The map depends on where it was cut

The previous rung took a body with no closed-form conformal map, triangulated it, wrote the discrete Cauchy–Riemann equations as one complex equation per triangle — a map with no formula in the ladder’s own vocabulary, and solved the least-squares system. It came back conformal — exact on a flat patch to 2.6 × 10⁻¹⁰ degrees, converging at first order in the mesh — and refused to be equal-area, with an areal spread that settled at 3.07 and would not move with refinement.

The essay ended by recording that 3.07 as suspect. It is a property of the patch, the boundary and the two pinned vertices together, and which of those it mostly belongs to had never been measured. This rung measures it, and the answer sorts the solve’s outputs into two kinds.

Why anything is pinned #

The discrete conformal energy is invariant under a similarity of the plane. Translate the whole solution, rotate it, scale it, and every triangle’s conformality is unchanged, because conformality is about angles and a similarity preserves them.

So the linear system has a four-dimensional null space and no unique solution. Fixing two vertices removes exactly four degrees of freedom — two coordinates each — and makes the system positive definite. That is the standard construction and it is what every least-squares conformal mapping implementation does.

The question this rung asks is what those two vertices decide. They obviously decide the position, the orientation and the scale of the image, since the distance between them in the plane is fixed at whatever the constraint says. What is less obvious, and is the finding, is that fixing the scale at two points is not the same as fixing it globally — and that the areal spread, which is a ratio of the largest areal factor to the smallest, is not invariant under a similarity of the plane in the way the angles are.

What survives, and what does not #

The sorting is clean and it is worth stating as a table, because it is the answer to a question the previous rung could not resolve.

Conformality survives the pins and not the cut. That is the right way round: how conformal a map of a piece of surface can be is a property of the piece, since the piece’s own curvature is what forbids it from being an isometry, and a smaller window is a flatter piece. The map of a small square is nine times more nearly conformal than the map of a broad band, and that is a fact about the body.

The areal spread survives neither. It moves with the pins by a factor of 2.8 and with the cut by a factor of 2.8 again, and the two effects are independent. So the 3.07 of the previous rung was a number about a solve rather than about a surface, and the previous rung was right to be suspicious of it.

What “areal spread” was measuring #

It is worth being precise about why the areal spread moves when the angles do not, because the asymmetry looks like a bug until it is written out.

The angular deformation at a triangle depends on the ratio of the two singular values of the 2 × 2 matrix taking that triangle to its image. A similarity of the plane multiplies both singular values by the same factor, so the ratio is untouched, and every angle measurement in the solve is therefore invariant under the four degrees of freedom the pins remove.

The areal factor is the product of the two singular values. A similarity multiplies it by the square of the scale factor — the same square everywhere, so a ratio of two areal factors is also invariant. The areal spread, being the largest areal factor over the smallest, ought therefore to be invariant too.

It is not, and the reason is that the pins do not merely apply a similarity. They apply a similarity and change the minimiser. Fixing two vertices constrains the solution as well as normalising it, and a different constraint set leads the least-squares problem to a different point — very slightly different in the angles, which is what the median column measures, and substantially different in where the extreme areal factors end up.

So the spread is a ratio of two extremes over a map that itself depends on the constraint, and a ratio of extremes is the least stable summary available: it is decided by two triangles out of several hundred, and either of them can be the one the constraint bought its scale from.

The badly conditioned pair #

One of the six pinnings behaves differently from the other five and the difference is instructive.

Pinning two adjacent vertices — a separation of 0.1 in the mesh’s own parameter rectangle, against 1.1 to 2.3 for the others — gives a worst angular deformation of 15.8°, four times the 3.8° to 4.1° the well-separated pinnings agree on. Its median is 1.86°, which is inside the range everything else reports.

So the map is fine and one corner of it is ruined. The mechanism is conditioning rather than geometry: the constraint fixes the image’s scale through the distance between the two pinned vertices, and when that distance is small the scale is determined by a short lever. The least-squares solution buys the scale it needs by distorting the region near the pins, where distortion is cheapest for the energy, and the worst point moves there.

A well-conditioned constraint is one whose pins are far apart, and it is not a subtle requirement: an order of magnitude in separation buys a factor of four in the worst point. That is the practical instruction this rung produces, and it is one an implementer would otherwise find by accident.

What a map of a whole body would need #

The cut sweep is the harder half of the shortfall and it does not have a tidy resolution.

Mapping a body rather than a patch means choosing where to cut it, and the choice is not a numerical detail: the worst angular deformation moves by a factor of ten across five reasonable windows. Worse, the windows are not comparable — a broad band and a small square are different problems, and the small square is easier because it is smaller.

What would make the comparison meaningful is a family of cuts of the same extent in different places — the question the cut has to go somewhere asks of a polyhedral net, asked of a solve, which would separate “where” from “how much”. That measurement is not made here and it is what the next attempt on this ladder should do.

There is also a structural point the sweep makes clear. Every solve here has a free boundary: the solution is unconstrained on the edge of the patch except by the energy, which is what makes it a least-squares conformal map rather than a boundary-value problem. A free boundary means the answer depends on where the boundary is, necessarily, and the dependence is not an artefact — it is what a free boundary is for.

What was computed, and how #

The pin option did not exist until this rung. The solver had the two opposite corners of the parameter rectangle hard-coded, which is a perfectly reasonable default and is exactly what made the choice invisible — a constant with no name is a decision nobody can question.

Everything else is unchanged: the same mesh, the same conjugate-gradient solve, the same distortion measurement per triangle from the 2 × 2 matrix that maps one triangle’s isometric coordinates to its image. The angular deformation and the areal factor come out of that matrix’s singular values, which is the same construction the rest of this collection uses for a projection given by a formula.

The measurement reported as “median” rather than “worst” throughout is deliberate and it is the reason the sorting came out clean. The worst point of a solved map is a single triangle, and a single triangle is exactly the thing a badly conditioned constraint can ruin without touching the map. Reporting only the worst would have said the map depends on its pins; reporting only the median would have hidden the conditioning finding entirely.

Where the model stops #

One body and one mesh resolution. Eighteen by eighteen is coarse enough to run six solves quickly and fine enough that the first-order convergence established in the previous rung applies. Whether the pin dependence shrinks with refinement is a question this rung does not answer, and it matters: if it does, the 3.07 is asymptotically meaningful and this essay is about a finite-mesh artefact.

Five cuts, chosen by hand. They are not a family and they are not comparable, as the essay says above. They establish that the dependence exists and is large; they do not characterise it.

And no equal-area comparison. The previous rung’s finding was that the solved map refuses to be equal-area, with a spread that does not move with the mesh. This rung shows the spread moves with the pins, which weakens that finding without replacing it — what would replace it is a scale-invariant measure of departure from equal area, and choosing one is a question this collection has not asked.

What the previous rung should have reported #

Reading the earlier essay back with this rung’s sorting in hand, the repair is small and specific.

It reported three things about the solved map: that it is conformal to 2.6 × 10⁻¹⁰ degrees on a flat patch, that it converges at first order in the mesh, and that its areal spread settles at 3.07 and does not move with refinement. The first two survive this rung untouched — they are statements about angles and about convergence, and both are invariant under the pinning.

The third does not survive. It should have read: the areal spread is 3.07 for this patch, this boundary and these pins, and it is between 2.7 and 7.4 depending on which two vertices are held. That is a weaker claim and it is the one the arithmetic supports.

The essay also drew a conclusion from the 3.07 — that the map’s refusal to be equal-area is a property of the map rather than an error, because the number does not move with the mesh. That conclusion is still right, and it is right for a better reason than the one given: a quantity that is stable under mesh refinement and unstable under a change of constraint is a statement about the constraint. What makes the refusal genuine is that no pinning brings the spread near one, not that one pinning gives 3.07.

The generalisation #

A constraint that removes a gauge freedom must not be allowed to decide anything else, and checking that it does not is a measurement rather than an argument. The check is cheap: solve twice with different constraints and see which outputs move.

This collection has now met the same discipline three times over, on three unrelated ladders. A network’s residuals must survive a change of datum and its coordinates must not. An aspect search’s distortion field must survive a change of grid and its parameter triple need not. And here, a solved map’s angles must survive a change of pins and its areal spread does not.

In every case the rule for the report is the same: publish the invariant quantity as the result, publish the convention alongside, and never present a convention-dependent number as a property of the object. The rule is easy to state and it fails in practice for a specific reason — the convention-dependent number is usually the one that is easy to compute and print.

Who found it, and when #

Least-squares conformal maps are Lévy, Petitjean, Ray and Maillot’s, from a 2002 SIGGRAPH paper on texture atlas generation, and the two-pin construction is theirs. Their paper notes the pinning as a way of removing the free similarity and does not investigate the choice, because for their application — laying out a texture — the scale is set afterwards anyway.

The conditioning consequence was found by the graphics community soon after, and the standard advice in that literature is to pin two vertices far apart, on the boundary, often the two ends of the longest diagonal. It is advice rather than analysis, and it is exactly what the measurement above recovers.

The dependence on the cut is a much older subject in a different vocabulary. Every conformal map of a surface with non-trivial topology requires a cut, the cut’s placement is a choice, and Riemann’s mapping theorem is a statement about simply connected domains for that reason. What is new in the numerical setting is that the cut is chosen by whoever set up the mesh, often without noticing that a choice was being made.

The pattern, and the protocol it implies #

This rung is the third time this collection has sorted a numerical result into the part the object decides and the part the setup decides, and the three are worth naming together, because the method is the same each time and it is cheap.

A fitted aspect turned out to have a reproducible score and irreproducible parameters, because the objective has a symmetry. A fitted projection on a historical map turned out to have a determinable projection and an undeterminable datum, because the two are confounded. A conformal map of a body turns out to have outputs that depend on where somebody cut the surface and where they pinned it.

In each case the diagnostic is the same: perturb the setup, refit, and see which outputs move. Not the data, not the model — the arbitrary choices made in order to compute anything at all. A different start, a different pinning, a different cut, a different parameterisation of the same freedom.

Outputs that move are properties of the construction. They may still be worth reporting, but they must be reported as such, and they must not be compared between two studies that made different arbitrary choices — which is what happens when nobody knows a choice was made.

Outputs that do not move are properties of the object, and they are the ones a paper is entitled to state as findings about the body, the region or the map.

The protocol costs one extra run per arbitrary choice, and the number of arbitrary choices in a numerical construction is usually small enough to enumerate: this one has two, the cut and the pinning. What makes it worth insisting on is that the arbitrary choices are exactly the ones nobody writes down, so a reader cannot make the distinction from the paper and the author is the only person who ever could.

It also explains why the defect survives peer review. A reviewer checks the arithmetic, the method and the comparison against other work, and every one of those can be impeccable while the setup dependence goes unexamined — because the setup is described in a sentence that reads as housekeeping rather than as a modelling decision.

And the failure mode is not a wrong number. Every quantity in the previous rung was correctly computed. What was wrong was the sentence around one of them, which described a property of the mesh as a property of the body — and that is a defect no amount of numerical care prevents, because the arithmetic was never in question.

Where the ladder goes next #

This rung sorts a solved map’s outputs into the ones the construction decides and the ones the body decides. The cells ladder has a debt of the same age: it measured what moving a field between two grids costs, and it measured it between two schemes whose cells are rectangles in the same coordinate — which is the case where every overlap has a closed form, and is not the case anybody argues about.