One face, one scale

The anamorph field reports a stretch: how far apart two design points a hair’s breadth apart land on the object. It is the quantity that decides whether a mark can be painted at all — an anamorph has one eye is where the design’s own geometry is established — a design point smeared over half a metre is not a picture anybody can execute — and the stretch decides the band is where it sets the extent of the design.

It was reported as one number, and that round recorded in its own shortfall that this was not 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.

This is the two-number version, and it changes what the parallel case says.

The two numbers #

The map from a design point to the object’s surface has a Jacobian: three rows for the point in space, two columns for the design’s two directions. Its singular values are the largest and smallest amounts the map stretches anything by, taken over all directions in the design plane.

They are the honest form because they are the extremes rather than a sample. The larger, σ₁, is the worst stretch in any direction; the smaller, σ₂, the best. Their ratio is the anisotropy, and their product is the area scale.

The distinction the single number was hiding is the one between the two. A design point smeared equally in both directions is a picture at a different scale, which a painter can work with by using a bigger brush; a design point smeared in one direction only is a picture nobody can paint, because the marks it needs are not marks. Anisotropy is what makes a face unusable, and a single directional difference cannot see it.

The parallel result #

A plane carried onto a plane along a direction is an affine map. Its Jacobian is constant, so both singular values are constants of the face, and their ratio and product are too.

Measured on the cluster of blocks with the same differencing the anamorph field uses, the largest stretch varies across a single face by a factor of 1.000000000. That is the machine’s floor.

A design cast from a point onto the same faces varies by a factor of 5.754 across the worst one. Same object, same faces, same sampling, one line changed — the centre moved to infinity.

The control matters as much as the number. A picture with no eye runs both casts on the anamorph field’s own seven objects with the anamorph field’s own design grid, so a difference between them cannot be a difference in the objects, in the sampling or in the machinery. It can only be where the rays come from.

What is not constant #

Two things the affine result does not say, and both are easy to conclude from it.

A parallel design is not isotropic. The median anisotropy on this object is about 1.95 from a direction, which is a design nearly twice as long one way as the other. Being affine says the stretch does not vary across a face, not that it is one.

And faces are not at the same scale as each other. A face nearly edge-on to the direction receives the design enormously smeared, and a face square to it receives it undistorted. The ratio between two faces can be anything, and is unbounded as a face approaches edge-on.

So the correct statement is narrow and worth the narrowness: within any one face, the scale does not change. A face is therefore 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.

And in the parallel case the two numbers are one #

There is a further collapse that the affine result makes available, and it is worth having because it turns a differencing into a dot product.

Casting a design along $\hat{\mathbf d}$ onto a face with normal $\hat{\mathbf n}$ leaves the direction along the two planes’ intersection untouched and stretches the perpendicular one by $1/|\hat{\mathbf n}\cdot\hat{\mathbf d}|$. So

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

and the anisotropy and the area stretch are the same number — the secant of the angle between the face’s normal and the direction, and nothing else about the face, the design or the object enters.

Two things follow that the differencing cannot say on its own.

The whole measurement is a statistic of the object’s normals. The median anisotropy of 1.95 is the median of $\sec$ over the object’s facing area, which puts the typical face 59° from the direction; the worst of 8.93 is a face 6.4° from edge-on. Neither number is about the design, so casting a different design on the same object from the same direction returns them unchanged — which is a stronger form of the essay’s control than running the two casts side by side.

And “unusable” and “expensive” stop being separate questions. In a perspective design the anisotropy and the area scale vary independently, so a face can be cheap and unpaintable or costly and clean. In a parallel design they are equal, so a face that costs twice the paint is smeared by exactly two, and one decision covers both. That is a genuine simplification of the painter’s problem and it is the practical content of “one face, one scale” — not merely that the scale is constant across a face, but that there is only one scale to know.

Why the two-number form matters here in particular #

On a perspective anamorph the single directional number was a reasonable approximation, because the stretch is large and varies a great deal and the largest of two numbers that both vary is not far from the larger of the two.

On a parallel design it would have been actively misleading. Both singular values are constants, so a single directional difference is also constant — and reporting “the stretch does not vary across a face” from a measurement that samples one direction would have been reporting the sampler. It happens to be true; the measurement would not have established it.

That is the general reason to prefer the honest form even where the rough one agrees. A quantity that is right for the wrong reason cannot be relied on when the arrangement changes, and this arrangement changed.

The cross-check with the earlier number #

The two forms agree where they should, and checking that is what makes the new one a refinement rather than a replacement.

The anamorph field’s worstStretch samples along one design direction and reports the largest value found; the median of that quantity over a cast comes out at 1.917 on the cluster of blocks, and the median of the larger singular value over the same cast is 1.924. They differ by four parts in a thousand, which is the sampled direction happening to lie near the map’s own major axis on most of the object.

The worst cases differ much more: 17.8 for the single number against 8.93 for the anisotropy, because the single number’s extreme is a face-crease artefact and the two-number form drops any sample whose neighbours land on a different face. That refusal is not a convenience — a derivative taken across a crease is a property of the object’s corner rather than of the map, and an early version of this measurement reported anisotropies of forty on a flat floor because of it.

How the Jacobian is taken #

The measurement is a differencing and the details of it decide two of the numbers above, so they are worth setting out.

A design point is cast onto the object and so are two neighbours, one a hair’s breadth away in each design direction. The two differences, divided by the step, are the Jacobian’s columns. Its singular values come from the two-by-two matrix of dot products between those columns: the eigenvalues of that matrix are the squares of the singular values, and a two-by-two symmetric eigenvalue problem has a closed form.

Two refusals are built into it.

A sample whose neighbours land on a different face is dropped. A derivative taken across a crease measures the corner, not the map. An early version without this refusal reported anisotropies of forty on a flat floor, which is a surface with no anisotropy at all — the samples at the floor’s own edge were differencing onto nothing.

And a sample whose neighbours miss the object is dropped. The same reason: a difference against a ray that escaped is a difference against infinity.

Both refusals remove samples rather than clamping them, which matters for the reported extremes. A clamped sample would appear in the distribution as a large finite value; a dropped one does not appear at all, so the reported worst is the worst of the map rather than the worst of the sampling.

Why a projective map varies and an affine one does not #

The result is worth one line of algebra, because “affine” and “projective” are being used as though they explained something.

A projective map of a plane to a plane divides by a linear function of position — that is exactly what makes it projective rather than affine. Differentiating a quotient gives a term with the divisor squared in it, so the Jacobian depends on where on the plane the point is, and the dependence is strong wherever the divisor is small. The divisor is the depth, so the Jacobian varies most where the surface is furthest from the eye.

An affine map has no divisor. Its derivative is the map’s own linear part, everywhere, and there is nothing left for position to enter through.

That is the whole difference, and it says immediately where a perspective design’s variation is worst: across a face that recedes steeply from the eye, where the near end and the far end are at very different depths. On the cluster of blocks the worst face is exactly that — a side wall running away from the design eye — and its factor of nearly six is the square of a depth ratio of about two and a half.

What a single number is right about #

It would be unfair to leave the impression that the earlier reporting was wrong, because on the arrangement it was made for it is very nearly right.

A perspective anamorph on a floor is cast from an eye above and in front of it, and the design’s own vertical direction is close to the direction in which the stretch varies most. So sampling that direction picks up something close to σ₁, and the median of the single number and the median of σ₁ agree to four parts in a thousand.

Where it goes wrong is at the extremes and on arrangements it was not made for. The extremes because of the crease samples described above; the arrangements because the moment the design’s own direction stops lining up with the map’s major axis — which is exactly what happens on a face at an angle, and on every parallel design — the sampled number is neither σ₁ nor σ₂ but something between them, with no way to know which.

That is the general shape of a measurement that is right for the wrong reason, and it is why the two-number form is worth the extra differencing even where the answers agree.

What this buys a drawing #

The affine result is the reason a parallel drawing can be measured with a ruler, and it is worth connecting the two statements because they look unrelated.

A ruler on an isometric drawing establishes what a ruler laid on such a drawing actually reads: a length along one of the three axes, divided by that axis’s own scale, and nothing else. That is a statement about the drawing.

The affine result is the same statement about the painting — about a design laid onto a surface rather than a surface projected onto a page. A face receiving a parallel design receives it at one scale, so a ruler laid on the face reads a length in the design divided by that face’s own constant, and a single number per face converts between the two.

A perspective design has no such number. Its conversion factor varies across the face by a factor of nearly six, so a ruler laid on the painted surface reads something different at each end, and the reading is a property of where the ruler was laid.

The area scale, which is the third number #

σ₁ times σ₂ is how much a small patch of design grows in area on the object, and it is the quantity a painter estimating materials would want.

Under a parallel design it is a constant of the face, like the other two. Under a perspective design it varies with the square of the depth ratio, which on a design spanning a corridor is a factor of several — the same depth ratio the anamorph that crosses a corner has to carry across a crease.

It is worth naming separately because it is the quantity the anamorph field’s cost arguments implicitly use — how much paint, how much scaffolding, how long — and those arguments have been made with the single stretch standing in for it. On a parallel design that substitution is harmless; on a perspective one it is out by the anisotropy, which is about two.

Where the honest form changes an earlier answer #

One place, and it is worth being explicit rather than quietly correcting the record.

The stretch decides the band sets a design’s extent by capping the stretch — everything above a stated cap is outside the paintable band. With one number, that cap is a cap on whichever stretch the sampling found. With two, it is a choice: cap σ₁, cap the anisotropy, or cap the area.

They give different bands, and the design that outruns the floor is the case where the choice is forced. Capping σ₁ excludes the far end of a design; capping the anisotropy excludes the parts landing on faces nearly edge-on to the design; capping the area excludes both. The earlier answer is a cap on σ₁ measured in one direction, which is a reasonable choice among the three and was not presented as a choice.

Naming it as one is the correction. The band an anamorph can carry depends on which of three quantities a painter cannot exceed, and a painter with a large brush and a painter working at a fixed scale have different constraints.

What a parallel design is actually good for #

Two arrangements, both real.

A design read from far away. A parallel anamorph resolves from any point along its direction, which is the limiting case of the room the eye may stand in, so it is what land art read from an aircraft or markings read from an approach path actually are. The affine result is what makes such a design constructible at all: each surface receives the picture at one scale, so it can be set out with a tape.

And a design on a machined part. A marking laid onto a faceted component so that it reads correctly from a fixed inspection direction is the same problem at a smaller size, and there the constant scale is not a convenience — a set cut for one eye is the same problem at building scale — it is what allows the marking to be specified as a drawing with dimensions rather than as a surface.

Both are cases where a perspective anamorph would need a different scale at every point, and both are cases where the object has faces at many angles, so the second half of the result matters as much as the first: constant within a face, and different between faces.

The short version #

The stretch a design suffers on its way onto a surface has two numbers in it, and the previous round reported one. In the two-number form, a design carried along a direction has both of them constant across any single face — 1.000000000 of variation, against 5.754 for the same design cast from a point.

That is not the same as saying a parallel design is undistorted. Its anisotropy is about two, and two faces at different angles receive it at wildly different scales. What it says is that a face is uniform, which is what lets a parallel design be set out with a tape and a perspective one not.