A long window and a square one

Cuts of the same size in different places settled a question the previous rung had raised and could not answer. A conformal map of a patch of an irregular body depends on where the patch was cut — but the first measurement of that had let the patches vary in size, and most of the factor of ten it reported was size. Holding the surface area to a quarter of a per cent left a factor of 1.36 in the areal spread, and the total Gaussian curvature each window enclosed predicted it at r = 0.969 over seven points.

That essay recorded what it had not held: shape. A long thin patch and a square one of the same area containing the same curvature are not the same problem, the conditioning of the solve on the first was visibly worse, and nothing in the run separated the two.

The separation turns out to be clean, and it puts the two quantities in different places.

Elongation has to be measured on the surface #

The windows in the previous rung are rectangles in the body’s own colatitude and longitude, and a rectangle in those parameters is not a rectangle on the body. A patch of Δθ by Δφ covers roughly Δθ by Δφ · sin θ on the surface, so a window that is square in the parameters is elongated by 1/sin θ near the poles and by more again wherever the body’s own radius varies.

So the sweep is parameterised by the ratio of the two surface extents. A single scale is then bisected until the measured surface area matches the target, exactly as the previous rung did, and the seven windows come out within two hundredths of a per cent of one area.

The U is the whole argument #

The cost curve has a minimum at the square window and rises symmetrically in both directions. That symmetry is what makes the conclusion available without a regression.

A window elongated by four along colatitude and a window elongated by four along longitude are the same shape rotated. They enclose very different amounts of curvature — 0.311 against −0.012 on this body at this centre — because the body’s own features run in a particular direction. And they cost the solve almost exactly the same: 265 steps against 275.

A quantity that runs monotonically across the sweep cannot produce a curve with an interior minimum. The curvature falls from 0.311 to −0.012 with only one small reversal at the far end, and its least value is at an end. The cost’s least value is in the middle. Whatever is making the solve expensive, it is not the curvature, and no amount of arranging the curvature can make it be.

What is expensive is the aspect ratio of the linear system #

The mechanism is standard numerical linear algebra rather than geometry. The conformal solve is a least-squares problem over the patch’s mesh, solved by conjugate gradients, and the number of steps such a method needs goes as the square root of the condition number of the system. For a discretised elliptic problem on a rectangular domain, the condition number rises with the domain’s aspect ratio: a long thin region has slowly-decaying modes along its length that a local update cannot damp quickly.

Which is to say the cost is a property of the discretisation rather than of the body, and it would be there on a flat plane with no curvature anywhere. Two consequences follow and both are worth stating plainly.

It is not a property of the map. The solved map on a long window is as conformal as the solved map on a square one — the median angular deformation runs from 0.027° to 0.137° across the sweep, and is worst on the window with the most curvature rather than on the longest. The Riemann mapping theorem promises a conformal map of any simply-connected patch and it delivers one at every aspect ratio tried.

It is a property that a different solver would change. A multigrid or a direct sparse factorisation would have a different dependence on the aspect ratio, and a preconditioned iteration would have a smaller one. The number 174 is a property of this code; the shape of the curve is a property of the problem.

Two predictors, two responses #

That is the shortfall’s question answered in one figure. The two candidate predictors are not competing for the same response; each owns one.

The regression on seven points is not strong evidence on its own and is not offered as such. What makes it convincing is the U — an unmistakable qualitative signature that no monotone predictor can produce — and the matched pair below, which needs no fitting at all.

The comparison the shortfall asked for #

The shortfall’s own words were a long thin patch and a square one of the same area containing the same curvature. Such a pair cannot be constructed at a single centre, because at a fixed centre the curvature enclosed varies with the shape. It can be found, by sweeping several centres and looking for coincidences.

Five such pairs come out of twenty-eight windows across four centres. Every one of them says the same thing: the longer window costs more and produces a map that is no worse, and on the sharpest pair it produces a map that is marginally better, which is the opposite of what a conditioning problem contaminating the answer would look like.

What this does to the previous rung’s number #

The factor of 1.36 in areal spread stands, and it stands better than it did.

Before this rung it was a measurement of “where the cut goes”, with two candidate mechanisms tangled inside it and a recorded suspicion that one of them was an artefact of the solver. It is now a measurement of the curvature the window encloses, with the solver’s own difficulty separated out and shown to live in a different quantity.

There is a small correction. The seven windows in the previous rung vary in shape as well as in place, because the equal-area bisection changes the extent in colatitude and in longitude by the same factor and a body whose radius varies does not respect that. Their aspect ratios on the surface run from 0.71 to 1.34, an elongation of at most 1.34 — well inside the flat bottom of the U, where the cost varies by under ten steps. So the reported correlation is not contaminated, and it is not contaminated for a reason that can now be stated rather than hoped.

The cost is not free, even though the map is right #

There is a temptation to file this as bookkeeping: the map comes out the same, so who cares how many steps it took. Two things make it worth the rung.

A solve that costs more is a solve that is stopped early. Every conformal patch on this ladder runs to a stated tolerance or to an iteration cap, whichever comes first, and a window that needs 275 steps at a cap of 250 returns a map that has not converged and says nothing about it. The previous rung’s seven windows were run at three thousand iterations and every one converged, so nothing was lost — but the margin was never measured, and a body with sharper features or a finer mesh eats it. The failure mode here is silent, which is the same shape as the relaxation that stopped three per cent short and looked like the equation’s own inaccuracy.

And the cost decides what can be attempted. A polyhedral scheme on an irregular body cuts it into faces, and the number of faces is a trade: more faces means less distortion per face and more cutting. This rung adds a term to that trade which was not there before. Faces that are long and thin — which is what a subdivision of an elongated body produces — cost the solver disproportionately, so the practical ceiling on face count is lower for such a body than the distortion argument alone suggests.

The aspect ratios are the reassurance; the curvature is the claim they were protecting. Both are worth seeing, because a correction that leaves a correlation standing has to say which of the two it touched.

What a square window is worth #

The flat bottom of the U is the useful part. Between aspect ratios of about 0.6 and 1.6 the cost varies by under twenty steps out of 174, which is a tenth; outside that band it climbs quickly.

So the advice this rung supports is weak and specific: when a patch can be chosen, choose one that is roughly square on the surface, and do not work hard at it. An elongation of 1.5 is free. An elongation of four is worth 60 per cent more solver time and buys nothing.

That is the same shape of advice as the aspect search’s near-optimal set — a broad basin where the choice does not matter and steep sides where it does — and it is worth noticing that the two are unrelated problems that produce the same kind of answer. A quantity with an interior optimum usually has a flat neighbourhood around it, and the practical question is almost never where is the optimum but how wide is the basin.

Where the cap actually is #

The warning that a slow solve is a solve that gets stopped early is stated above without a number, and the number is available from two measurements this ladder has already made.

The iteration count of conjugate gradients on the normal equations grows with the mesh. Measured on the refinement study of a conformal map of a body that is not a quadric, it ran 75 steps at 128 triangles and 467 at 2,048 — a sixteenfold refinement in triangle count, which is a fourfold refinement in linear resolution, for a 6.2-fold rise in steps. That is n^1.32, sitting between the n a well-conditioned elliptic system gives and the n² the normal equations threaten.

Take that exponent and the aspect penalty measured here, and the cap has a position. The square window at sixteen by sixteen takes 174 steps. A mesh eight times finer in each direction takes 174 × 8^1.32 ≈ 2,700, which is at the three-thousand cap these runs use. The most elongated window of the sweep costs 1.6 times as much, so it reaches the cap at a mesh about a third coarser — six times finer rather than eight.

So the margin is not eleven-fold as the raw step counts suggest; it is a factor of eight in resolution for a square window and six for a long one, and both are inside the range a real shape model would want. A spacecraft’s shape model of a small body has hundreds of thousands of facets against this sweep’s few hundred, which is a linear refinement of twenty or thirty — comfortably past the cap on any window shape.

The consequence is that the aspect ratio’s real cost is not solver time, it is the resolution at which the solve silently stops converging, and it moves that resolution by about a third. A third of a mesh refinement sounds negligible and it is the difference between a map that has converged and one that has been truncated at an iteration count with no error reported, which is the failure this section exists to name.

Two repairs follow directly and neither is exotic. Report the iteration count beside every solved map, so that a run at the cap is visible as a run at the cap rather than as an answer. And scale the cap with the mesh rather than fixing it — a cap of 3,000 is generous at sixteen by sixteen and is a truncation at a hundred and twenty-eight, which is exactly the shape of a constant chosen at one size and carried to another that this collection keeps finding elsewhere.

The third repair is the one the numerical literature would reach for first and it is worth naming for completeness. A preconditioner changes the exponent rather than the constant, and for this class of problem a good one takes the step count close to independent of the mesh — which would remove the cap question entirely and would also remove the U, since the aspect-ratio penalty lives in the same condition number. That is a different solver rather than a tuning of this one, and the shape of every curve in this essay is a statement about the unpreconditioned method it was measured on.

Where the model stops #

One body and one solver. The sweep is on the star-shaped test body this ladder has used since the map depends on where it was cut, and the cost is measured as conjugate-gradient steps at a stated tolerance. Both are the natural things to hold constant and neither is general: another body’s features run another way, and another solver has another dependence on the aspect ratio.

The mesh is uniform in the parameters. Every window is meshed at sixteen by sixteen regardless of its shape, so a long window has long thin triangles and a square one has near-equilateral ones. That is a second mechanism for the U — badly-shaped triangles condition a finite-element system worse — and it is not separated from the domain’s own aspect ratio here. The two are hard to separate because a uniform mesh on a long domain is either anisotropic or has more elements, and changing the element count changes the problem size.

Seven points, two predictors. The regression has four degrees of freedom and would not survive a demand for significance. It is reported because it agrees with the U and with the matched pair, and the three together are the argument rather than any one of them.

And the curvature is discrete. The total Gaussian curvature is the angular defect summed over interior vertices — exact for the polyhedron, an approximation to the body — and a long window has a different distribution of vertices from a square one. The agreement of the matched pair to one per cent is therefore an agreement between two discrete estimates rather than between two exact quantities.

The step from “a formula, evaluated” to “a system, solved” is what makes this rung necessary and it is worth naming as a change of kind. Everywhere else on this site a map is a function: it has a closed form, its derivatives are exact, and the only errors are the ones the projection itself commits. On an irregular body the map is the output of an iteration, and an iteration has a tolerance, a step count, a conditioning and a failure mode — none of which is a property of the body, and all of which end up in the numbers unless somebody separates them.

That is the same distinction a coordinate is the output of a solve draws in the practice field, and it lands here for the same reason. Once the answer comes from a solver, the solver is part of the instrument.

Where the ladder goes next #

The bodies anchor has now taken a conformal map of an irregular body apart by where it is cut, by how large the cut is, and by what shape the cut is. Two things it still treats as given are worth naming.

The pins are still two corners. The conformal energy is invariant under a similarity of the plane, so two vertices are held to make the solution unique — and rung eight found that the pins decide the areal spread the solution reports even though they cannot decide whether it is conformal. Every number in this essay uses the two opposite corners of the parameter rectangle, which on a long window are much further apart than on a square one. Whether the U survives a pin choice that scales with the window is not measured.

And nothing here is a whole body. Every patch is simply connected, which is what the Riemann mapping theorem requires and what makes the solve well posed. A map of the whole of an irregular body is the impossibility the field opposite this one is about, and the cut exists precisely to make the theorem apply.