The thread: The viewer is in the geometry
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.
The rectangle behind the lensA 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.
The second projectionThe screen sets the distance
Every viewing distance quoted for a picture on a page is conditional on an assumed figure width. Replace the assumption with an actual chain — focal length, sensor width, display width — and the same 50 mm frame is correct from 9 cm on a phone, 83 cm on a monitor and 16.7 m in a cinema. Nobody is standing at any of them.
Mirrors that are not camerasA 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.
What survivesWhere 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.
Where to standWide 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.
The rectangle behind the lensStepping 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.
The second projectionA wide field on a small screen
A picture rendered at a hundred degrees and shown on a screen that subtends forty-nine is being read from two and a half times its own station distance, so the depicted space is two and a half times too deep. The stretch at the edges everybody complains about is correct; the complaint is really that nobody is sitting where it would be invisible.
Where to standAnamorphosis 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.
The other systemsThe eye taken to infinity
A parallel projection is a photograph from infinitely far away with the lens lengthened to match. That is not an analogy — it is the limit, it can be watched happening, and it explains why a long lens flattens a scene and why an isometric drawing has no viewing distance to state.
Constructing a viewThe horizon is at eye level — if the picture plane is vertical
The horizon cuts every standing figure at the same fraction of its height however far away it is, which is the most immediately usable fact in the subject. It holds when the camera is level, and a twelve-degree tilt is enough to spread the fractions by more than a percentage point.
Drawn confidentlyThe sixty-degree cone of vision
Every book says keep the subject inside a 60° cone. Measured, the marginal stretch the rule is nominally about is exactly zero from the station point — 1.000000000000 to 1, over 720 sampled points. The rule is a statement about the reader, and books do not obey it.
Systems that kept the measureA picture with no size–distance signal
In a system with no diminution the drawn size of an object falls at exactly zero pixels per metre, so nothing in the picture says how far away anything is. Depth has to be carried by something else, and what carries it is height on the page — linearly, and without a horizon.
Constructing a viewAlberti draws a pavement, and chooses where the reader stands
The costruzione legittima of 1435 is exact. Run as a construction — with drawn rays and drawn intersections rather than the formula it turns out to satisfy — it agrees with a pinhole camera to six parts in a hundred trillion of a pixel. And it has one free parameter that the recipe never names, which is the distance from the eye to the panel.
Measuring from one pictureThe one thing a single view cannot give
Make the world a hundred and thirty-seven times larger and move the eye a hundred and thirty-seven times further away, and the picture does not change by a measurable amount. Every ratio in a scene is recoverable from one photograph and no size is, and that is not a caveat about the method — it is the shape of the method.
Where to standWhen 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.
The real instrumentThe principal point is not the centre
Every textbook that computes a focal length from two vanishing points supplies the middle of the frame as the principal point. On a shifted or cropped picture that is wrong, and it costs one and a half per cent of the answer at a fifth of a frame's shift.
Drawn confidentlyThe third point put where it looks right
Three-point perspective is taught as two vanishing points plus a third placed by judgement. The third is not free: two points and the centre of the picture fix it exactly. It survives being placed by eye because in the layout a book draws it belongs thousands of pixels off the paper, where 400 px of error costs less than a degree — and in a photograph taken looking up at a tower the same 400 px costs ten.
Systems that kept the measureA carpet and the people on it
A Persian miniature draws the ground as though from above and the figures standing on it as though from in front. The two views want optical axes exactly ninety degrees apart, and the plan view does not shorten a standing figure — it replaces its height with its distance from the point under the eye.
Where to standAn anamorph at true size, on paper
An anamorph is the shadow of the intended picture, cast from the reader’s own eye. Every claim about a figure on a screen is quoted against an assumed display width, because nobody can know how wide a screen shows it; this one is not, because it ships a sheet in millimetres and states where to put an eye.
Constructing a viewThe distance point is the viewing distance, drawn
There is exactly one place in the whole classical apparatus where the distance from the eye to the picture appears as a length on the page, and it is the offset from the centric point to the distance point. Everything this site exists to compute was drawn in the fifteenth century, on the horizon, and nobody said what it was.
The real instrumentThe eye is a place, not a point
Rotate a camera about the wrong point and the sky still stitches perfectly while the foreground slides. The misregistration falls as one over the distance, exactly — which is what says the fault is the pivot and not the lens.
Mirrors that are not camerasThe cone that reads the floor
A conical mirror standing on a design turns it into a picture, and the map it performs is exact, one-to-one, and not a projectivity — a homography fitted to four of the marks returns those four to 7e-13 mm and puts the rest 2480 mm away, on a design 369 mm wide. The reason is that the cone turns the annulus inside out: the middle of the picture comes from the far edge of the floor and the rim from the near one.
Drawn confidentlyBoth vanishing points on the paper
Putting the two vanishing points on the sheet is presented as a composition rule. It is a statement about the reader: with the two points one page-width apart the picture is a 90° view, correct from 80 mm, and a reader holding it at arm's length is shown a room five times as deep as the one drawn. The layout that is honest at arm's length puts both points four and a half pages off the sheet.