Every essay — page 5
What survives Constructing a view Where to stand Surfaces that are not flat The other systems Light and mirrors 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
The eye that moves
A handscroll is drawn by an eye that travels, imaging one line at a time — orthographic along the roll and perspective across it, so a mile of river holds its scale while a single pavilion still recedes. Its rays miss their own best centre by metres, and the miss is exactly the length of track you unroll.
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.
A map along, and a picture across
The midpoint of a segment lying along a scroll's length images to the midpoint of its image, exactly. The midpoint of a receding one lands 22% of the way off. One projection, two answers — and the direction that keeps measure keeps something a perspective picture never offers.
A scroll is not a panorama
Both draw straight world lines as curves, and one of them is a projection. A rotating eye keeps its centre exactly however far it turns; a translating eye has none at all. Curvature and centrelessness are independent properties, and conflating them is the standard mistake about both objects.
The rectangle behind the lens
A focal length is not an angle until a piece of silicon of a stated width is named. And that silicon is a grid of samples that need not be square, read over an interval rather than at an instant, mostly a row at a time — so a frame is neither one place nor one moment, and the site's own round trip cannot tell.
A focal length is not an angle
Fifty millimetres means nothing until a rectangle is named behind it. The same lens is 39.6° across full frame, 26.6° across APS-C and 8.7° across a phone sensor — and the distance the resulting print is correct from depends on the ratio of the two, so two cameras matched on angle agree exactly whatever their formats.
Stepping closer is not zooming
Changing the focal length leaves the ratio between any two things in a picture exactly alone — to twelve decimal places, at every focal length there is. Moving changes it. Hold the subject's drawn size across a step from 3 m to 1.5 m and the background halves, which is the whole of the shot everybody knows and nobody derives.
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.
A frame is an interval
An exposure is not an instant, so a frame is an integral of projections and every moving point draws a streak. The streak is straight, because the image of a straight path is straight — and its length goes as one over the depth, so two objects at 3 m and 6 m blur by lengths in the ratio 2.000. No single kernel describes the frame.
The pixel that is not square
A camera with two focal lengths is a real thing — anamorphic cinema optics, non-square photosites, a stretched video format. Hand this site's own round trip a picture from one and it returns a focal length 49.6% out, a principal point far from the truth, three independent estimates agreeing to 1e-16, and bundle residuals at the noise floor. Every alarm the site has stays silent.
The second eye
One picture fixes a ray; two fix a point. Everything a pair of pictures determines about the eyes that made them — the image of one eye in the other's picture, the line a match must lie on, and the camera pose recovered from correspondences alone — with the recovery never shown a camera.
The image of the other eye
Two photographs of one courtyard, and in each of them a point that is the other camera. It is computed from forty-four matched marks and nothing else, and it lands on the projection of the other eye to about a billionth of a pixel.
A point is a line over there
Knowing where a mark sits in one photograph does not say where it sits in the other. It says which line it is on, which removes one of the two unknowns and leaves one — and the five lines drawn here meet at a point to within a thirtieth of a billionth of a pixel.
Eight points and the basis they are read in
The linear system that recovers a fundamental matrix is written in whatever coordinates the marks were read in, and pixel coordinates are a bad choice. Centring and scaling them first is worth nothing at a quarter-pixel reading and a factor of thirty at four.
Four cameras fit, and one of them can see
The essential matrix does not determine a camera pair. It determines four, all of which reproject every correspondence exactly, and the thing that picks one is not more algebra — it is the assumption that the photographer could see what was photographed.
Two views give shape and no size
Every pairwise distance ratio in a courtyard recovered from two photographs matches the real one to fourteen digits. The courtyard's actual size is not merely uncertain — it is absent, and a reconstruction three and a half times larger fits the same two pictures exactly as well.
Where to stand
A perspective picture is correct from exactly one point in the room. Every figure here computes where that point is, which turns the oldest complaint about wide-angle photographs into arithmetic.
The point you have to stand at
A perspective picture is a projection through a centre, and scaling that centre's distance to the width the picture is actually shown at gives a distance in centimetres. Shown 160 mm wide, a 40° picture is correct from 22 cm and a 90° one from 8 cm. Nobody stands there, and that single fact explains most of what gets called distortion.
Wide angle is not distortion
A wide lens stretches shapes at the edge of the frame by exactly 1/cos θ — 3% at 28° across, 41% at 90°. Every bit of that is what a correct rectilinear projection must do, and every bit of it disappears if the picture is viewed from the point it was made for. Nobody views it from there.
Anamorphosis is only a viewpoint
A smear on a wall that resolves into a skull when the viewer stands in one particular place off to the side. It is not a trick added to perspective — it is perspective, with the centre of projection put somewhere nobody would naturally stand, and it is the clearest demonstration there is that every picture has such a point.
When the picture surface is not flat
A flat picture plane keeps straight lines straight and stretches the edges without bound. A cylindrical one spreads the stretch evenly and bends every straight line that is not through the axis. Neither is the distorted one — they are answers to different questions, and the choice decides what a wide view can be.
An anamorph at true size, on paper
An anamorph is the shadow of the intended picture, cast from the reader’s own eye. Every claim on this site is quoted against an assumed display width because the site cannot know how wide its figures really are; this one is not, because it ships a sheet in millimetres and states where to put an eye.
The cylindrical mirror unbends it
Put a mirrored cylinder in the middle of the sheet and the light path from the eye to the paper bends once. The map that results is not a homography and not a projection in the plane sense at all — it varies its scale by a factor of seven across the design, which is why the smear is unreadable and why the mirror can put it back.
Standing in the wrong place
A picture read from twice the distance it is correct from depicts a scene twice as deep — and not one mark on the paper moves. The error is invisible in the picture, which is why it survives everywhere.
A floor anamorph is three numbers
An anamorph is described everywhere as a picture stretched until only one viewpoint can read it, which says what it looks like and nothing about what it is. Cast one onto a floor and fit a map to the marks, and the map turns out to be a planar homology — a line of fixed points, one point off it, and a single ratio. Those three numbers are not a description of the eye. They are the eye.
Where the anamorph still works
A picture correct from one point raises an obvious question that nobody answers with a number — how far may the eye move. The answer here is exact rather than approximate: a wrong viewpoint composes the intended picture with a central collineation whose axis is the line the picture stands on, so the error is zero along that line and grows linearly upward, and a step sideways costs precisely as much as the same step upward.
The marks name the place, not the height
Run the site's own round trip on an anamorph — hand the machinery the floor marks and ask for the eye back — and it returns the spot on the floor to eleven decimal places with nothing assumed at all. It does not return the height. What the marks fix is the product of the eye's height with the design's aspect ratio, and no amount of looking at the floor separates the two.
The ceiling that is not a plane
Paint the same design for the same eye onto a floor and onto a barrel vault, then fit the best possible homography to each set of marks. On the floor it misses by femtometres, because the map is a collineation and four marks determine every other. On the vault it misses by half a metre, and no choice of four marks helps — which is where the projective machinery this site has used since its first phase stops applying.