Visual Plane
every picture states the point it is correct from
Concept

Projective limit — where it appears

The value a projected quantity approaches as its object recedes, which a vanishing point is the clearest instance of. It is reached only at infinity, which is why a vanishing point is a real point of the picture and never a point of the scene.

Named by 7 essays across 4 fields — each of them below, with the objects they name alongside it.

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

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.

parallel · Limit 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

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.

conventions · Aspective the eyenear 0.90 mfar 4.20 mfocal 740 pxprincipal 345, 210

Four numbers and a window

A projection matrix is built from six numbers and one of them is not a number at all. Four sides carry the focal length and the principal point; the near and far planes move nothing a reader can see; and the bottom row, (0, 0, 1, 0), is the only place the depth divides — set it to (0, 0, 0, 1) and the same machine draws a parallel projection.

pipeline · Frustum what was drawnthe solid it depicts — reading 1 of 2xyzlooked at 0.00° off the normalcube edge 1.0000 of the drawn unitresidual 1e-16

The drawing does not say which corner is nearer

The Necker cube is filed under optical illusion, as though the flipping were something the eye does. It is not: a parallel drawing of a cube is a drawing of exactly two cubes, mirror images of each other, and they project to the identical picture to the last bit. Perspective rules the second one out at a rate exactly inverse in the eye's distance, and never entirely.

parallel · Reversal 01230123the plane moved by a factor of ten to the …depth resolution, relative (powers of ten)near plane, brought infar plane, pushed out1/near − 1/farone term does all the work

One plane is nearly free

The near and far planes enter a depth buffer's precision through 1/near − 1/far, and one of those reciprocals is enormous. Pushing the far plane out by a factor of a thousand costs a tenth of a per cent; bringing the near plane in by the same factor costs a factor of a thousand — and an infinite far plane is the limit of the first rather than a separate case.

pipeline · Zbuffer band 1band 2band 3every figure the same height, by constructionno horizon

A picture in bands

A register picture stacks its scene in horizontal bands, each with its own ground line and every figure drawn at one height. The feet line and the heads line of any pair are then parallel to the arithmetic floor — 0° against 15.4° in a photograph of the same figures — so the picture has no horizon anywhere in it, and what a reader recovers is an ordering with no metre attached.

conventions · Register centrefaint dots: before · solid: afterhomologyratio 2.4000

What a flat map leaves alone

A projectivity of the plane is eight numbers in a matrix, and reading them tells a reader nothing. What it does is decided by its fixed points, and there are exactly three cases: three isolated fixed points, or a whole line of them with one point off it, or a whole line of them with the point fallen onto it. The middle case has five numbers instead of eight and every point slides along a line, and it is what most of the maps this site builds turn out to be.

foundations · Fixedstructure

Named alongside it

The objects these essays reach for when they reach for this one.

degrees of freedomDemonstrationHomogeneous coordinatesOrthographicOrthographic limitParallel projectionCamera matrixClip spaceConditioningDegeneracyDepth cueDrawing system

All concepts