The Groundtruth Seam · a scorecard, re-derived
The Line That Isn't in the Paper
In 2007 Goldberg and Gott scored world maps on six errors and the Winkel Tripel came first with 4.5629. The paper says its samples are uniform on the sphere. The authors' archived code multiplies every drawn latitude and longitude by 0.995 first. Two of the paper's own numbers need that line. Two others rule it out.
This page does not argue about map projections. It hands you the arithmetic. The first two panels are exact: closed forms and long division, no simulation anywhere, and you can check them on paper. Only after they land does anything here run a random number.
Two cells that must be twice each other
Take the unit sphere. The equirectangular map is x = λ, y = φ. Along a meridian it neither stretches nor shrinks; along a parallel it stretches by 1/cosφ. So the product of the two principal scale factors is ab = 1/cosφ. Mercator is x = λ, y = arsinh(tanφ), conformal, with both scale factors equal to 1/cosφ, so its product is 1/cos2φ.
The paper's area error A is the root mean square of ln(ab) about its own mean. Doubling a random variable doubles that. So whatever points you sample, however you sample them, seeded however you like, AMercator = 2 Aequirectangular. It is not a modelling choice. Table 1 of the paper prints A = for equirectangular and A = for Mercator.
Rounded to two places, the equirectangular cell pins the true value to , so the Mercator cell is pinned to . The printed Mercator cell lies in . Those two intervals are computed here and compared here; no sampling scheme can bridge them.
What the exact values actually are
Both cells have closed forms. With longitude uniform and sinφ uniform, which is what "uniform on the sphere" means, the expectation of any even function of latitude is ∫0π/2 g(φ) cosφ dφ, and the integrals close:
Now the archived code. The authors' downloadable map_random.pro draws a uniform point and then multiplies both the latitude and the longitude by a factor, written in the source as 0.995. The paper does not mention it. Move the factor and watch which printed cells come into range and which leave. Every value in the third column below is a Gauss-Legendre quadrature run in your browser when you move the control, not a stored value.
| exact quantity | printed | exact value | rounds to | verdict |
|---|
contraction factors that reproduce the printed cellthe archived 0.995
The two equirectangular cells need the contraction. The two Mercator cells forbid it. That is the finding, and it took no random numbers to reach. The equirectangular row matters more than the others, because the paper takes its six values as the normalizing constants Ni … Nb that divide every other map's errors before they are squared and summed.
Adding up the paper's own column
Before any new number, the instrument has to reproduce a published one. The paper's combined score is a plain sum of squares, and the paper prints six of them. Take the printed six-tuples from Table 1, divide by the printed normalizers, square, add. The whole of Table 1 as it appears in the PDF is shipped with this page and parsed in your browser.
| projection (as printed) | I | A | F | S | D | B | Σ re-added | Σ printed | difference |
|---|
The aggregation formula is confirmed. Four of the six ranked scores fall out of the printed components exactly. Two do not, and the shortfall is far larger than any rounding of the printed cells can produce. That says the ranked list and the table were not written from the same numbers, which is a separate defect from the sampling convention above.
The same instrument, run again
Only now does anything here draw a random point. The engine below implements the six 2007 definitions directly: singular values of the differential from an orthonormal north-east frame, a fourth-order centred stencil in geodesic arclength for the curvature terms, and logarithmic scale errors with the mean removed. Every forward map in it was checked against PROJ 9.5.1 on a unit sphere; the worst disagreement found anywhere on a global grid is .
| projection | I here | I printed | A here | A printed | F here | F printed | S here | S printed | worst gap |
|---|
Flip the convention and run it again. The median gap is the number to watch: it is several times smaller under the contraction, and pushing the sample count up shrinks it very little, which says the residual is a difference of convention rather than of sample size. The heavy-tailed columns are F and S: under the paper's literal reading the second moment of the skewness diverges for any map with a polar singularity, so the estimate wanders and no honest error bar exists. Contracting the latitude by 0.995 cuts the last 0.45 degrees off each pole and makes every moment finite. Whatever else it is, that line is also a regularization.
The distance column, and the one number that will not come back
The distance error D is a different experiment: random pairs of points, the logarithm of the ratio of the plotted separation to the great-circle arc, root mean square about its own mean. The paper prints D to three places for thirteen maps this page can also draw. Twelve come back. One does not.
| projection | D printed | D here | difference | pairs |
|---|
The Winkel Tripel distance value is the single number that decides the paper's headline. Push the pair count up and the estimate here settles; it does not settle on the printed value. This page cannot say why. It can say that the same code, the same seeds and the same definitions return the other printed distances, including the Winkel Tripel variant on the very next line of the table.
Now make the dismissal fail
The catalogue is not a curated shortlist. The pinned library exposes operations. of them carry no projection class line in PROJ's own operation table and are transformations rather than maps. Of the rest, a mechanical filter removes anything that will not construct on a unit sphere without further parameters, anything whose forward map is not finite across a global grid, and then a reviewed manifest removes interrupted layouts, polyhedral tilings, hemispheres, regional grids and constructions that are transverse or oblique by design. Four more are removed on measured evidence. configurations survive. Every exclusion and its reason is listed at the bottom of this panel.
on the frontier of the weighted componentsdominatedcurrently first
| # | projection | PROJ | I | A | F | S | D | Σ | P(first) | frontier |
|---|
The eligibility ledger: every operation in the pinned build and why it is in or out
| operation | name | class | stage | status and reason |
|---|
That is the honest answer to the dismissal, and it cuts both ways. The convention argument of the first two panels is real and it changes what the numbers mean, but it does not by itself change who wins. What changes who wins is the weighting, which the paper itself called not unique and offered only as a simple illustration. Move the flexion weight up and the conformal maps take the top. Score area alone and every equal-area map ties at exactly zero, which is a fact about their definitions and not a measurement, so the panel says so when it happens.
The number we are claiming
Two quantities on this page are ones we went looking for and did not find in print.
First. The exact closed forms for the equirectangular normalizers, A = √(1 − π2/12) = and I = √(2 − 2ln2 + ln22 − π2/12) = , together with the pointwise identity that forces AMercator = 2 Aequirectangular and therefore makes the printed pair 0.41 and 0.84 mutually inconsistent. We could not find this published as of 2026-08-01, having searched the 2007 paper and its arXiv v1 and v2, the companion distance paper astro-ph/0608500, the 2021 restatement arXiv:2102.08176, the authors' archived code and project page at physics.drexel.edu, the PROJ documentation, and web searches for the closed forms, for the Goldberg-Gott score with the words reproduce, errata and normalization constant, and for GitHub implementations of flexion and skewness.
Second. The six errors, composite and rank for all single-sheet whole-world configurations the pinned PROJ 9.5.1 build will draw, under both sampling conventions, with an exclusion reason recorded for every one of the operations in that build. The configuration that comes first on this instrument under the paper's own printed normalizers and equal weights is , at against for the Winkel Tripel measured the same way on the same points. We could not find this published as of 2026-08-01, having searched the same places plus the PROJ 9.5.1 operation table itself.
Three things that claim is not. It is not a claim to have found the first map that beats the Winkel Tripel on this measure: the authors published a Gott-Wagner variant at 4.497 against 4.563 in 2021, and a double-sided azimuthal map at 0.881. It is not a comparison with the published 4.5629, because that number and ours come from different instruments; the two numbers quoted above are both measured here, on the same sample, which is the only comparison this page will make. And the confidence that it is unpublished is moderate rather than absolute: a thesis, a private notebook or a table under different terminology would change it, and the 2007 authors wrote plainly that they had not done all known projections. This page did not originate the question and does not claim to have found the best map.
The check
Everything numeric above is computed in your browser when the page loads or when you move a control. The offline verifier recomputes all of it a second time with independent code, then runs this page's own engine and compares, then corrupts inputs on purpose to prove the comparisons can fail.
| primary record | Goldberg and Gott 2007, arXiv:astro-ph/0608501v2, PDF sha256 ; Table 1 and the ranked list shipped verbatim and parsed here |
|---|---|
| which version | v2 of 2007-04-12, which reversed v1's winner; v1 ranked Kavrayskiy VII first with a different Winkel row |
| sphere | unit radius, geographic latitude, no ellipsoid and no authalic or geocentric latitude |
| sampling | longitude uniform, sin(latitude) uniform, azimuth uniform in physical bearing; contraction factor is a reader control |
| logarithms | natural throughout |
| derivatives | fourth-order centred stencil, step radians of geodesic arclength |
| quadrature | -node Gauss-Legendre on panels graded toward the pole |
| catalogue | PROJ via pyproj ; operations, scored |
| offline run | local samples and point pairs per configuration per convention, generated |
| boundary term | B = 1/4 for every scored row, by the single pole-to-pole seam; it is assigned, not measured |
- The boundary column is vacuous for ranking. Every scored configuration here has a single 180 degree seam, so B is 0.25 for all of them and its weight shifts every composite by the same constant. The B slider is present because the paper's formula has the term, but it cannot reorder anything. The columns that can are I, A, F, S and D.
- One normalizer setting is true by construction. Choosing "this run's own equirectangular row" makes the equirectangular composite exactly 6.000000 whatever the errors are. That row is then not evidence about equirectangular. The honest comparison is between "as printed" and "exact".
- F and S have no trustworthy error bar under the uniform convention. They are means of absolute second-derivative quantities whose second moment diverges for maps with a polar singularity. The stability replicates use the offline batch-means standard errors, which understate that tail. Treat P(first) as a spread under the stated errors, not as a confidence statement.
- Four projections were removed on measured evidence, not taste. Eckert I, Eckert II and PROJ's van der Grinten have a corner along a curve of the sheet, and PROJ builds Robinson by interpolating a nineteen-point table. For all four the measured flexion changes by more than two percent when the stencil step changes by a factor of ten, so what it measures is the kink or the interpolant. The measured spreads are in
data/stencil-screen.json. - The offline pipeline discards a sample whose stencil would cross the 180 degree seam, because PROJ wraps longitude and the two banks are different points of the sheet. The discard rate is about five in 1.2 million. The browser engine has no such discard: its analytic maps continue the longitude branch through the seam instead. Both paths are compared in the verifier.
- Configurations that score identically are the same map. The offline run finds exact ties between Werenskiold I and Putnins P4', between Winkel I and Eckert V, and between Foucaut sinusoidal at its default and the Lambert cylindrical equal area. All six errors are invariant under a uniform rescaling of the plane, so identical vectors mean affinely identical maps, not a bug.
- Free choices we made and you cannot see from the numbers: PROJ defaults are used for every parameterized entry except two the paper itself tabulates, the cylindrical equal area at 45 degrees and the Winkel Tripel at 40 degrees; the sphere is the unit sphere rather than any Earth radius; the distance experiment uses the shorter great-circle arc rather than the chord; and D is centred on its own mean log, which is equivalent to choosing the one global map scale that minimizes it.
- What is transcribed rather than computed: Table 1 and the ranked list, character for character from the paper's own text layer. The composites beside them are recomputed here.
Run the independent checker with node research/map-projection-ranking/verify-map-projection-ranking.mjs. It reproduces the closed forms by an independent series, re-parses Table 1 from the shipped excerpt, re-adds every composite, re-measures the six errors with its own Monte Carlo and its own random generator, then extracts and executes this page's engine and asserts the two agree. It finishes by mutating coefficients and asserting the checks go red.
What this page does not settle
It does not explain the Winkel Tripel distance value. The measurement here is stable and repeatable and lands at about against a printed 0.374, while the neighbouring printed distances come back to within about . Chord distance instead of arc, contracting one point of the pair, the Times Atlas standard parallel and periodic wrapping of the sheet were all tried in the blinded re-derivation kept in this repository, and none of them lands on 0.374. The archive the authors still host carries 2006 file dates and a Winkel formula that reproduces the superseded v1 row, so it is evidence about v1 rather than about the corrected table.
It does not settle whether the two ranked composites that fail to re-add, Kavrayskiy VII and the Times Atlas Winkel Tripel, are typesetting damage in the table or arithmetic done on different numbers. Either is consistent with what is visible.
It does not optimize anything. Continuous parameters, central meridians, oblique aspects and interruption layouts are all left where PROJ's defaults put them, and the maps that need a cut manifest to be scored honestly are excluded rather than approximated. A projection is not one map, and this page measures one canonical configuration of each.
Primary records: Goldberg and Gott, arXiv:astro-ph/0608501, journal form Cartographica 42(4), 297-318; the companion distance paper Gott, Mugnolo and Colley, Cartographica 42(3), 219-234; the 2021 restatement Gott, Goldberg and Vanderbei; the authors' projection page and archived IDL, which is inspected here and not copied; and PROJ, X/MIT licensed.