Cartographic Projection
every map is wrong, and by exactly how much
Concept

Objective function — where it appears

The quantity an optimisation minimises, which has to be stated before any map can be called better than another. Two objectives applied to the same projections over the same region give different orderings, and there is no ranking underneath them.

Named by 14 essays across 3 fields — each of them below, with the objects they name alongside it.

Angular deformation against latitude, four projections. The same quantity for mercator, gallPeters, winkelTripel, robinson, from the equator to 80°. A conformal projection sits flat on zero in the first of these plots and runs away in the second; an equal-area projection does the reverse. Nothing is flat in both.

Every projection minimises something

A projection is the solution to an optimisation problem, and naming the objective explains more than naming the family. Some objectives are exact constraints, some are least-squares fits, and one is a table of numbers a man adjusted until it looked right.

choosing · Choosing The scale along the Madrid–Tokyo corridor. The scale factor of four projections along the great circle from Madrid to Tokyo, each normalised to its own average over the route so the comparison is of variation rather than of size. The oblique Mercator whose own equator is laid along the corridor holds the scale to 0 parts per million; Mercator varies by 118.7% over the same line. A corridor is a curve, not a region, and the projection an area criterion picks is not the one a curve wants.

Choosing for a line, not a region

Every criterion in this subject integrates over an area. A pipeline, a railway or a coastal survey is a curve, and the projection an area criterion picks for it is not the one it should have — measurably, by a factor of five thousand.

choosing · Choosing Every mixture of Equirectangular at 50.46° and Aitoff. Airy's criterion over the whole sphere, for every weighted average of the Winkel tripel's own pair. Each mixture is normalised to its own best constant scale first, so the comparison is about shape rather than size. The curve has its minimum at w = 0.45, where the criterion is 0.3037 against 0.4114 and 0.4290 at the two ends. That is a projection nobody constructed beating both projections somebody did, by 26 per cent.

The average of two projections

The Winkel tripel is literally the arithmetic mean of two other projections, and this site's implementation of it agrees with that mean to the last bit. Averaging beats both ingredients by 26 per cent — and it preserves conformality exactly, destroys equal-area completely, and can turn eighteen per cent of the world inside out without either distortion measure saying so.

choosing · Choosing The minimum-distortion conformal map of an elongated region, 30° by 10°. The conformal projection of an elongated region, 30° by 10° whose scale is constant on the boundary, which is Chebyshev's criterion, obtained by fitting eight terms of a series rather than by choosing a named projection. The scale factor runs from 0.98640 to 1.00000, a spread of 1.01379; on the boundary itself the largest departure from constancy is 2.13e-7 in the log, which is what the fit achieved and not what it was told. Each dot is an interior sample shaded by its own departure from the boundary's scale.

Solving for the map instead of choosing it

Chebyshev's criterion has sat on this site since its second phase with one case it could be applied to: the spherical cap, whose answer is the stereographic projection. For any other region the site stated the criterion and stopped. It is a linear least-squares fit, and the fitted map beats every named projection over the region it was fitted to.

choosing · Condition Every aspect of Mercator over Japan, and the line the old sweep searched. The regional distortion of Mercator over Japan for every position of the projection's pole — darker is better — with the meridian the site's one-dimensional sweep searches drawn on it. The sweep's best is a gain of 24.4× over the normal aspect; the two-parameter search finds 61.4×, which is 2.51 times better again, at a pole 45° of longitude away from anything the sweep could reach. The shortfall recorded when the sweep was written said this would happen for a region whose long axis runs diagonally, and this is the measurement of it.

The aspect has three numbers, not one

The site's aspect search has swung the projection's axis through one plane for a long time, and recorded that a region whose long axis runs diagonally has its optimum somewhere that plane never reaches. Searching the whole sphere of pole positions finds 2.6 times more improvement over Japan and 3.4 over the conterminous United States.

choosing · Choosing One slice of the aspect objective, at the best γ. The Kavrayskiy score for Robinson over Japan, as the pole is moved over the whole sphere with the third rotation held at the value the search settled on. Dark is good. The marks are local minima of the full three-dimensional grid that happen to lie in this slice: there are 6 of them here and 58 in the cube, and a search that walks downhill from a random start reaches the best of them 7 per cent of the time.

The landscape the search walks on

The three-parameter aspect search was run and its answer recorded with a note admitting nothing proved it global. Mapping the objective finds 26 to 34 local minima for every projection and region tried, a downhill walk from a random start reaching the best of them 6 to 35 per cent of the time — and one seed from the coarse grid the search already uses reaching it in all four cases. The score is reproducible to two per cent across a sevenfold refinement; the pole it names moves 60 degrees.

choosing · Choosing The parameters are not reproducible and the map is. The three-parameter aspect search run at three grid resolutions and compared with the finest, twice over. Compared on the numbers it returns, the answers are 65° of pole apart. Compared on what they do to the region — the root-mean-square difference in angular deformation at every sample — they are 0.29° apart, against a map whose own deformation over that region averages about a degree. The disagreement recorded as a shortfall is a disagreement about coordinates for one map.

Report the map, not the parameters

The previous rung found the aspect search returning the same score to 2.3 per cent from poles sixty degrees of latitude apart, and recorded that as a shortfall: the answer was not reproducible. The shortfall assumed the disagreeing triples make disagreeing maps. They do not — the three answers agree on the distortion field to a quarter of the deformation the map already has.

choosing · Choosing The set of aspects within a stated distance of the best, for Robinson over Japan. Each row takes every point of a 36 × 19 × 24 grid in the three aspect parameters that scores within (1 + t) of the best, joins neighbouring points, and identifies the pieces the exact degeneracy relates. At t = 3 it is one connected piece spanning 170° of pole; by t = 1 it has broken into 14 pieces; and by t = 0.3 the largest of them spans 11°. So it is not one valley and it is not one basin — it is a sheet that fractures.

The shape of the valley

An aspect search returns three numbers, two searches return triples that differ by a hemisphere, and the maps they produce agree. One cause is an exact degeneracy and the rest was called a valley and left unmeasured. Sampled densely, it is neither a valley nor a basin: a connected sheet spanning 170° of pole that fractures into fourteen pieces once the threshold tightens.

choosing · Choosing Where Robinson's valley breaks, against how large the region is. The threshold at which the set of near-optimal aspects stops being one connected piece, for square regions of growing size at 38° north. It falls from 2.13 at 6° to 0.27 at 30°, a factor of 8.0. The previous rung measured this at one region and quoted "about twice the optimum"; that value belongs to a small region, and the prediction that it should fall as the region grows is what this tests.

Where the valley breaks in two

The previous rung found a near-optimal set that is one connected sheet at a loose threshold and fourteen basins at a tight one, and explained the transition without testing it. The explanation is a prediction about region size and projection sharpness: swept over both, the threshold falls from 2.13 to 0.27 as a region grows from 6° to 30°, three projections lie on nearly one curve, and a fourth declines to join for a reason worth having.

choosing · Choosing The part of a request no conformal map can supply — scale falling with distance from the centre. The difference between the requested scale field and the nearest achievable one, over the patch, with the sign shown by colour. The RMS is 1.81e-1 in the logarithm of the scale and the worst single point is 4.91e-1. This is not an error: it is the part of the request that no conformal map of any kind can grant, and its SHAPE is the answer — it says where the request was impossible, which a single number cannot.

The nearest map to an impossible request

Rung seven found that not every distortion can be asked for. It never asked what happens when one is asked for anyway — and the answer has a shape: the achievable fields are the solutions of an elliptic equation, a request is a point off that set, and the nearest point leaves a residual whose floor is the curvature rather than the size of the ask.

choosing · Condition The threshold, and the two things it is a ratio of. The near-optimal set's fracture threshold on Robinson, against the size of the region, with the two quantities it is a ratio of drawn beside it. The threshold falls with fitted slope -1.293. The best score a region admits at all rises with slope 0.923 — a bigger region is harder to map — and the absolute score of the pass falls with slope -0.370. The first is the sum of the other two by construction, and the arithmetic says which of them is doing the work: the denominator carries 71 per cent of it.

The first break is mostly its denominator

Three rungs have fitted the near-optimal set's fracture threshold against region size and read the answer as a statement about the landscape. It is a ratio, and separating it takes one multiplication: the pass's own depth is constant to 12 per cent below twenty degrees of span, and the whole of the threshold's movement there is the denominator — the best score the region admits at all — rising with exponent 0.92.

choosing · Choosing One quantity, one region, and the exponent left free. The scale departure of six projections over the world, aggregated as a p-norm, against p on a logarithmic axis. At p = 1 the best is Eckert IV; at p = 64 it is Winkel tripel. Nothing about the maps changed between the two ends of the axis — only how much of the region a bad point is allowed to spoil. Drawn in no projection: the axes are an exponent and a score.

The average was a choice of norm

Ten projections, one region, one measured quantity, and the only free decision left is how to turn a field into a number. Over the world's scale departure the ordering at the mean and the ordering at the worst case have a rank correlation of −0.04, all ten maps change position, and the exponent that produced each answer is stated nowhere.

wrong · Audit Two regions, three answers. Britain and New Zealand, 166° apart, with the pole of the best oblique conic under each of three objectives. Pooling the samples and taking an area-weighted score puts the pole in one place; refusing to let either region be worse than the other puts it somewhere else. The regions are drawn on Mollweide so that equal ground areas are equal page areas.

The pooled score abandons a region

Fifteen rungs optimise for one region. An atlas is several, and pooling their samples into one area-weighted score is what everybody does — which on Britain and New Zealand serves Britain 1.2 times worse than it could be served alone and New Zealand 125 times worse. The worst-case objective makes them equal at 33 and 59, and the cost of sharing rises with separation from 1.4 to 59.

choosing · Choosing One field, one classifier, two sets of breaks. A field that varies with latitude put into 5 classes by area-weighted quantiles, on Mercator. The upper panel weights each region by its ground area and the lower by the area it occupies on this page, which is what a classifier handed projected geometry does. 60 of 150 regions land in a different class, marked in the lower panel. The data has not changed and neither has the number of classes.

The class breaks were computed on the page

The three rungs below price what a reader does with a finished map. A classifier is software, it runs on the geometry it has, and the geometry it has is projected: a five-class quantile classification of one stated field puts half of the three hundred and eighty-four regions in a different colour on Mercator, and 87.5 per cent of them at nine classes.

distortion · Thematic

Named alongside it

The objects these essays reach for when they reach for this one.

OptimisationAspectDegeneracyRegional distortionConformalityKavrayskiy's criterionPurposeReproducibilitySearchVerificationAggregationAiry's criterion

All concepts