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

Shortfall — where it appears

Something a phase of this collection could not finish, recorded at the time rather than discovered later. Each one names what is owed and why it was deferred, and a later rung pays it — sometimes by measuring the suspect and ruling it out.

Named by 8 essays across 4 fields — each of them below, with the objects they name alongside it.

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 -1.60 at a coefficient of determination of 0.913. The previous rung's level-set measurement gave −1.33 and the argument predicts −1, so measuring the height directly moves the answer FURTHER from the prediction rather than towards it.

The height of the pass between two basins

The previous rung recorded a shortfall: the fracture threshold falls as the −1.33 power where the argument predicts −1, and the difference was supposed to be the height of the pass. It is not. Measuring the pass directly gives −1.60, which is further from the prediction, and the two basins turn out to be exact symmetric copies with nothing to decompose.

choosing · Choosing The corner against where along the seam the feature crosses. A feature crossing the seam of a cube at right angles, drawn on each face's own map and unfolded, with the crossing moved along the edge. Every curve starts at zero, because the face's mirror through the edge's own midpoint reverses the along-edge direction and forces the shear term in the Jacobian to be odd. Away from it the gnomonic — the map with no corner at all in the rung below — reaches 44.4°, three times the equal-area map's and far past the conformal one's. The edge's half-length is 35.3°, so the right-hand end is still well inside it.

The corner is not at the midpoint

The rung below measured the corner a feature gets crossing a polyhedral seam, and found none at all under the gnomonic face map. It crossed at the edge's own midpoint every time — the one point on the edge where a face's own mirror symmetry forces the corner to vanish. Two fifths of the way to the vertex the gnomonic gives 20.15°, the equal-area map 7.28° and the conformal map under a degree, which reverses the ordering entirely.

families · Polyhedral 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

The previous rung measured the height of the pass and recorded a shortfall: 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 23 at 1.256, and returns to one as the set shrinks onto the single best aspect. The set first disconnects at 2.244, which is above the peak: the pieces keep multiplying after the first break. This is the sweep the rung below could not afford and it costs one grid, because every threshold reads the same 4992 evaluations.

The threshold is not a percolation

The rung below found the near-optimal aspect set breaking into twelve pieces rather than two, called the transition a percolation, and recorded that it had not measured the exponent. Swept finely, the piece count rises from one to twenty-three and falls back to one — and refining the grid by a factor of fifteen does not move the peak, while an uncorrelated field on the same lattice grows by a factor of twelve.

choosing · Choosing Three face maps on a cube, over a shrinking span. The corner a feature gets crossing the seam of a cube, measured by reading the tangent over an arc and then shrinking the arc by a factor of sixteen. A chord differs from a tangent in proportion to the arc, so a perfectly smooth join reports a corner that HALVES when the span halves — a slope of one on these axes. None of the three lines has a slope of one. The fitted slopes are 0.000, 0.000, -0.018, which is a flat line in each case, and a flat line is a real corner. The three differ in size and not in kind: 20.1513°, 7.2772°, 0.7687°.

The span ladder, run on all five

A recorded shortfall said the exact map's span ladder was one loop away from settling the seam on the five Platonic solids. The loop was run and the answer is the other one: the series conformal face map has a corner that does not shrink with the measurement span on any of them — 4.10° on the tetrahedron, 0.77 on the cube, 0.004 on the icosahedron — and it belongs to the truncation rather than to conformality.

families · Polyhedral A compacting basin, and the tilt it produces. The stated vertical velocity field — a bowl of subsidence 120 km across, with no mass leaving — drawn as circles proportional to the rate, with the tilt of the ground surface as the arrows. The subsidence is largest at the centre and the tilt is exactly zero there, because a smooth bowl has no gradient at its own bottom. The largest tilt is 125.5 nanoradians a year, on a ring at the bowl's own scale length over root two, and it is the quantity a levelling network measures.

A vertical rate needs a height system

Four rungs of this anchor measure the two-by-two horizontal tensor, because that is what a tangent chart returns. The larger signal in a subsiding basin is vertical — 126 nanoradians a year of tilt against 0.76 nanostrain a year of horizontal strain — and it is not a measurement at all until the surface it is measured against is named, because that surface is moving too.

datums · Strain Two maps of one density: one costs nothing and one costs forty-four degrees. Both drawings meet the same areal request — an exponential ramp of contrast 79.8 to one, whose logarithm is harmonic — to arithmetic noise. The first is a conformal map written down in closed form, the conformal map that meets an exponential ramp, log-harmonic, whose angular deformation is 2.4e-8 degrees. The second is the triangular construction the previous rungs use, at 43.8° on the same request. A grid of squares is drawn through each: the first keeps every angle and the second does not.

The cheapest map that meets its areas

An earlier essay bracketed a cartogram's least cost between a construction charging eighty degrees and a bound valid only for symmetric densities, and recorded the gap as a shortfall. One request settles it: a density of contrast eighty whose least cost is exactly zero, met by a map written down in closed form, while the standard construction charges 43.8° for it.

distortion · Cartogram The density every real cartogram is handed. Two islands in an ocean that is not lightly populated but empty: the density is exactly zero over 83.7 per cent of the sphere. Four rungs of this anchor assume a positive density, because the construction divides by it — and the conditional cumulative of a column with no mass in it is zero over zero. Drawn on the equal-area base, so a cell's ink is a density and its area is ground.

A density that asks for no room at all

Five rungs assume the density is positive everywhere, because the construction divides by it. Every cartogram anybody draws has an ocean, and an ocean is not sparsely populated but empty — 83.7 per cent of the sphere, exactly zero, and the construction returns nothing at all for a third of the probes.

distortion · Cartogram

Named alongside it

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

Aspect searchBasinConformalityExponentFracture thresholdJacobianLevel setOptimisation landscapeVerificationAngular deformationAreal factorAspect

All concepts