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

The projection itself, as one generator with a mode for each question

One object: a map of the sphere. projection-map draws one; projection-grid draws several side by side; world-cells puts equal patches of ground on it and labels what happened to them; radial-family, conic-family, aspect-sweep and pole-treatment sweep the construction parameter that defines a family; and interrupted-map and interruption-tradeoff are the same map with the cuts made explicit.

One object: a map of the sphere. projection-map draws one; projection-grid draws several side by side; world-cells puts equal patches of ground on it and labels what happened to them; radial-family, conic-family, aspect-sweep and pole-treatment sweep the construction parameter that defines a family; and interrupted-map and interruption-tradeoff are the same map with the cuts made explicit.

Every one of these is the same generator answering a different question, which is why they share a file, a set of colour roles and a set of assertions. Changing it changes all 16.

43 essays call it. Every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about.

What it draws

One picture per question the family answers. Each has a page of its own.

Mollweide. The graticule of the Mollweide projection at 30° of longitude and 15° of latitude. equal-area in an ellipse, at the cost of the corners. It is equal-area.

projection-map

Mollweide 6 projections of the same sphere. The same graticule under Equirectangular, Mercator, Mollweide, Sinusoidal, Robinson, Winkel tripel. Every one of them is a faithful drawing of the same object and no two agree, because each has chosen a different thing to preserve.

projection-grid

6 projections of the same sphere Five identical cells on Mercator. Five patches, each 20° of longitude by 10° of latitude. On the sphere the higher ones are genuinely smaller, because the meridians converge. On Mercator the cell at 70° comes out 15.4 times larger than the equatorial one relative to its true size.

world-cells

Five identical cells on Mercator The azimuthal family, as one function of one variable. Every azimuthal projection is a rule for how far from the centre to draw a point at angular distance ρ, and nothing else. five named projections are plotted as their radial functions — tan ρ, 2 tan(ρ/2), ρ, 2 sin(ρ/2), sin ρ — and the two dashed curves are the solutions of the two conditions, integrated from the centre with no more than "true scale at the centre" to start from. They land on the stereographic and the Lambert azimuthal to 8e-10. The names are the answers to the equations, not descriptions of viewpoints.

radial-family

The azimuthal family, as one function of one variable 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.

pseudo-search

Three named members of the family, and one the family has no name for What each extra degree of freedom in the family is worth. The best equal-area pseudocylindrical the family holds, against how many numbers its shape function is allowed. One number is the sinusoidal stretched — still equal-area, because the two axes are scaled by reciprocal factors, and not the same angles, because an anisotropic scaling is not a similarity — and it reaches 38.36° against the unstretched sinusoidal's 38.86. Two numbers reach 27.04, which already beats every named member. Six reach 25.03. The dashed lines are the three named maps, and the family passes the last of them between its first and second free number.

pseudo-dimensions

What each extra degree of freedom in the family is worth The free function, drawn. The quantity the family leaves free, shown as what it decides: how long each parallel is drawn relative to the equator's. The equal-area condition C·Y′ = cos φ fixes one of C and Y from the other, so this curve IS the map. The sinusoidal's is cos φ exactly, which is why its parallels are true to scale; Mollweide's falls faster and reaches zero at the pole; Eckert IV's stops short of zero, which is its pole line. The searched member's is a curve none of the three names, and it is a curve rather than a choice among curves — which is the whole of what this rung is about.

pseudo-shape

The free function, drawn Every named equal-area pseudocylindrical, and one the family found. Mean angular deformation over the world, area-weighted, for the three named equal-area pseudocylindricals in this collection's library and for the best member a search over six free numbers finds. The named maps are scored as themselves rather than through the basis, so no part of the comparison depends on how well the basis reproduces them — that is reported separately and runs from four parts in 10¹⁵ for the sinusoidal, which the basis holds exactly, to seven parts in a thousand for Eckert IV.

pseudo-scores

Every named equal-area pseudocylindrical, and one the family found The test the ladder asked for, and it refutes the conjecture. How much better the best asymmetric projection is than the best symmetric one, under weightings of four different symmetries, with every symmetric map allowed to re-aim its axis at twelve candidate poles. The conjecture rung eight recorded was that the seven earn their place by PLACING distortion where a symmetric map cannot, so their advantage should collapse under a criterion with no place preference. It does the opposite: the advantage is largest at 1.343 under the uniform weighting and smallest at 1.144 under a band, with the fully asymmetric concentration at 1.204 in between. The winner is named on each row and the map it beat is Equirectangular throughout. The seven are simply better maps.

symmetry-criterion

The test the ladder asked for, and it refutes the conjecture Every projection in the library, ranked under the flat weighting. Kavrayskiy's criterion — ½[(ln a)² + (ln b)²] in the two principal scale factors, so shape and area are charged together — weighted by the flat field, with every projection allowed its best of twelve axes. The seven with no continuous symmetry are marked. three of them come before the first symmetric map, and the best symmetric map is Equirectangular at 0.3539 against Winkel tripel's 0.2635. The ordering is not the one this collection's other criteria produce, and it is not meant to be — it is the one the conjecture had to be tested on.

symmetry-ranking

Every projection in the library, ranked under the flat weighting How many maps want to move their axis, and under which criterion. Every projection here was scored at twelve candidate axes and kept its best. Under the uniform weighting 5 of 13 symmetric maps and 1 of 7 asymmetric ones prefer an axis other than the poles — which is nearly none, as it should be, since a uniform criterion has no direction in it. Under the concentration, 8 of 13 and 2 of 7 move. That is the freedom the conjecture said the symmetric family could not use, and it uses it — and it still loses by 1.204 times.

symmetry-aspect

How many maps want to move their axis, and under which criterion Sinusoidal, cut into 6 lobes. six lobes, cut through the oceans so each continent stays whole. Each lobe is the same projection about its own central meridian, so each point is near a line where the shape distortion vanishes: the mean angular deformation over the mapped world falls from 38.6° uninterrupted to 17.8° here. What is given up is that the map is no longer one surface — the graticule stops at every cut because there is nothing on the other side of it.

interrupted-map

Sinusoidal, cut into 6 lobes What interruption buys and what it costs. Mean angular deformation over the mapped world, and the total length of the tears as a fraction of the map's width, against the number of lobes. Cutting the map into eight gores drops the mean shape distortion from 38.6° to 5.5° and adds 2.2 map-widths of edge running through the middle of it. Both curves are real and only the first is usually drawn.

interruption-tradeoff

What interruption buys and what it costs Mercator swung through every tilt, over Chile and the tropics. The regional distortion of one projection as its axis is tilted from the normal aspect at 0° to the transverse at 90°, each curve divided by its own value in the normal aspect so the two regions can share an axis. Chile is best at a tilt of 90°, a factor of 183.5 better than north-up; the tropics is best at a tilt of 0°, which is north-up. Rotating the sphere costs nothing and changes none of the projection's own properties, which makes this the cheapest improvement available and the one most often left unmade.

aspect-sweep

Mercator swung through every tilt, over Chile and the tropics The equal-area conic, from cylinder to plane. The largest distance between the conic at cone constant n and each of its two limits, over a shared grid, with the free scale and offset removed. Both fall as the FIRST power of the distance from their end — the slope on these axes is one — so the conic is never nearly cylindrical: halving n only halves the difference. At n = 0.999999 the conic is the Lambert azimuthal to 1.3e-6, and at n = 0.000001 it is the Lambert cylindrical to 1.2e-6.

conic-family

The equal-area conic, from cylinder to plane What a pole line buys, and what it costs. four pseudocylindrical projections placed by the two numbers the decision moves. Across the bottom, the angular deformation averaged over the whole sphere, where the pole-line projections — Eckert IV and Robinson — are the better maps. Up the side, on a log axis, the factor by which the last parallel is stretched, where they are worse by more than tenfold: 49× at best against 4.6× at worst for a pole drawn as a point. A pole line represents one point of the globe by a line of the map, and that is the price.

pole-treatment

What a pole line buys, and what it costs

Where it is called

Changing this changes every one of these figures.

Construction against property. Every projection in the library, sorted by the shape it was notionally rolled from and by what it actually preserves. The families do not line up with the properties: cylindrical and pseudocylindrical and pseudoazimuthal and azimuthal and conic each contain projections of more than one kind, which is why the cylindrical–conic–azimuthal taxonomy answers a question nobody has. Three entries are picked out: Mercator, Albers equal-area conic, Orthographic. The families

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.

 Angular deformation against latitude, four projections. The same quantity for mercator, gallPeters, winkelTripel, robinson, from the equator to 80°. A conformal projection sits flat on zero in the first of these plots and runs away in the second; an equal-area projection does the reverse. Nothing is flat in both. What each projection optimises

Every projection minimises something

A projection is the solution to an optimisation problem, and naming the objective explains more than naming the family. Some objectives are exact constraints, some are least-squares fits, and one is a table of numbers a man adjusted until it looked right.

 Every projection in the library, measured against both properties. Maximum angular deformation across the bottom, maximum areal error up the side, both on logarithmic scales and both measured over several hundred sample points rather than taken from the projection's description. Conformal projections lie on the left edge, equal-area ones along the bottom, and the corner where both vanish is empty because a projection there would be an isometry of the sphere onto the plane. All 26 are plotted; the four named here are Mercator, Mercator (ellipsoidal), Web Mercator, Gall–Peters. What is taught wrongly

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.

 6 projections of the same sphere. The same graticule under Equirectangular, Mercator, Mollweide, Sinusoidal, Robinson, Winkel tripel. Every one of them is a faithful drawing of the same object and no two agree, because each has chosen a different thing to preserve. What each projection optimises

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.

 A cylinder unrolls exactly; a sphere does not. Both pictures show the same grid. On the left it is wrapped round a cylinder of radius 1, on the right it is laid flat, and every distance in the grid is the same in both — the circumference is 2π and so is the width of the rectangle, checked to 10⁻⁹. This is possible because a cylinder has zero Gaussian curvature. No corresponding picture exists for a sphere. The impossibility

What can be unrolled

A cylinder is obviously curved and is intrinsically flat, so it lays out on a table with nothing stretched. A sphere is not. The distinction is exactly zero Gaussian curvature, and it is why paper tubes exist and paper globes do not.

 Angular deformation around three standard parallels. Three equal-area cylindrical projections differing only in where they are exact — standard parallels at equator, 30°, 45°. Each has zero angular deformation at its own standard parallel and grows away from it in both directions. Choosing a standard parallel is choosing which latitudes to treat well, and there is no choice that treats them all well. The families

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.

 The indicatrix at 30°, 40° on Gall–Peters. A circle on the sphere becomes this ellipse on the map. Its semi-axes are the two principal scale factors, a = 1.083 and b = 0.923; their product is the areal factor 1.000; and the maximum angular deformation is 9.16°. h and k are shown too, and depend on the coordinates rather than on the map. Measuring distortion

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.

 New York to Madrid on Mercator. Two routes. The great circle is 5768 km and is the shortest path on the sphere. The rhumb line holds a single compass bearing the whole way and is 5939 km — 171 km further, or 3.0 per cent. On Mercator the rhumb line departs from straight by 9.6e-16 of its own length. Paths and directions

Why Mercator exists

A ship can hold a compass bearing and cannot easily hold a great circle. Mercator is the answer to one question — what must a map do so that a constant bearing is a straight line — and it answers it exactly.

The whole library