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

Aspect search — where it appears

A search over the three angles of a rotation for the aspect that scores best over a region. Its answer is a map rather than a set of parameters, because the objective is nearly flat along one direction and two searches that agree about the map need not agree about the angles.

Named by 4 essays across one field — each of them below, with the objects they name alongside it.

Also named here as basin, optimisation landscape — the same set of essays touches all of them, so they are one junction rather than several.

The pass between the basins, measured rather than bracketed. The height of the lowest path from one near-optimal basin to the other, for regions of growing size. It is found by sorting the score surface and joining cells in order, so it is the minimax path's own height rather than the level at which a bisection stops finding two pieces. The fitted exponent is -1.34 at a coefficient of determination of 0.986. The earlier level-set measurement on the same projection gave -1.57 and the argument predicts −1, so measuring the height directly moves the answer towards the prediction, though not onto it.

The height of the pass between two basins

One thing was left owed: the fracture threshold falls faster than the argument predicts, and the difference was supposed to be the level set's blindness to the height of the pass. Measured directly, the pass between the two deepest basins falls as the −1.34 power at R-squared 0.986 — nearer the predicted −1 than the level set's −1.57 on the same projection, and not onto it. It is not the pass that breaks the set first, and at five of six sizes the second basin is the first one's mirror twin.

choosing · Choosing The basin has three widths, and they differ by a factor of 4.6. Two sections through the near-optimal basin of the Robinson aspect over Japan, drawn at one scale. Each ellipse is the set of aspects whose score is twice the optimum's, from the objective's own second derivative at the optimum: 32.1°, 16.0°, 6.9° along the three principal directions. A single number for "the width of the basin" is the cube root of their product, 15.3°, and it is not any of them.

The basins have widths as well as depths

Measuring the height of the pass left one thing owed: shape means widths too. Measured, the basin has three of them — 32°, 16° and 7° at Japan — it gets wider rather than narrower as the region grows, and the exponent it predicts overshoots the measured one by half again.

choosing · Choosing The piece count rises and falls. The number of connected pieces of the near-optimal aspect set for Robinson over Japan, swept finely through the threshold rather than sampled once below it. It is one piece at a wide threshold, reaches 13 at 1.010, and returns to one as the set shrinks onto the single best aspect. The set first disconnects at 1.941, which is above the peak: the pieces keep multiplying after the first break. This is the sweep the earlier measurement could not afford and it costs one grid, because every threshold reads the same 4992 evaluations.

The threshold is not a percolation

The near-optimal aspect set was found breaking into twelve pieces rather than two, the transition was called a percolation, and the exponent went unmeasured. Swept finely, the piece count rises from one to thirteen and falls back to one — and refining the grid by a factor of fifteen moves the peak from ten to nineteen, at the rate the surface's own minima grow, while an uncorrelated field on the same lattice grows by a factor of twelve.

choosing · Choosing Twelve cells of the grid's pole row are one map. The pole row of the aspect grid for Robinson over a 10° square region centred at 38° N: 24 pole longitudes across, 12 turns of the third angle down, with the projection's pole on the south geographic pole. There, moving the pole in longitude and turning the third angle are the same rotation, so every cell whose longitude less its turn is the same is the same map. The twelve filled cells are one map, the lowest on the row at 1.71 times the region's best score, and they score alike to 2e-11. No two of them are neighbours. The hollow cells are the maps one 15° step round the axis, the lowest at 2.33 times the best, and on the grid as it was the copies could join only through them.

The pass was a map meeting itself

Four measurements fitted a power law to the depth of the pass that splits a near-optimal set of aspects, and found it tight — R-squared of 0.98 for Robinson, 0.996 along most of a family of maps. On the grid those aspects were searched over, a pole row holds every normal-aspect map twelve times at cells that are not neighbours, and in 28 of Robinson's 36 regions the published pass was two copies of one map meeting through the map one step round the axis. Joined as the one map they are, Robinson's curve falls to R-squared 0.008, and the sinusoidal's famous refusal turns out to be the one measurement the copies never reached.

choosing · Choosing

Named alongside it

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

BasinExponentOptimisation landscapeAspectFracture thresholdLevel setShortfallDegeneracyReproducibilitySaddleVerificationAnisotropy

All concepts