Map Projections Minimizing Distance Errors
Bibliographic record
Abstract
Maps convey important information about distances between pairs of points. It is therefore desirable to minimize the errors made in representing distances between pairs of points on maps. Since it is just as bad to have two points on the map at twice their proper separation as to have them at half their proper separation, it is the root-mean-square (rms) logarithmic distance between random points in the mapped region that we will minimize. The best previously known projection of the entire sphere for distances is the Lambert equal-area azimuthal, with an rms logarithmic distance error of σ = 0.343. By way of comparison, the Mercator projection has σ = 0.444 and the Mollweide, σ = 0.390. We present three new projections – the Gott equal-area elliptical, with perfect shapes on the central meridian; the Gott-Mugnolo equal-area elliptical; and the Gott-Mugnolo azimuthal, with rms logarithmic distance errors of σ = 0.365, σ = 0.348, and σ = 0.341 respectively – that improve on previous projections of their type. The Gott-Mugnolo azimuthal projection has the lowest distance errors of any map and is produced by a new technique using “forces” between pairs of points on a map, which make the points move so as to minimize σ. The Gott equal-area elliptical projection produces a particularly attractive map of Mars, and the Gott-Mugnolo azimuthal projection produces an interesting map of the Moon, both of which we also show.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.011 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.001 | 0.004 |
| Research integrity | 0.002 | 0.002 |
| Insufficient payload (model declined to judge) | 0.006 | 0.003 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".