Visual Plane
every picture states the point it is correct from
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The thread: One machine used twice

A shadow is a projection from the lamp. A reflection is the view from a camera on the far side of the mirror. A parallel drawing is a photograph from infinitely far away. Three subjects usually taught as three recipes are one operation with the centre moved.
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.

 elevationx 1.000y 1.000z 0.000cavalierx 1.000y 1.000z 1.000cabinetx 1.000y 1.000z 0.500isometricx 0.816y 0.816z 0.816dimetricx 0.943y 0.943z 0.471axis scales measured from the drawingall 5 preserve midpoints The other systems

Parallel projection is not primitive perspective

Isometric and oblique drawing are not what people used before they worked perspective out. They are a different answer to a different question, and the difference is one measurable quantity — a parallel projection preserves the ratio in which a point divides a segment, and a perspective projection destroys it by 7% of the segment's drawn length at a comfortable depth, rising to 13% over the range the slider covers.

 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 Mirrors that are not cameras

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.

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

Where shadows vanish

The shadows of parallel posts under the sun meet at a point, and that point must lie on the horizon. Under a lamp they meet at the lamp's foot instead. Both are checkable in any photograph, and a picture whose shadows fail the check was not lit by anything.

 3 m, 50 mm1.50 m, 25 mmsubject ×1.000000 · background ×0.526 · zoom alone would give ×1 for bothnear-to-far ratio 10.00 → 19.00changing the focal length leaves it at 1.000000000000 The rectangle behind the lens

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.

 grey: the reflectiontwo routes, agreeing to 0e+0 px after one flip Light and mirrors

A mirror is a second camera

Reflect the scene and photograph it, or reflect the camera and photograph the scene. The two routes disagree by 315 px and agree to the last bit once one axis of the image is reversed — which is the whole of why a mirror is said to swap left and right, written down.

 eye, 74° offgrey: the word before the projectionblack: the same word, projected Where to stand

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.

 halfway along the drawn linethe actual midpointcorrect from 26 cm, at 160 mm wide7 px apart The other systems

The 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.

 no single viewpoint — the rays miss by 4.5 px, depth-dependentf recovered from it: 396.88 px Through water and glass

What survives a pane of glass

A slab of glass moves every point of a picture and moves no direction at all. So the camera recovered from a photograph taken through a display case is exactly the camera that took it — out of a picture in which nothing is where it was.

 focus, R/2 = 0.800 m184 mm of causticvertex radius 1.60 m, aperture 1.24 m5e-9 mm against 184 mm Mirrors that are not cameras

The one shape that focuses

A paraboloid takes every ray parallel to its axis exactly through one point — 4.8e-9 mm of envelope across a 1.24 m aperture, which is arithmetic. A sphere with the same vertex curvature spreads the same bundle over 184.1 mm. So there is exactly one curved mirror that is a projection through a centre, it is a projection for exactly one bundle of rays, and every telescope in the world is built out of that sentence.

 the picturethe same plane, rectifiedacross — 0.6000 widths (true 0.6000)along — 0.7667 widths (true 0.7667)diagonal — 1.2023 widths (true 1.2023)worst error 1e-15 relativethe probes were not used to build the map Measuring from one picture

The plan hidden in the photograph

Rectifying the ground is the same operation as rectifying a wall, and it turns a photograph into a site plan. The horizon is not an input to it and comes out as a consequence — the plan's points at infinity land on it, at first order, which is the check that the plan is a plan and not a plausible warp.

 flat picture plane — straight lines stay straightcylindrical picture surface — even stretch, bowed linesone scene, two picture surfacesneither is the distorted one Where to stand

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.

 source, 35 cmthe occluder's edgefraction of the source visiblepenumbra 17.5 cmprojection: 17.50 cmsampled: 17.41 cm Light and mirrors

The penumbra is the lamp's image

The soft edge of a shadow is a picture of the light, projected through the occluder's edge as through a pinhole. That gives its width without any integration — and it is why the dapples under a tree go crescent-shaped during an eclipse.

 3 m · 47 px6 m · 24 px12 m · 12 pxnear ÷ far = 4.000 against a depth ratio of 4.000exposure 33.3 ms · 6 m/s across the frameeach streak is straight; the set of them is not one kernel The rectangle behind the lens

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.

 k₁ = -0.28 · the round trip closes to 8.0e-13 pxpale: what the eye receives · dark: what the renderer drawsthe inner 86% of a 72° frame, where the inverse is exact The second projection

The render is distorted on purpose

A headset renders a bent picture so its lens can straighten it, which is a lens's distortion polynomial run backwards, and the one case in which distortion is introduced deliberately. The round trip closes to a thousandth of a millionth of a pixel, and the price is that one rendered pixel becomes 0.493 delivered pixels at the edge of the field and one at the centre.

 00.2500.5000.7501050100150angle of the direction from the camera's own axis (degrees)where it lands in the picture, as a fraction of the picture's radiusequisolidequidistantorthographica ball 24 cm across, camera 24 radii offequal-area within 0.82% · equidistant 21.5% Mirrors that are not cameras

A mirror ball is an equal-area fisheye

Photograph a mirror ball from far enough away and its rule is ρ = R·sin(θ/2), which is the equal-area fisheye — not an approximation to it, the rule. Measured, the departure falls from 4.27% of the picture's radius at 3 radii to 0.01% at 2000, while the next-best named rule stays 21% out at every distance. And the ball reflects 100.0% of the directions there are, which no designed surface does.

 the sheet, 150 × 105 mmthe eye, 52 mm up and 80 mm backgrey: the word standing upright · black: the same word on the paper142 mm from the sheet's middle Where to stand

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 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.

 centric pointdistance point, 12 px off the sheet →430 pxcorrect from 10 cm at 160 mm wide47° across Constructing a view

The 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.

 no picture at 300°the plane is unbounded at 180°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%300° across in every panelsame scene, same angle, six surfaces Surfaces that are not flat

What a 360-degree photograph actually is

The format every spherical camera writes preserves nothing — not straightness, not shape, not area — and it is the right choice anyway, for a reason that has nothing to do with looking at it. An equirectangular file is a lookup table of directions, and the picture only exists at the moment something re-projects a piece of it.

 an anchorcorrect from 19 cm, at 160 mm wideflat 1e-13 px · rolled 30.5 px What survives

A projection of a projection

Photograph a photograph and the composite map is a homography — four marks determine it and every other lands where they say, to 1e-13 px. Roll the print and the same four mispredict the rest by 30.5 px, which is why straightening a page of a thick book never quite works.

 the far field — where the stitch was fitted2.2 m — 3.2 px out24 m — 0.3 px outthe sky registers to 1e-13 pxthe foreground does not — up to 3.2 px The real instrument

The 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.

 00.50011.50205101520the dome's centre, off the entrance pupil (mm)worst departure from the pinhole it would be in air (degrees)centred: exactly zero6 mm → 0.635°a 100 mm dome in acrylic, n = 1.4910.106° per mm of centring error Through water and glass

The port that is not there

A flat window into water costs a lens a third of its field. A sphere centred on the entrance pupil costs nothing at all — not nearly nothing, exactly nothing — and six millimetres off centre costs 0.635°.

 head0.668profile — the outline that names a faceeye0.669frontal — an eye in profile is a wedgeshoulders0.669frontal — the width that says two armslegs0.668profile — a stride is a side viewpond0.326plan — a rectangle of water is a rectangle1.000 — what its own aspect keepsshare of each part that reaches the picture, from the best single direction|d · n| for each part3.000 of 5, swept and in closed form Systems that kept the measure

Assembled from several views

An Egyptian relief takes each part of a figure from the direction that identifies it — head in profile, eye and shoulders frontal, a pond in plan. What that buys is exactly measurable: any single viewing direction keeps at most √k of k perpendicular aspects, so the best compromise view retains 58% of each.

 15° of yaw, 6° of pitch, 1.50 throw ratio81.3% of the panel reaches the corrected rectangleouter: the thrown quadrilateral · inner: what correction can keep18.7% of the panel discarded The second projection

A projector is a camera run backwards

Turn a projector fifteen degrees from square and it throws a trapezium; keystone correction cannot add light outside it, so it shrinks the picture until it fits and discards a sixth of the panel. And the instrument itself comes back out of the picture it threw — 2880 panel pixels recovered against 2880, by the function the wrong field wrote for hand-drawn cubes.

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