to graduate the radius evenly. Read that way, the arcs miss a straight line's true image by up to 3.75 pixels on a 300-pixel disc. Read at the stereographic scale instead, the same arcs are exact to 4.3e-13 pixels — the construction was always drawing one projection, and the taught scale was never it.
The hole a rig cannot fill
A two-lens spherical rig covers every direction between its two lenses and still cannot see 0.470 per cent of the sphere directly beneath it, reaching 12.8 degrees from straight down — its own tripod, standing exactly where neither lens can look. No arrangement of lenses removes it, because it is not a gap in coverage; it is the rig occluding itself.
A projector that is not at the dome's centre
A projector 0.40 of a dome's radius off centre puts its own picture up to 23.6 degrees from where it belongs, and the pre-warp that corrects it is exact for one seat and only one. Two metres from that seat costs 11.5 degrees of the same displacement, wherever the projector itself stands — because the correction never knew where the projector was in the first place.
What a pair is for
Depth from two views is a reciprocal, so a fixed error in what is read maps to an interval that is not symmetric and eventually is not bounded. What a stereo pair can and cannot measure, computed rather than described, including the range past which one pixel reaches infinity.
Depth is a reciprocal
Two eyes measure a shift in the picture, and depth is that shift divided into a constant. So a fixed error in what is read maps to an interval in what is reported that is not centred on the answer, and at forty metres runs sixteen metres nearer and eighty-six further.
The range a pair cannot see past
A stereo rig has a distance beyond which it cannot say "no further than", and the distance is fixed before anything is built. It is the focal length times the baseline divided by the reading precision, and for a human pair of eyes it is fifty-eight and a half metres.
Two rays that do not meet
Triangulation is described everywhere as the intersection of two rays, and two rays in space do not intersect. Read the same two marks to a whole pixel and they miss by 2.77 mm at seven metres, which is a real length and is the part a residual will not report.
A turn of the head is not a step sideways
The textbook says a short baseline makes reconstruction ill-conditioned. Measured, the algebra does not notice — a two-millimetre baseline recovers the courtyard to nine digits from exact marks. What fails is the ratio of baseline to reading error, and it fails without refusing.
A wrong match is not a small error
Move one correspondence of forty-four by thirty pixels and the recovered geometry is wrong for every other point — the typical one by half a pixel, from a fit that was exact to a part in ten trillion. Least squares has nowhere to put a bad row except across all of them.
The depth a pair calls zero
Two eyes verged on a point agree — the same coordinate in both pictures — not on a plane at the fixation distance but on a circle through both eyes and that point. Found by bisection along 121 azimuths and fitted rather than assumed, it is a circle to 0.0000 cm; at 26° off centre it lies 23 cm nearer than a flat wall does.
The midpoint is a choice of ruler
Two photographs do not change when the world is measured with a different ruler, so an answer that belongs to the photographs cannot change either. The midpoint of two skew rays does: a threefold stretch moves it 0.203 mm and a projective frame 1.503 mm, while the point that minimises reprojection error stays put to 10⁻¹⁵ m. Both are 15.5 mm from the truth, which is the part a choice of route does not touch.