A picture with no eye

What an eye can paint asks a question of a faceted object and an eye: cast a design from the eye through a picture plane onto the object, and see which faces receive marks. The answers were all about the eye — the paintable subset is a function of it, there is a best height for it, two of them can carry two disjoint pictures.

The parallel field’s whole subject is drawing systems with no such point. Isometric, oblique, elevation and the military projection carry a design along one direction rather than from one place, and the same three questions therefore have a different shape.

Two of the answers change. One does not, and it is the one everybody assumes away.

The arrangement #

The convention has to move slightly and the move is itself a finding.

In the anamorph field the design plane is pinned: the intended picture stands upright at the origin and the eye is in front of it. That is right there, because the picture is a thing on a wall and the eye is a person in the room.

A parallel design has no eye to stand in front of the plane, so the plane is free — perpendicular to the direction, placed far enough back that every ray starts outside the object. That is exactly a parallel projection’s picture plane, and giving it that freedom is what lets the direction range over the whole sphere instead of over the half the pinned convention allows.

What does not change: occlusion #

Start with the negative, because “no centre of projection” is routinely read as “nothing hidden” and it is not what it means.

Occlusion is a statement about which face a ray meets first. A ray has a first face whether or not the rays are concurrent, so a block still hides a block, a riser still faces the wrong way, and a doorway still lets rays through into the next room.

Drawn isometrically, a cluster of blocks loses nearly thirty per cent of the area that turns toward the direction. The control is a corner, which hides nothing from itself and loses exactly none — without it the measurement would be reporting the sampler rather than the objects.

The measurement is taken on the object’s surface rather than by casting a design, and that is deliberate. Casting a design makes the answer depend on how finely it was sampled, because a face receiving no sample is indistinguishable from a face nothing could reach. Sampling each facing face over its own area and asking whether the ray leaving it backwards escapes the solid asks the same question with the design taken out of it.

What changes: the map onto each face becomes affine #

This is the difference that matters, and it is exact.

A plane carried onto a plane from a point is a projective map. Its scale varies across the face: a design point and its neighbour land closer together on the near part and further apart on the far part, and the variation can be large.

A plane carried onto a plane along a direction is an affine map. Its scale is a constant of the face. Not nearly constant — constant, to the arithmetic floor.

Measured on the same object with the same differencing, a design cast from a point varies by a factor of nearly six across the worst single face; the same design carried along a direction varies by 1.000000000, which is the machine’s floor.

That is the reason a parallel drawing can be measured with a ruler and a perspective one cannot, arrived at on the anamorph field’s own machinery rather than asserted. A ruler on an isometric drawing is where this collection establishes what the ruler actually reads; this is the same fact from the painter’s side rather than the reader’s.

The two singular values, which is the honest form #

A local map from a plane to a plane has two stretches, not one, and reporting a single number means reporting whichever the sampling direction happened to pick up.

The honest form is the two singular values of the map’s Jacobian: the worst stretch in any direction and the best. Their ratio is the anisotropy, which is what makes a mark unpaintable rather than merely large — a design point smeared equally in both directions is a picture at a different scale, and one smeared in a single direction is a picture nobody can paint. Their product is the area scale.

Both are constants of the face under a parallel design and neither is under a perspective one. Measured on the cluster of blocks, the median anisotropy is about 1.95 either way — a direction does not make a design isotropic, and nothing suggested it would — but from a direction that number is the same everywhere on a given face, and from a point it runs from one to nearly nine across the object.

And the two are not independent #

Under a parallel design the two singular values are not merely constant across a face; they are locked together, and the lock is worth stating because it removes a design freedom a reader might look for.

Projecting along $\hat{\mathbf d}$ onto a plane perpendicular to it, a face with normal $\hat{\mathbf n}$ is unchanged along the line where the two planes meet and compressed by $|\hat{\mathbf n}\cdot\hat{\mathbf d}|$ across it. So the two singular values are

$$\sigma_1 = 1, \qquad \sigma_2 = |\hat{\mathbf n}\cdot\hat{\mathbf d}|,$$

which makes the area scale $|\hat{\mathbf n}\cdot\hat{\mathbf d}|$ and the anisotropy its reciprocal. Area scale times anisotropy is exactly one, on every face, in every parallel system.

That is a genuine constraint rather than an observation. A face drawn at half its area is necessarily smeared by a factor of two; a face drawn isotropically is necessarily drawn at full area, which happens only when the direction is along its normal. There is no parallel drawing in which a face is both foreshortened and undistorted in shape, and no choice of direction buys one at the expense of the other, because the trade the two are usually in is not available: one of the two stretches is always exactly one.

A perspective design has no such lock. There both singular values vary over the face and vary independently, so a point-cast design can be badly anisotropic at full area or nearly isotropic at a tenth of it, depending on where on the face the mark falls.

The isometric case is the familiar number arriving from a third direction. All three coordinate faces have $|\hat{\mathbf n}\cdot\hat{\mathbf d}| = 1/\sqrt3$, so all three are drawn at 0.5774 of their area with an anisotropy of 1.7321 — and 0.5774 is the ellipse ratio the drawing office cuts its templates to. The template’s number, the face’s area scale and the design’s anisotropy are one quantity counted three ways.

The dimension of the answer #

The third change is a count, and it is the one that most alters what “the family of pictures an object supports” means.

An eye ranges over a volume: three numbers. A direction ranges over a sphere: two. So the family of paintable subsets an object offers drops a dimension when the centre goes to infinity, and every parallel drawing system in the field is a choice of two numbers rather than three.

That is visible in the field’s own results without anybody having said so. Which axis scales are possible finds the axonometric systems lying on a surface rather than filling a volume, and the surface is two-dimensional for exactly this reason: an axonometric projection is a direction and a rotation of the paper, and the rotation does not change what is reachable.

The direction has to be computed, not read off #

A practical point that turns into a finding.

Each drawing system in this collection states its geometry differently — a pair of rotations, a shear angle and a depth factor, an azimuth. Only one of them names a direction. So the direction a system projects along is computed here as the kernel of its own linear map: the map is two rows by three columns, each row is a plane, and the kernel is where they meet.

For the orthographic and axonometric systems the answer is the picture plane’s normal, as expected. For an oblique system it is not, and that is the whole of what an oblique projection is: cavalier draws the front plane true and pushes depth away at forty-five degrees from the normal, so “the way the picture faces” and “the way it looks” are two different vectors.

Oblique is a shear, and the shear is the whole system is where this collection establishes the algebra. The direction computed here is the same fact arriving as a vector, and it matters because the next rung’s claim — that nothing moves when a part slides along the view direction — is about the second vector and not the first.

Which objects lose most #

The seven objects the anamorph field measures behave differently under a direction, and the ordering is informative.

A corner loses nothing: it is convex from outside, so nothing on it hides anything else on it, and its loss is exactly zero at every direction. A flat floor likewise. A seating rake loses nothing from above and everything from below, because its risers face one way.

A cluster of blocks loses about thirty per cent, and a corridor with a doorway rather more, because a doorway is a hole and a hole lets rays past every wall they were supposed to land on. A flight of stairs with its ends loses most of all from a shallow direction, where each tread hides the riser behind it.

The pattern is that convexity is what protects an object and holes are what damage it, which is unsurprising and worth measuring anyway — because the claim being tested is that a parallel design has no occlusion at all, and a single object losing area refutes it.

What the design band becomes #

A small technical difference with a consequence, because it is where the free plane pays for itself.

An anamorph cast from a point has a band of design heights whose rays meet the object — below it the rays hit the floor short of the object, above it they clear the top and go into the room. Finding that band is a real problem: the design that outruns the floor is the case where it is unbounded, and the anamorph field bisects for it rather than assuming it.

A parallel design has no such difficulty. The design plane is perpendicular to the direction, so the object’s silhouette on it is the object’s outline, and the band is the outline’s bounding box — computed in one pass from the object’s own corners with nothing to bisect.

That is a fair summary of the whole difference between the two arrangements. A point makes the design’s extent a question and a direction makes it a measurement, and the reason is that a parallel projection has no near plane and no far one, so nothing can be behind the design or in front of it.

Two things this does not license #

Both are easy to conclude from the affine result and both are false.

It does not mean a parallel drawing has no distortion. The map onto each face is affine, and an affine map still stretches — the median anisotropy on the cluster of blocks is about 1.95, which is a design nearly twice as long one way as the other. What is constant is the stretch’s variation across the face, not its size.

And it does not mean every face is at the same scale. Two faces at different angles to the direction receive the design at different scales, and the ratio between them can be large: a face nearly edge-on to the direction receives a design smeared out enormously, which is exactly the case a drawing cannot use. What the affine result says is that within any one face the scale does not change, so a face is either usable throughout or unusable throughout — which is a considerably more convenient situation than a perspective design’s and is a long way from no distortion at all.

The distinction is the same one conformal is not undistorted makes about picture surfaces: a map can preserve one thing exactly and destroy another, and naming the thing it preserves is not a description of the whole map.

What a parallel anamorph would actually be #

The arrangement measured here is a real object, not a thought experiment, and it is worth saying what it is.

An anamorph cast from a point is a picture that resolves from one place in the room. An anamorph cast along a direction resolves from any point along that direction, arbitrarily far away — which is to say it resolves for a viewer at a distance, or through a telescope, or in a photograph taken with a long lens from across a field.

Those exist. Large land art read from an aircraft, markings on a runway read from an approach path, and any painted device meant to be read from a great distance are parallel anamorphs in exactly this sense, and their design problem is the one measured here: which faces of the terrain the direction reaches, and at what stretch.

The difference from the pinned-plane case is that such a design has no viewing distance at all, only a viewing direction, and so it has nothing to state where every other figure on this site states a number. That is why the figures in this family print the collection’s no-station-point strip rather than a distance: when the picture surface is not flat is the other place that strip is used, and for the same reason — there is no point from which the picture is correct, so printing one would be inventing it.

Where the objects came from #

Worth a paragraph, because the seven objects measured here are not chosen for this essay and that is the point of them.

They are the anamorph field’s own: a flat floor, a corner, a cluster of blocks, a seating rake, a corridor with a doorway, and a flight of stairs in each of its two directions. Each was built to make one claim about an eye — that occlusion is not facing, that a doorway lets rays past, that a flight’s two ends reach disjoint sets of faces — and each is measured here by the identical machinery with the centre moved to infinity.

Reusing them is what makes the comparison a comparison. A parallel result on a new set of objects would be a result about the objects as much as about the projection; the same objects, the same design grid, the same differencing and one line changed is a difference that can only be the centre.

That discipline is the reason a stair does not use all its faces and this essay can be read against each other at all, and it is the same discipline the screen field keeps when it measures a flat panel first at the arithmetic floor before saying anything about a curved one.

What the next rungs take from here #

Three things, each measured in its own essay.

The affine map has a consequence about movement. If a face’s scale is a constant, then sliding the face along the direction changes nothing about the picture on it — which is why exploded and cutaway drawings are drawn in parallel systems, as an identity rather than an approximation.

The limit has a distance. A parallel projection is the limit of a perspective one, and the eye taken to infinity argued that without putting a number on it. There is a number, and there are two of them.

And the free design plane buys a picture. A flight of stairs with step ends carries a picture on those ends that no pinned design plane can reach, which is the shortfall this collection recorded and the rule it corrects.

The short version #

A design carried along a direction rather than from a point maps onto each face affinely, so its scale is a constant of the face rather than a function of position — 1.000000000 across the worst face, against nearly six from an eye. The family of pictures an object supports drops from a three-parameter volume to a two-parameter sphere.

And occlusion is unchanged. A picture with no centre of projection still cannot paint what is hidden, because a ray has a first face whether or not the rays are concurrent, and a cluster of blocks drawn isometrically loses nearly thirty per cent of the area facing it.