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

Region — where it appears

The ground a map has to cover, which every distortion measure has to be stated over and which changes the answer when it changes. Every distortion statement on this site names one, because a projection's ranking, its optimal aspect and its worst point all change when it changes.

Named by 11 essays across 5 fields — each of them below, with the objects they name alongside it.

The taught rule against the measurement, on 30 regions. Each cell is a region built to order — a box of the stated height and width-to-height ratio, centred at the stated latitude — with the family that actually scores best over it, and whether that is what the rule says. Every family is given its own parameters for the region: the conic its cone constant, the cylindrical its standard parallel and the choice of a normal or transverse axis, the azimuthal its centre. The rule is right on 19 of 30, and where it is wrong it is wrong in one direction: it keys on latitude, and what decides the answer is shape.

The rule of thumb, scored

Cylindrical near the equator, conic in the middle latitudes, azimuthal at the poles. It is the most repeated piece of practical advice in cartography and it has never been run against a population of regions. Run against thirty, it is right nineteen times, and every one of its failures has the same shape.

wrong · Audit Three distances, two degrees of freedom. The Chamberlin trimetric construction with its three centres marked. A point is placed by intersecting circles of its true distance from each centre — but three circles in a plane do not meet in a point, and the three candidate positions obtained by taking the conditions two at a time are 17.0 km apart at the sample point. They are drawn here magnified 120 times about their own centroid, which is where Chamberlin's rule puts the point. The residual is not an error in the arithmetic; it is the amount by which the sphere refuses to be a plane, and it grows as the cube of the size of the region.

Three conditions are one too many

Two distances fix a point in a plane and a third has no freedom left to be satisfied with. Chamberlin's trimetric construction averages the three positions that satisfy two conditions each, and the spread between them — never zero anywhere, 22 km over North America, growing as the cube of the region — is the price of the extra clause.

choosing · Condition Eight world maps ranked by their second derivative. Each projection's root-mean-square flexion and skewness over the whole sphere, combined and ranked. The first-order ranking of the same eight by Kavrayskiy's criterion is given beside each bar; Spearman's rank correlation between the two orderings is 0.69. The two criteria are measuring different derivatives of the same maps and there is no reason for them to agree.

The second derivative has its own ranking

Rank eight world maps by how much they stretch and Mercator comes sixth of eight. Rank the same eight by how much they bend and it comes third. The two halves of the second-order score disagree with each other more sharply than either disagrees with the first-order one — Spearman 0.45 against 0.69.

distortion · Flexion Which of eight places is nearest, decided on the ground and decided on the page. Every cell of the window shaded by which of eight places across Europe is nearest on the ground, with the cells the page's own answer would hand to a different site drawn in the failure colour. Plate carrée misassigns 13.67 per cent of the window's ground area — 3,058 thousand square kilometres. The misassigned cells are not scattered: they lie in bands along the boundaries, which is what a systematic error looks like and what a sampled test of a few query points is least likely to find.

Nearest of many is a partition

One reach question with one source is a disc. With several sources it is a division of the whole surface, every place belonging to whichever source is nearest — and computing that division in the plane the data is stored in hands away between 0.75 and 22.16 per cent of the ground, in unbroken strips up to 1,591 kilometres across.

paths · Reach Two rules, three populations. The taught rule keys on latitude; the replacement keys on shape. On the thirty regions the replacement was read from it scores 83 per cent against the taught rule's 63. On forty-five different regions built the same way it scores 91 — higher, not lower, so the generalisation the shortfall doubted is real. On the seven named regions this collection actually uses, both rules score 29 per cent, and following either costs a mean factor of 14.6. The third bar of each group puts the taught rule's polar clause back into the shape rule, which is what the out-of-sample failures ask for: it repairs four of them, breaks two that were right, and reaches 43 of 45.

The rule scored out of sample

A replacement rule was read off thirty regions and scored on the same thirty, and this collection recorded that as not being evidence about any other thirty. It is: on forty-five different regions the rule scores 91 per cent against the 83 it managed at home. What it cannot do is the seven regions the collection actually uses, where both it and the rule it replaced name the winner twice out of seven and cost a mean factor of 14.6.

wrong · Audit Where the projection's pole should go, and the answer the third rotation moves it to. Every dot is a pole position the search tried, sized by the best score it can reach there when the third rotation is also free — the score is the distortion of Robinson over Europe by Kavrayskiy's criterion, so smaller is better and the large green dots are the good regions. The circled mark is the two-parameter optimum and the square is the three-parameter one: they are 74 pixels apart on this map, which is a different aspect rather than a refinement of the same one. The search costs 8 times the evaluations of the two-parameter one. Drawn in Mollweide.

The third parameter, run

An aspect has three numbers and this site has been searching two of them, with a note admitting it. Searching all three is worth up to 2.1 times — and the obvious way to do it, starting from the two-parameter answer and letting the third move, finds a fraction of that or nothing at all.

choosing · Choosing The scale spread of every grid Britain could have adopted. A grid is conformal by requirement, so its angular deformation is zero everywhere and the whole design problem is the spread of its one remaining number: the largest scale factor over the region divided by the smallest, in parts per million. The grid's own scale factor does not enter — multiplying every scale by a constant leaves the ratio alone, which is why it is chosen last. The bottom bar is Chebyshev's optimum, the conformal map of this region whose scale is constant on its boundary, which no map of any family can beat; the adopted grid sits 1.98 times above it. Measured over a stated box rather than a coastline, because a coastline would put the vendor's generalisation into the answer.

The best grid a country could have had

A national grid is a conformal map chosen for one region, so its whole design problem is one number: the spread of its scale factor. That number has a theoretical floor, this site can now compute it, and the adopted grid turns out to be either exactly optimal or half as good again — depending entirely on which box the country is declared to be.

practice · Grid 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 shortest route between two regions, and the pair it runs between. Europe and the conterminous United States, with the shortest route between them: 3,581 km, attained at one point on each boundary. The pair is not a corner, not a centre and not anything a reader could name — it is wherever two edges happen to come closest, which depends on the whole shape of both. The dashed route is the pair Gall–Peters makes look nearest: 4,085 km, which is 505 kilometres long, and its two ends sit 2135 kilometres from the true pair between them.

The shortest route between two coasts

Ten rungs find the shortest path between two points. Between two regions the answer is attained at a pair of boundary points, and which pair is not the pair any page makes look nearest — a cylindrical map picks a pair 2,135 kilometres from the right one across the Atlantic, and 1,716 kilometres wrong between Chile and New Zealand, where its route is 22.5 per cent long.

paths · Paths 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 How aligned a region's ellipses are, for every projection and every region. The resultant length of the doubled indicatrix orientations, over 12 projections and 9 regions. One means every ellipse in the region points the same way; zero means they are spread evenly and cancel. Three cylindrical rows are 1.00 throughout, and everything else varies down the row and across it — which is the answer to the question the number was first stated without: alignment is a property of the pair. Mercator's row is the exception that is not a measurement, because a conformal projection's indicatrix is a circle and a circle has no orientation.

Whether the ellipses point the same way

The previous rung found that alignment decides whether the average of a region's deformations is above or below the deformation of its average, and stated it at ten projections over one region. Swept over a hundred and eight pairs, the answer is that alignment belongs to the projection 28 per cent, to the region 38, and to neither 33 — and two rows of the table turn out not to be measurements at all.

distortion · Tissot

Named alongside it

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

OptimisationKavrayskiy's criterionAspectPurposeAzimuthalConformalityProjection selectionRankingBoundaryClosed formCompromise projectionConic

All concepts