Angular deformation — where it appears
Named by 41 essays across 7 fields — each of them below, with the objects they name alongside it.
Tissot's indicatrix
A tiny circle on the sphere becomes an ellipse on the map, and the ellipse's two axes are the whole story. Almost every published indicatrix is drawn without them, which discards the content and keeps the decoration.
Web Mercator is not conformal
It carries almost every map on the internet, it is named after the projection whose entire purpose is preserving angles, and it does not preserve angles. The machinery here found that without being told to look.
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 scale of a screen map is not one number
A zoom level prints one scale for the whole world, and the map is at that scale along exactly one line. At 60° north the picture labelled 1:136,495 is a 1:68,765 map — and the factor is not quite sec φ either, because the projection puts a geodetic latitude into a spherical formula.
Mercator against Peters
The most-argued question in cartography, conducted almost entirely without anyone measuring anything. Both projections are exactly what they claim, each destroys what the other keeps, and the numbers are computable in either direction.
The trade-off is two lines
Conformal means the two principal scales are equal. Equal-area means their product is one. Both at once forces both to one, which is an isometry, which the curvature forbids. That is the entire argument.
Compromise projections
A projection that preserves nothing exactly can distort everything less than one that preserves something exactly. For a general-purpose world map that is the right trade, and it is why the two most widely used ones today have no exact property at all.
A scale bar is right in one place
The bar in the corner of a world map is a picture of a distance, and it is a true picture along one line. On Mercator it reads 500 kilometres for a thousand at 60° north — and on an equal-area map it reads 500 one way and 2,000 the other, so the projection recommended for measuring is the one on which no single correction exists.
The two ways a map is wrong
Angles and areas fail independently. A projection can be perfect about one and catastrophic about the other, and a single number summarising "how distorted" a map is has already thrown away the distinction that matters.
Scale distortion is the third failure
A map's stated scale is its scale at one place. Everywhere else it differs, by a factor that no conformal projection can avoid and that neither of the two usual distortion measures reports. It is the failure everybody uses and nobody counts.
Giving up continuity
Cutting a map into lobes really does reduce the distortion, by a factor that can be measured. What is paid is that the map stops being one surface — and the size of the tear is the number that pictures of interrupted maps never carry.
The pyramid did not have to be Mercator
The usual defence is that a quadtree needs a square world and Mercator supplies one. So does the cylindrical equal-area with standard parallels at ±55.654° — the solution of π cos²φ₀ = 1 — and it needs no polar cut at all. What Mercator actually buys is conformality, and the price of giving it up is 13.8° of shear at 60° north.
Where a pseudocylindrical puts its error
Every projection in this family has to decide what to do at the pole, which on the globe is a point where every meridian meets. Drawing it as a point and drawing it as a line are the two answers, and the trade between them is measurable in both directions.
Conformal does not mean the angles are right
A conformal projection preserves angles between curves at a point. Draw a triangle on one with a ruler and its angles are wrong by degrees — and the total error is fixed before the projection is chosen, because a plane triangle sums to 180° and the real one does not.
A projection between two projections
Every distortion measured on this site so far compares a map with the sphere. The operation a machine actually performs compares a map with another map — and that map has its own two principal scales, its own areal factor and its own angular deformation, none of which is the difference of the two it was built from.
An error ellipse is an indicatrix
A positional covariance pushed through a projection is the same matrix sandwich that produces Tissot's indicatrix, so the error ellipse drawn on a map and the distortion ellipse drawn beside it are the same ellipse. A five-metre circular accuracy is drawn at an axis ratio of 3.04 on one common projection — and the same projection draws a genuinely lopsided 304 by 100 metre error as a perfect circle.
Every equal-area map is every other one
Take Mollweide and slide every row of the page sideways by an amount that depends on the row. The result satisfies the equal-area condition to 6 × 10⁻¹², exactly as well as Mollweide does, and it is a ruin — the angular deformation at one ordinary point has gone from 11° to 60°. Equal-area is one equation, and one equation leaves a whole function free.
Two charts are enough, and one is not
The topological minimum for an atlas of the sphere is two sheets, and the counting fixes a price nobody chose: the worst point of any two-chart atlas is 90° from a chart's centre, where a conformal chart's areal factor is exactly 4 and an equal-area one's angular deformation is 38.94°. A national series has a hundred and twenty thousand sheets, and a hundred and twenty thousand minus two of them are bought by accuracy.
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.
A map of a body with three axes
Every projection on this site is written in a latitude, and a body with three unequal axes has none: the coordinate lines are not perpendicular, Lambert's equal-area formula spreads areas by 28 per cent on Vesta, and the equal-area map that does work has a different one on every meridian.
What a cut buys
Six rungs count cuts and none measures one. A cut is a curve on the sphere with a length in kilometres, the shape distortion it removes is a falling function of that length, and the interruption everybody prints spends a hundred thousand kilometres to reach a figure the even-lobed curve reaches with sixty.
A projection defined by a table has an interpolation in it
One member of this library has no formula: Robinson set nineteen pairs of numbers by eye and the table is the definition. Three published interpolations of it draw graticules that differ by four parts in a thousand of the map's span and report angular deformations that differ by 8.4 degrees.
The nearest equal-area map to an impossible request
Every number in the previous rung is inside the conformal achievable set, because Liouville's is the conformal condition. The equal-area set is one equation on two functions and never refuses: the same four requests are met to nine parts in a billion, and charged for in angle instead.
An average of ellipses is not an ellipse
Rung three integrates distortion over a region and finds the weighting is somebody's opinion. It never asked what was being averaged. An indicatrix is a positive-definite matrix, matrices form a cone rather than a vector space, and the arithmetic average of an equal-area map's indicatrices comes back inflating area by up to 11 per cent.
Which projection a weighting can make best
Rung eight finds the seven world projections nothing beats on both counts and tells a reader with a preference that one of them is their answer. Four of the seven are not: they sit in dents of the front, undominated and unreachable, and no weighting of angle against area can ever put them first. Robinson can be first, on 2.8 per cent of the weight range.
The map depends on where it was cut
The previous rung solved the discrete conformal equations on a triangulated body and reported an areal spread of 3.07, then recorded that the number might belong to the patch, the boundary and the two pinned vertices rather than to the surface. It belongs to the pins: hold a different pair and it runs from 2.68 to 7.44, while the typical angular deformation does not move at all.
The ranking is not an order
The previous rung showed that a weighting can make almost any projection best. Remove the weights entirely, let each of the seven criteria vote once, and the answer is worse: over the whole sphere Mercator beats the sinusoidal, the sinusoidal beats Eckert IV, and Eckert IV beats Mercator — four such circles, every margin four to three, with a Condorcet winner sitting above them all.
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.
The ground has an indicatrix too
Two hundred and forty-six essays hold the Earth still while the page is measured. The ground is moving at tens of millimetres a year, and the motion is a map with four partial derivatives — so the same construction that draws Tissot's ellipse draws one for the ground, and the fastest plate in the model returns exactly nothing.
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.
A map drawn to a density it was handed
Two hundred and twenty-six essays measure distortion after the fact. This one specifies it: a density is handed to a map as a boundary condition, the areal scale factor comes out equal to it to five parts in a million, and every other invariant the site owns becomes the price.
Every density can be met and none is free
Four maps of the same data, all of them correct, charging between 57.6° and 104.4° of angular deformation for it. There is no such thing as the cartogram of a density — there is an infinite family, and somebody picked a member of it without saying so.
Everything else on the page pays for the areas
A cartogram gets one quantity exactly right and every other reading a page supports is collateral. A ruler on it is out by 35 per cent after the most generous calibration available, and ten of sixty triples of places change which one is in the middle.
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.
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.
The ellipses are a sample, drawn at a size somebody chose
Thirteen essays measure with the indicatrix and none audits it as an instrument. A published field has a gauge nobody states and a placement nobody states: on Mercator the standard convention draws twenty-five identical circles while the areal factor runs over a factor of 14.9, and the average a reader takes off any of these fields is between 22 and 64 per cent too high.
The best compromise for angle is not the best for bending
Every compromise projection in the library is an average of two others, and averaging is a first-order operation — so the second derivative was never part of the bargain. Swept along five ordinary blend paths, the weight that minimises angular deformation and the weight that minimises flexion are between a quarter and a half of the axis apart, and how much a compromise buys at one order predicts nothing about the other.
The worst point is not on the grid
Every maximum distortion this collection has printed is a maximum over a sample, and a maximum over a sample is a lower bound. On Mercator over a sixty-degree band a twelve-by-twelve grid reports 3.464 where the answer is exactly 4, and the shortfall does not go away with refinement so much as decay at a rate that says where the extreme is hiding.
Which of these numbers are the sampler's
Four rungs have shown that a sampled maximum understates, a sampled mean can be a report on the sampler, no arrangement of points is neutral, and refining until the answer settles proves nothing. So the collection re-measured itself. The means move by at most 1.1 per cent, the worst points by up to 19, and the rankings — which is what the essays actually argue with — do not move at all.
The sample was drawn on the page
Every mean in this collection integrates over the sphere, because that is where the ground is. A raster, a pixel loop and any figure that walks its own canvas integrate over the page instead, and the difference is exactly the covariance between the quantity being measured and the map's own area distortion — 7.2° of mean angular deformation on Miller becoming 18.0°.
Two indicatrices do not make a third
Reprojecting is composing, and the composite's indicatrix is not a function of its parents'. Mollweide followed by Hammer is gentler than either map over eighty per cent of the sphere; the sinusoidal followed by Gall–Peters is worse than both everywhere. What separates them is one angle, and it is the number rung ten showed the ellipse does not carry.
Named alongside it
The objects these essays reach for when they reach for this one.
Equal-areaConformalityAreal factorVerificationPrincipal scale factorsTrade-offJacobianMercatorPurposeScale factorTissot's indicatrixCartogram