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

Parameter search — where it appears

A systematic exploration of a projection's free parameters to minimise a stated criterion over a stated region. Its answer is reproducible in the map it produces and not in the parameters it reports, which is a distinction worth making before the parameters are published.

Named by 8 essays across 3 fields — each of them below, with the objects they name alongside it.

Every candidate fitted to one map's graticule, ranked by what is left over. The map is drawn in Conformal conic over a region 40° tall centred at 45° north, and the projection is not told to the fit. Each candidate is evaluated at the same 121 graticule crossings, its own parameters are searched, the best plane similarity between its output and the picture is removed, and what remains is drawn as a proportion of the map's own width on a logarithmic scale. Conformal conic fits to 1.3e-10, which is the arithmetic's floor; the next candidate is 3.7e+7 times worse.

A map does not say what it is

Every map on this site is one the site drew, from a projection it chose. Every map a reader has ever used is the other kind — a picture whose projection is a sentence in a corner, a legend, or nothing at all. The graticule is enough to recover it: fit every candidate to the crossings and rank what is left over.

wrong · Identify 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 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 2 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 splits in two at twice the best score and into thirteen pieces by 1.7 times it.

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 2.10 at 6° to 0.27 at 30°, a factor of 7.9. The earlier measurement was made 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

A near-optimal set is one connected sheet at a loose threshold and thirteen basins at a tight one, and the transition was explained without being tested. The explanation is a prediction about region size and projection sharpness: swept over both, the threshold falls from 2.10 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 Two members, their average, and the family's best answer to it. Two equal-area conics at cone constants 0.25 and 0.85, the average of the two, and the member of the family nearest that average — at 0.540, which is not the parameter midpoint. The average is not a conic at all: its parallels are still arcs but they are arcs of circles about different centres, so no single cone constant reproduces it and the residual is 0.1722.

A family is not closed under averaging

A family is a set of maps with a parameter running through it, and the compromise projections here are averages of members. An average of two members is not a member: two conics far apart average to something 31 per cent of their own separation outside the family — and averaging within a family makes the map worse every time, while averaging across two makes it 16 per cent better.

families · Families Three named members of the family, and one the family has no name for. The equal-area pseudocylindricals are not a list of maps. They are the solutions of one equation — C(φ)·Y′(φ) = cos φ — in two unknown functions, so one function is free and the named members are points in a space of them. Writing Y as six numbers and searching over them finds a map at 25.03° of mean angular deformation against Eckert IV's 28.08, Mollweide's 31.83 and the sinusoidal's 38.86 — 12.2 per cent better than the best map the family has a name for, and every one of the four is equal-area to the same precision.

A family is a function, not a list

The equal-area pseudocylindricals are the solutions of one equation in two unknown functions, so one function is free and the named members are points in a space of them. Writing that function as six numbers and searching over them finds a map at 25.03° of mean angular deformation against Eckert IV's 28.08 — and two numbers are already enough to beat every map the family has a name for.

families · Families Three solved maps span a plane of maps, and the band's best map lies well outside their triangle. Every conformal map of a 25° spherical square whose ln k is the p = 1 map's plus s times its difference from the least worst map plus u times its difference from the mean-square map, for s and u from −4 to 4, contoured by the share of the square it keeps inside a band of ±0.74 per cent — 0.5 of the least worst departure. The three maps are the corners of the shaded triangle at (0, 0), (1, 0) and (0, 1); the line between the first two is its lower edge, and the exponents from 1 to 32, dashed, run from one corner to the other through the mean-square map and out past it. The best map of the twelve exponents keeps 66.4 per cent, the best on the line 66.5, the best inside the triangle 66.5. The best in the plane sits at s = 0.72, u = −2.81, outside every one of them, and keeps 75.8.

Three maps make a plane, and the band's best map lies outside their triangle

Every conformal map whose ln k is a fixed blend of three solved maps — the p = 1 map, the least worst map and the mean-square map — is itself a conformal map, so three solved maps are a plane of them. Inside their triangle the third map recovers most of what the line between the first two gave up on thin regions: 0.63 points short of the best exponent on the thinnest ellipse, where the line was 1.48 short. But the band's best map in the plane is not between the three. On 172 of 182 regions and bands it lies outside the triangle, and it keeps up to twenty points more than any exponent; started from it, a search over all twenty-five coefficients adds about a point on four regions of five and seven on a hexagon. Its price is a worst departure up to nearly five times the least worst.

choosing · Condition Each cap on the worst departure is a long convex band of maps, and the best map under it rides the band's edge. The plane of conformal maps of a 25° spherical square whose ln k is the p = 1 map's plus s times its difference from the least worst map plus u times its difference from the mean-square map, for s and u from −4 to 4. Dashed: the edges of the maps whose worst departure is at most 1.2, 1.5, 2, 3 times the least worst — convex, nested, long and narrow, each running diagonally through the least worst corner. Thin contour: the maps keeping as much of the region inside a band of ±0.74 per cent as the best exponent's 66.4 per cent; heavy contour: three points under the uncapped best, 72.8 per cent. Dots: the map keeping the most inside the band under each cap — 1.2: 59.8 per cent at (−0.30, 1.56); 1.5: 61.3 per cent at (−0.17, 1.03); 2: 65.8 per cent at (0.50, −0.36); 2.5: 69.1 per cent at (0.51, −1.00); 3: 72.1 per cent at (0.55, −1.63); 4: 75.8 per cent at (0.72, −2.81); the large dot is the uncapped best, 75.8 per cent at a worst of 3.82 times the least worst.

A cap on the worst departure keeps the thin regions' gain and gives back the square's

The conformal map that keeps the most of a region inside a band can sit far out in the plane of three solved maps and let its edges go to nearly five times the least worst departure. Capped at twice the least worst, the plane still matches or beats the best exponent on 69 region-and-band pairs of 91: on the thin ellipses it keeps the whole of its gain, because that gain sits in a small pocket just past the exponents' own worst departure and arrives all at once; on the square and the bays it gives most of it back, because there the gain is a ramp that runs out to four times the least worst. Under any one cap the plane still beats the exponents that cap admits, by up to eighteen points — and it can name maps that no weighting of area against worst departure would ever choose.

choosing · Condition

Named alongside it

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

Objective functionOptimisationAspectIdentifiabilityLocal minimumRegionReproducibilitySearchVerificationCone constantDegeneracyLeast-squares

All concepts