Visual Plane
every picture states the point it is correct from
Concept

skew rays — where it appears

Rays that pass one another without meeting, whose closest point is a fit rather than an intersection. They are the ordinary case rather than the exception, because two measured rays never quite meet, and the miss is what says whether the measurement is any good.

Named by 10 essays across 5 fields — each of them below, with the objects they name alongside it.

eyethe best fit — no ray goes through itno single viewpoint — the rays miss by 2.46 mmover 20 cm of a 2.00 m ball

A curved mirror has no eye

A flat mirror is a second camera — reflect the eye in the plane and every line of sight passes through the reflected point, to 2.8e-12 mm. Curve the mirror and the point is gone. Over 20 cm of a mirror ball two metres across, the lines of sight miss their own best-fitting point by 2.8 mm, and by 52.9 mm on a ball half a metre across. What replaces the eye is not a worse eye; it is nothing.

mirrors · Curvedmirror midpoint — 2.77 mm gapfrom the left eyefrom the right eyegap 2.77 mm at 7.22 mexact marks: 2.2e-16 m

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.

depth · Triangulation 0.0 ms8.3 ms16.7 ms25.0 ms33.3 mslean 1.052° drawn against 1.055° predicted · a still world leans 0.000°no single viewpoint — the rays miss by 33.3 ms of travelthe frame is a stack of projections indexed by row

Every row is a different camera

A shutter that reads its rows one after another images each of them from wherever the camera was at that instant, so a frame is a stack of projections indexed by height — a handscroll with the roll running down the picture. Its rays miss their own best centre by the spread of the eye's track, at a ratio of 0.988, and a global shutter's meet to 2 × 10⁻¹⁶ m.

sensor · Rolling the needle: along the rays' bisectorreprojection minimum, both framesmidpoint, world framemidpoint, changed framemidpoint moved 0.203 mm · minimum moved 1.0e-15 mtruth 15.5 mm along the needle

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.

depth · Triangulation horizonthe pointits shadowcorrect from 22 cm, at 160 mm widerays cross at 28.5° · recovered to 9e-16 m

The lamp is the second eye

One photograph, one lamp whose position is known, and a point's place in space comes back to 9e-16 m — the camera's ray through the point, the lamp's ray through the image of its shadow, and the intersection of two lines. It is triangulation with one of the two eyes replaced by a light, and it degrades exactly like a stereo pair: 5.9 mm of depth per pixel at 39° between the rays, 1 mm at 15.4°.

light · Shadowdepth 050100204060how far the third eye stands from the point, in metreshow far the answer is from the truth, in millimetresnearest in metresleast reprojection error120 noise draws averaged at each station132 vs 34 mm

A third ray is worth what its picture is worth

Three eyes on one point, two at seven metres and one walked back to seventy. The point nearest all three rays in metres is 132 millimetres from the truth and the point of least reprojection error is 34 — the same 34 the near pair gives alone — and the first is pulled 12 millimetres along the line to the distant eye. And arrangement beats count outright — two rays spread over fifty-five degrees beat eight rays inside four, by a factor of 4.4.

depth · Triangulation eyethe stickcorrect from 18 cm, at 160 mm widerays miss by 3.13 mm

The stick a stereo pair puts back

Two eyes side by side reconstruct a submerged stick exactly as the sagittal image — kinked 14.96° — and two eyes one above the other exactly as the tangential one, kinked 9.59°. Roll the baseline between them and the two rays to a point miss each other by up to 3.13 millimetres, past the 2.85 a pixel covers at that range, and the reconstruction is a third stick that is neither — 551 millimetres of a one-metre stick, with its tip at 0.405 metres against a true 0.866.

refraction · Refraction the stickreconstructedfitted straightbaseline rolled 45°residual 0.2 mm

The residual does not warn

Fit a straight line to the stick a stereo pair puts back and the fit looks best exactly where the reconstruction is least supported: the residual is 0.535 mm at a level baseline, where the two rays meet perfectly, and 0.038 mm at sixty degrees of roll, inside the band where they miss by more than a pixel covers. Over the same sweep the fitted line is 285 to 584 mm short of the stick's metre — fourteen thousand times its own residual at worst.

refraction · Refraction 00.50011.502020406080roll of the baseline, degreesmillimetresunweighted residualfloored at a pixelweighted, no floorthe rays' miss, rmsa one-metre stick, the baseline rolledthe weighting runs the wrong way harder

A fit weighted by the miss trusts only the surface

Weight each point of a reconstructed underwater stick by how well its two rays meet, and the fit hands all but a ten-billionth of its trust to the one point where the stick enters the water — and reports a residual of nothing at every rolled baseline. Floored at the reading error, the weighting changes the answer by a few thousandths of a millimetre. And the miss itself, the one honest number, is exactly zero at a level baseline where the stick comes back 285 mm short.

refraction · Refraction 024020406080how deep the point is, cmthird ray's miss of the pair's point (px)one pixelraised 25 mmraised 50 mmraised 100 mmraised 200 mma one-metre stick, 30° into waterzero at the surface, every offset

A third eye sees the water the pair cannot

A level stereo pair looking at a stick in water gets two rays that meet exactly and a stick 285 mm too short, with nothing to warn it. A third eye raised ten centimetres above the pair sees a disagreement of 2.6 pixels at the stick's tip. Asked to agree with the other two as if there were no water, it hides nearly all of that and makes the stick shorter still. Told only that there is a flat surface, the same three eyes find its refractive index and put the metre back.

refraction · Refraction

Named alongside it

The objects these essays reach for when they reach for this one.

TriangulationBaselineResidualRefractionReprojection errorAstigmatismcentre of projectionleast squaresMidpointStereo pairApparent-depthCorrespondence

All concepts