Aspect — where it appears
Named by 22 essays across 4 fields — each of them below, with the objects they name alongside it.
Cylinders, cones and planes
The standard taxonomy sorts projections by the shape of paper they were notionally rolled from. It is memorable, it is how everyone is taught, and it says almost nothing about the properties anyone actually chooses on.
Which projection is best
An incomplete question, and the incompleteness is the answer. Every projection preserves something and destroys something else, so the comparison worth making is between a projection and a purpose, not between two projections.
What a standard parallel buys
A standard parallel is a line where the projection is exact. Choosing one does not reduce the distortion — it decides where the distortion is zero and lets everything grow away from it.
What survives a change of coordinates
The scale along the meridian is a property of the map and the grid together. The principal scale factors are a property of the map alone. Only the second kind describes the projection, and the two are routinely quoted as though they were the same thing.
The aspect is a free choice
A projection's distortion pattern is fixed relative to its own axis, and where that axis points is entirely up to the cartographer. Rotating it is the cheapest available improvement and it is the one most often left unmade.
Transverse Mercator and the series that computes it
The projection most of the world's survey data lives in has no closed form. What every national grid actually computes is a truncated power series in the flattening, and how far it can be trusted is an engineering parameter rather than a property of the projection.
Fitting the aspect to the region
Choosing a projection is a choice among a few dozen named things. Choosing its aspect is a choice among a continuum, it costs nothing, it changes none of the projection's own properties, and for a long thin country it is worth a factor of 183.
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.
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.
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.
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.
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.
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.
The family is a symmetry, not a shape
Rung seven dissolves the taxonomy: eleven named maps fitted to one perspective formula, six reproduced exactly and five refused, so the developable surface describes how the projections were discovered rather than what they are. What is left is the question it does not ask. If the shapes are not the classification, what is?
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.
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.
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.
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.
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.
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.
The maps with no family are simply better
Rung eight found seven projections with no continuous symmetry and noticed they are almost exactly the set anybody would choose for a world map, then offered a conjecture with a test attached: their advantage should collapse under a criterion that does not care where anything is. Run, it does the opposite — 1.343 times under a uniform weighting and 1.204 under a concentration. The conjecture is refuted.
The light comes from a page direction
A hillshade's illumination azimuth is declared from the top of the sheet, and the top of the sheet is grid north. On a conic the light therefore swings 52.6° across eighty degrees of longitude, and 72.8 per cent of the sheet is shaded differently from what the legend claims.
Named alongside it
The objects these essays reach for when they reach for this one.
OptimisationObjective functionMercatorDegeneracyKavrayskiy's criterionPurposeLocal minimumProjection selectionSearchSymmetryAzimuthalConic