Visual Plane
every picture states the point it is correct from
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The thread: Projected, not constructed

Every point in every figure here comes out of a camera with a focal length, not out of a line run to a vanishing point that was placed where it looked right. A picture no camera could produce fails to build rather than appearing with nothing to say so.
horizon — the eye's own heightbase — 0 mhorizon crossing — 1.62 mtop — 3.40 m recoveredthe vertical vanishing point is 8586 px above this framerecovered 3.400 m · true 3.400 m1.8 cm per pixel of click error Measuring from one picture

A height, out of one photograph

Four points on a vertical, one cross-ratio, and the height of something nobody measured. The only metric input is the photographer's own eye height, because the horizon is at eye level and that is the one piece of perspective folklore that is exactly true.

 horizonshadows meet at x = -58, off the frameon the horizon, as it must be Light and mirrors

A shadow is a second projection

The construction that puts a shadow on the ground is the construction that puts the scene on the picture plane, with the lamp where the eye was. Shadow drawing is taught as a separate set of recipes and it is one operation with the centre moved, which is why the same code draws both.

 plane — bend 0.0%cylinder — bend 6.0%stereographic — bend 3.5%equidistant — bend 4.5%equal-area — bend 5.1%equirect. — bend 5.7%120° across in every panelsame scene, same angle, six surfaces Surfaces that are not flat

No picture surface keeps everything

A picture has to be cast onto something, and every candidate surface destroys something different. Six of them are measured here on the same three properties, and the corner of the plot where a surface pays nothing is empty — not because nobody has thought of one, but because a theorem says there is none.

 horizon3 vanishing points at a finite distance2564 px · 8129 px · 531 px Constructing a view

One, two and three point are one construction

The names count how many vanishing points sit at a finite place in the picture, and the count is a fact about how the object is turned and the camera is aimed. Nothing about the method changes between them, and a vanishing point does not appear — it arrives from infinity.

 the picture, 160 mm wide22 cm40°the eyefocal length 948 px22 cm at 160 mm wide Where to stand

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.

 horizonABCDon the groundin the picturelength AB1.00011.3930ratio AB:CD0.56670.6837cross-ratio1.31681.3168correct from 26 cm, at 160 mm wide34° across What survives

What a projection destroys

A projection loses length, angle, area and the ratio in which a point divides a segment. Exactly one quantity comes through untouched, and almost everything that can be checked about a picture is checked with it — including, as it turns out, some things it cannot check at all.

 principal pointk₁ = -0.320, k₂ = 0.110 — barrel distortioncentre line 0e+0 px of sag, outermost 17.8 px The real instrument

Straight lines that are not

Everybody says the edges of a wide-angle frame bow. Nothing is special about the edge. A radial map moves every point along its own radius, so the only line it leaves straight is one through the principal point, and the bend of every other is decided by how far it passes from that one place.

 water, n = 1.333eyetruly 1.50 m downappears 0.818 m downh/n would be 1.125 mno single viewpoint — the rays miss by 17.1° of bend at the surfaceapparent depth 54.5% of the true one, not 75.0% Through water and glass

What a ray does at a surface

A pool looks three-quarters as deep as it is — but only if you look straight down. At sixty degrees the same bottom appears at half its depth, and at eighty at a fifth, which is why the far end of a pool looks shallow enough to walk in.

 epipoleepipoleleft pictureright pictureepipole from 44 correspondences vs the projected eye: 1.1e-9 px2.60 m between the eyes The second eye

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.

 the eye's trackthe eye at x = -10.5 mevery point is drawn by the one position of the eye that is level with itplan — the eye's track and the scans it makes28 m of travel The eye that moves

A scroll is a camera that moves

A Chinese handscroll is not a picture with a wandering viewpoint or a picture with no viewpoint. It is the image of an eye that travels along a track and records one vertical line at a time, and that object has an exact geometry — orthographic along the roll, perspective across it.

 they meet at y = 924, off the pagefar edge ÷ near edge = 1.32a one-point construction, swept through its splaythe point is below the near edge What each system gave up

The picture whose lines spread

In a Byzantine icon the sides of a table diverge with depth. The standard account says the vanishing point is behind the viewer. It is not — it sits below the near edge, in front of the eye, and it is the vanishing point of a direction running down and away.

 x/z, y/z — the pinholeM·p, then divide by wworst disagreement 4.0e-14 px over 8 verticescorrect from 21 cm, at 160 mm wide42° across · near 0.1 m, far 1000 m What a machine computes

The divide is postponed, not avoided

A renderer does not divide by depth. It multiplies by a four-by-four matrix that carries the depth in a fourth coordinate and divides later, and the postponement is not an optimisation — it is what makes clipping and texture interpolation possible at all. The matrix and this site's pinhole put every point on the same pixel to five parts in a hundred trillion.

 the picturethe same plane, rectifieddoor — 0.5068 widths (true 0.5068)sill — 0.3026 widths (true 0.3026)diagonal — 1.1766 widths (true 1.1766)worst error 4e-16 relativethe probes were not used to build the map Measuring from one picture

Flattening a façade out of the photograph

Four corners of a rectangle whose proportions are known are enough to undo the projection of one plane. After that the plane can be measured with a ruler — lengths, angles, areas, all of it — in units of the rectangle's own width, and lengths the map was never given come back to fifteen digits.

 fitted conic: 1.467 from being a circlecylinderthe samples, fitted Surfaces that are not flat

The cylinder, and the price of going all the way round

A cylindrical picture can hold three hundred and sixty degrees, keeps every vertical vertical, and bows every horizontal. Its cost is a stretch of sec φ in elevation, which is also the equirectangular surface's cost exactly — two surfaces that are always described as different and are identical in the one respect anybody notices.

 horizon — the image of the line at infinityvanishing point at x = 1464 — off the framecorrect from 26 cm, at 160 mm wide34° across What survives

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.

 horizonequal stepshalve thetaper byworst error, in metres of depthequal steps to the horizon14.72 mhalve the remaining gap228.13 mtaper by eye350.80 mcorrect from 26 cm, at 160 mm wide34° across Constructing a view

The measuring point, and the step the method leaves out

Laying out equal depths correctly needs a second vanishing point that most treatments never introduce — the one belonging to the diagonals. With it the construction lands on the projected divisions to eighty femtopixels. Without it, depth is placed by judgement and the drawing depicts something nobody chose.

 1234512345a point is chosen hereand must be on this linematch to its own epipolar line: 1.4e-13 px5 of 44 correspondences drawn The second eye

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.

 near edge 155 pxfar edge 196 pxrecovered: ratio 1.000000, corners 1e-14° from squarecorrect from 12 cm, at 160 mm wideplane tilted -45° What each system gave up

An inverse perspective is a leaning plane

Ask a divergent construction what solid it depicts and it answers: a rectangle, four right angles, near edge equal to far. What the splay encodes is not the shape but the plane's tilt — and a real square on a plane leaning toward the camera really does photograph with its far edge wider.

 clipped at the near planedivided without clippingdirection cosine -1.0000 — the far end is drawn 2142 px away20 grid segments, 2.4 m of them behind the eyea reversed line is not a large error, it is a different picture What a machine computes

What happens behind the eye

A point behind the camera has a perfectly plausible image. Dividing by a negative fourth coordinate flips both signs, so the point lands through the principal point on the far side of the frame, and a segment crossing the eye plane is drawn straight, inside the frame, and running in exactly the opposite direction — a direction cosine of −1.0000.

 184 mm of causticvertex radius 1.60 m, aperture 1.24 m184 mm of envelope Mirrors that are not cameras

Where the focus went

If the reflected rays do not meet at a point, they meet each other in pairs, and the curve they are all tangent to is what the mirror has instead of a focus. On a sphere of 1.6 m vertex radius it is 184.1 mm long and has a cusp; near the axis it sits at R/2, which is why a small enough spherical mirror passes for a good one. This is the bright shape in the bottom of a coffee cup, computed rather than admired.

 fitted conic: a circle to 1e-9stereographicdashed: the fit, not the samples Surfaces that are not flat

Stereographic keeps every angle, and only stereographic

One surface in the family preserves shape exactly — every right angle stays a right angle and both its arms are magnified equally, to the last bit the arithmetic has. It also sends every circle in the world to a circle in the picture, which the site's existing conic fit can be pointed at and asked to confirm without being told what it is looking at.

 recovered principal pointused to drawrecoveredgapfocal length853.90853.904e-15principal x345.0345.02e-12angle44.0°44.0°—correct from 20 cm, at 160 mm wide44° across What survives

Recovering the camera from the picture it drew

Draw a box from a known camera, forget the camera, and get it back from the twelve drawn edges alone. Agreement to one part in 10¹⁵ is a statement about the geometry, because the only thing that crossed between the two halves was a list of line segments.

 the panelhorizon — the centric point's heightthe section — the eye, the panel, the ground470 px — the viewing distancethree routes agree to 6e-14 pxsection, distance point, and a pinhole camera Constructing a view

Alberti 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.

 02468020406080angle off the optical axis (degrees)area scale, relative to the centreplanecylinderstereographicequidistantequal-areaequirect.measured by differencing on the spherethe plane leaves the frame Surfaces that are not flat

Every fisheye is a different rule

The word "fisheye" names a shape of lens and not a projection. There are several, they disagree with each other by tens of per cent at the frame edge, and each is the right answer to a different question — one is a protractor, one is a counting instrument, one preserves shape. Which one a lens implements is a fact about that lens, and it is rarely printed on the barrel.

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