Cuts of the same size in different places

The map depends on where it was cut established that a body has to be cut before it can be mapped and that where the cut goes decides how conformal the answer can be, by a factor of ten. It did that with five windows — a broad band, a narrow band, a tall window, one off centre and a small square — and it recorded, in its own list of what it had not done, that the five are of different extents.

That is a confound and it is a fatal one for the claim as stated. A patch’s own size affects how well any map can serve it, so a factor of ten across five windows of five sizes could be entirely a statement about size. What was owed is a family of cuts of the same extent in different places.

Holding the size, properly #

Equal extent in the body’s own parameters is not equal size on the body. A rectangle in colatitude and longitude covers less surface towards the pole by a factor of sin θ, and a body with lobes on it varies again on top of that — so seven windows of identical parameter extent have surface areas differing by a factor of two, and the confound would still be there.

So the longitude extent is divided by sin θ, which keeps each window roughly square on the surface, and a single scale is then bisected per window until the measured surface area matches a target. The seven windows come out at surface areas between 0.73042 and 0.73215 in the body’s own units: a range of 0.24 per cent.

With that held, the conformal solve is run on each and its areal spread — the ratio between the largest and smallest area factor over the patch — is recorded.

The result, and what it is not #

The areal spread across the seven windows runs from 1.034 to 1.363, which is a factor of 1.32.

That is much less than the factor of ten reported without the size held, and the difference is the measure of how much of the original number was the confound. Most of it was: five windows spanning a factor of several in area produce a spread dominated by their sizes, and holding the size removes the bulk of the effect.

What remains is not nothing. A factor of 1.32 between the best-placed and worst-placed window of identical size is a real difference in what a map of that patch can do, and it cannot be a statement about extent because the extents agree to a quarter of a per cent.

The angular deformation, meanwhile, stays small everywhere: a median between 0.088° and 0.840°, on a discrete conformal solve whose residual is what that number mostly measures. That is the expected result and it is worth stating as a control — a conformal map of a simply connected patch always exists, by the Riemann mapping theorem, so a solve that reported large angular deformation somewhere would be reporting its own failure rather than the body’s geometry.

What predicts it #

The seven windows differ in one measurable way that is not their size: how much curvature they contain.

Total Gaussian curvature over a patch is computed here as the summed angular defect of its own triangulation — the discrete Gauss–Bonnet statement, which needs no derivatives of the surface and is exact for a polyhedral approximation of it. Across the seven windows it runs from 0.234 to 1.479, a factor of six.

Plotted against the logarithm of the areal spread it gives a correlation of 0.969, with more curvature meaning more spread.

So where a body should be cut is not a matter of taste and it is not a matter of the solver’s convergence. It is a matter of how much curvature the piece contains — which is the quantity the impossibility has always been about, doing the predicting rather than merely forbidding.

The mechanism is the one this collection keeps arriving at. A conformal map of a patch exists and is essentially unique, so the only freedom left is how its scale factor varies — and the scale factor’s variation is forced by the curvature it has to absorb, through the same Liouville relation that makes a scale field unaskable-for. More enclosed curvature, more variation, larger spread.

Why the question has no answer on a sphere #

It is worth noticing that this rung could not exist for the Earth.

A sphere has constant curvature, so the total curvature of a patch is proportional to its area by Gauss–Bonnet, exactly. Hold the area and the enclosed curvature is held automatically; every window of the same size is equivalent to every other, up to a rotation. There is no where to separate from how much, because the two are the same variable.

That is the whole reason the irregular-body ladder has to ask questions the terrestrial ladders do not. On a body whose curvature varies by a factor of six between one part and another, area and curvature content are independent, and a decision that is empty on a sphere — where to put the patch — becomes a decision with a measurable cost.

It also says which bodies will show the effect most strongly: not the largest or the most irregular in outline, but the ones whose curvature contrast is greatest. A smooth triaxial ellipsoid has a modest contrast; a body with a crater rim or a lobe boundary has a sharp one, concentrated in a small area, and a patch that contains it will be much worse than one beside it of the same size.

Where the windows are #

The pattern in the table is legible on the body. The three windows with the most curvature are the ones facing the body’s lobes — the raised regions the shape model puts on it — at 1.479, 1.384 and 1.115. The two with the least are the ones between lobes, at 0.234 and 0.376. The high-latitude window sits between at 0.613.

That is a satisfying result for a practical reason: a body’s lobes are visible. Anybody choosing where to cut an irregular body has a shape model in front of them, and the rule that falls out is to cut so that each patch avoids containing a whole lobe — which is what a person would do by eye, now with a number attached and a reason.

It is also the same rule the interrupted projections of the Earth follow. Giving up continuity puts the cuts in the oceans, and the reason usually given is that nobody lives there; the reason available here is that the lobes of the thing being mapped are the continents, and a lobe inside a patch is curvature inside a patch.

What the number is against #

A factor of 1.32 in areal spread needs something to be measured against, or it is a number with no size.

The best-placed window has a spread of 1.034, which means the area factor across the whole patch varies by three and a half per cent — a map that is very nearly equal-area as well as conformal, which on a curved body is as good as it gets. The worst has 1.363, so its area factor varies by more than a third across the patch.

For a reader deciding whether that matters: a third of a variation in area factor across one tile is roughly what Mercator does between the equator and 30° north, on a map nobody would use for area. So the difference between a well-placed cut and a badly placed one of the same size is the difference between a tile that is nearly free of areal distortion and one carrying as much as a familiar bad case.

Stated as a choice: cutting an irregular body into patches of equal area and letting the boundaries fall where they may gives some tiles that are excellent and some that are mediocre, and the mediocre ones are predictable in advance.

What was computed, and how #

The body is an irregular one built from a stated rule: a triaxial ellipsoid with axes 1, 0.94 and 0.86 and a bump function on it of stated amplitude, so its shape is a formula rather than a scan. Nothing here is a measured asteroid, for the same reason the curve work builds its curves: a real shape model has a resolution, and a measurement made on one is partly a measurement of that.

The conformal map is the discrete Cauchy–Riemann least-squares solve this collection already uses: one complex equation per triangle, conjugate gradients on the normal equations, two vertices pinned to remove the similarity the equations cannot see. Its residual is what the median angular deformation above mostly reports.

The pins are held to a well-separated pair throughout, because the pinning experiment established that two adjacent pins fix the similarity through a short lever and cost one corner a great deal — a conditioning effect that would have entered this measurement as a difference between windows.

The area targeting is by bisection on the measured area rather than by a formula, at forty iterations, because the body has no closed form for the area of a parameter rectangle and using one from the ellipsoid would have left an error that varies exactly where the lobes are.

The two-part check: the areas must agree to within a per cent, and the spreads must not. If the areas differed the whole measurement would be the old confound in new clothes; if the spreads agreed, the earlier factor of ten would have been shown to be entirely size and this rung would have had a different result to report.

What the correlation does not settle #

One thing the fit above deliberately does not claim is a functional form.

The slope of the fitted line is 0.222 in log spread per unit of enclosed curvature, over a range of curvature from 0.234 to 1.479. A slope fitted over one octave of one variable on seven points is a local description, and reading it as a law would be reading far more into it than the data holds.

What the data does hold is the sign and the strength: more curvature is worse, monotonically across all seven windows, with a correlation of 0.969. That is enough to use for ranking candidate cuts against each other, which is what the practical recommendation above needs, and it is not enough to predict a spread from a curvature.

Getting a form would need windows spanning a much wider range of enclosed curvature at fixed area, which on this body means either a more extreme shape or much smaller windows — and smaller windows bring the solve’s own residual up towards the effect being measured. That is the experiment this rung would have to run to say more, and it is a different experiment rather than more of this one.

Where the model stops #

One body. The correlation is over seven windows of one irregular shape. A body with sharper features would give a wider spread and probably a steeper slope; a nearly spherical one would give almost no variation at all, because there is no curvature contrast to enclose.

Curvature is one predictor and it is not proved to be the mechanism. A correlation of 0.969 over seven points is strong evidence and it is seven points. The mechanism argued above — through the Liouville relation — is a reason to expect it rather than a derivation of the slope, and the slope itself is not predicted from anything.

Equal area is not equal shape. The windows are held to the same surface area and made roughly square by dividing the longitude extent by sin θ, but “roughly square on a lumpy body” is not a precise statement and the residual variation in aspect ratio is not measured. A long thin window and a square one of the same area would not be expected to behave alike.

The solve is discrete. The angular deformation reported is dominated by the mesh, so the statement “conformal wherever the cut is put” is a statement at the resolution used. A refinement study is what established that it converges and it is not repeated here.

The generalisation #

The shortfall is paid, and the payment changes the claim rather than confirming it.

Where a cut goes matters, and it matters by about a third rather than by an order of magnitude. The larger number was mostly size.

What decides it is the curvature the patch encloses, and that is measurable in advance. A cut can be chosen by computing the angular defect of each candidate patch — which needs only the shape model and no solve at all — rather than by solving a conformal map for each and comparing.

That second half is the useful one. The solve is expensive and the defect sum is cheap, so a body can be partitioned by curvature and the maps computed afterwards, in the confidence that the partition was chosen on the quantity that decides the answer.

And it is a small vindication of the whole approach this collection takes to irregular bodies. The question “where should this be cut” sounds like a matter of judgement, and it turns out to have a scalar attached that anybody can compute — which is what happens whenever a decision in this subject is examined instead of made.

One consequence of the correlation is worth stating for anybody choosing where to cut rather than measuring what a cut cost. The enclosed curvature is an integral, so it can be computed for a candidate window before any solve is run — Gauss–Bonnet gives it from the window’s boundary alone, as the boundary’s turning subtracted from 2π. The predictor is therefore available a great deal more cheaply than the thing it predicts, which is what turns a correlation of 0.969 from a description into a design rule: sweep the candidate windows, integrate the curvature round each boundary, and solve only the one that encloses the least.

Who found it, and when #

Gauss–Bonnet is 1848 in Bonnet’s form, and the discrete version — curvature as an angular defect summed over vertices — goes back to Descartes and is the same statement the polyhedral work uses.

The Riemann mapping theorem is 1851, and its consequence here is the one that makes the measurement possible: since a conformal map of any simply connected patch exists, the interesting quantity is not whether the map is conformal but how badly its scale varies, which is what the areal spread measures.

The practice of cutting irregular bodies into patches is modern and is driven by spacecraft imaging: shape models of asteroids and comets are mapped in tiles, and the choice of tiling is made by mission planners on grounds of coverage and viewing geometry. Whether curvature enters those decisions explicitly is not something this collection can say; that it predicts what the tiles can achieve is what is measured here.

Where the ladder goes next #

Nine rungs have taken a coordinate onto another body, put a projection on it, made the body a non-quadric, solved a conformal map on it without a formula, found that the pins and the cut both matter, and now separated where the cut goes from how big it is.

What is still unmeasured is the shape of the cut at fixed area and fixed curvature: a long thin patch and a square one containing the same curvature are not the same problem, and the conditioning of the solve on the first is visibly worse. That is a third variable in the same experiment and it is one this rung deliberately holds rather than explores.