least-squares intersection — where it appears
Named by 15 essays across 8 fields — each of them below, with the objects they name alongside it.
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.
Where parallel lines meet
They meet at a point that is not in the plane, and the horizon is the line made of all such points. Treating that as bookkeeping misses what it buys — a vanishing point becomes an ordinary point of the picture, findable from the drawn lines, checkable, and enterable into any calculation the others are.
The centre a scroll does not have
Fit a common point to the rays of one section of a handscroll and it misses by metres. The miss is not a residual to be tightened — it is exactly the standard deviation of the eye's own track, it grows linearly with how much is unrolled, and it goes to zero only for a section of no width.
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.
The lamp, out of the picture
Two posts and their shadows put the light's position in a photograph, exactly, with nothing given but the drawn lines and the camera's own horizon. Two posts. One gives a residual of zero and an answer that is a whole one-parameter family — the sharpest counter-example there is to the idea that a small residual means a right answer, met again in a new field.
The wall under the paint
An anamorph's design is a set of rays from a known eye, fixed before any paint is applied — so the marks are a calibration target whose rays are known exactly and whose shape is not. One ordinary photograph supplies the second ray for each mark and the wall comes back at the arithmetic floor, on a dished floor, a ridged one, and a floor with a step in it. What it says between the marks is nothing, by the sagitta.
A mirror ball does not know its size
The outline of a mirror ball in a photograph gives the ratio of its radius to its distance and stops there — a ball three and a half times bigger, three and a half times further away, draws an outline identical to the last bit. What breaks the tie is a point of the room, and only a near one: the sensitivity falls as one over the room's distance, so a mirror ball photographed against a landscape has no recoverable size at all.
One surface, two images
A water surface reflects what is above it and refracts what is below it in the same photograph, from the same plane. The reflected half has a centre of projection to 1 × 10⁻¹⁴ m and every theorem about central projection applies to it; the refracted half misses its own best-fitting point by 28.6 mm and none of it does. And the landscape manual's rule for drawing a reflection turns out to be the epipole placed at infinity, which is why it costs nothing at zero tilt and 11.5 px at twenty-two degrees.
Two pictures of a ball
Two outlines give two tangent cones, and where their axes cross is the ball's centre — with the radius following from either half-angle. There is no correspondence problem, because a ball's outline in one picture and its outline in another are guaranteed to be the same ball; and there is a degeneracy, which is the two eyes and the ball in a line, and the solver refuses it rather than returning something plausible.
A ball of water has no eye either
A flat interface is not a projection through a centre and misses by ten millimetres. A sphere of water misses by more than that on a ball the size of a plum — 1.3 mm on a fifty-millimetre radius, and the axis crossings spread over 18.6 mm at seven tenths of the aperture. But at two per cent of the radius the same fit returns 24 nanometres, so a ball does have a centre — one at zero aperture and none by the time it is gathering any light.
The 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.
A 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.
How many lamps a drawing has
The shadow field recovers a lamp by intersecting drawn lines. Two lamps make that a partition rather than an intersection — and two centres fit any bundle better than one, on a one-lamp drawing as readily as on a two-lamp one, so a count is a decision that needs a noise level before it exists. A criterion built on a penalty instead of a noise level returns four.
The drawing does not run out of lines
Every post supplies a line to every lamp, so a drawing of five posts offers ten lines to two lamps and twenty to four — the unknowns and the constraints grow together and two posts fix any number of lights. What runs out is the partition, whose margin falls from 251 pixels to six as the share of lines assigned correctly falls from all to just over half.
The lamp a low shadow cannot locate
Five posts, their shadows, and the line from each foot through its own shadow's tip meet at the lamp standing over them — at 52 degrees up that meet moves 0.11 m for half a pixel of marking error; at 11.5 degrees, 0.64 m. The two eigenvalues of the same pencil of lines part by a factor of 8849084 across the sweep, and half a pixel becomes more than a metre of lamp below 14.4 degrees.
Named alongside it
The objects these essays reach for when they reach for this one.
centre of projectionConditioningResidualDemonstrationPoint lightCausticleast squaresNot a projectionReconstructionReflectionscale ambiguityTriangulation