Figure

6 projections of the same sphere

Drawn here at the parameters it defaults to, with every essay that calls it.

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.

It is drawn by projection-figure with show: "projection-grid" — one member of a family of 16 figures that share a generator, so the drawing above is what that generator returns when it is asked for this one and given nothing else.

13 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.

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.

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.

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.

6 projections of the same sphere. The same graticule under Robinson, Winkel tripel, Mollweide, Mercator, Gall–Peters, Eckert IV. 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

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.

American polyconic. The graticule of the American polyconic projection at 30° of longitude and 15° of latitude. a different tangent cone for every parallel — true to scale along all of them, and along the central meridian. It is neither conformal nor equal-area.

The families

The projections that gave up being one thing

The polyconic is built from a different cone for every parallel, which means it is built from no cone at all. It preserves nothing the usual tests look for, it has an exact property neither of them measures, and its sheets do not fit together — a defect discovered in the field rather than at the drawing board.

How Equirectangular distorts, by latitude. Angular deformation and areal scale plotted against latitude along the meridian at 0°. On Equirectangular the angular deformation reaches 114.2° and the areal factor reaches 11.5.

What is taught wrongly

The plate carrée, the projection nobody chooses

Plotting latitude against longitude on ordinary axes is a projection. It preserves nothing, its angular deformation reaches 108° and its areal error eightfold, and it is probably the most widely produced map in the world because it is what happens when nobody decides anything.

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.

The families

The conic is the whole family

Cylindrical and azimuthal are usually presented as two of three families beside the conic. They are the two ends of it. One parameter runs from the cylinder to the plane, and both limits are exact rather than suggestive — which is measurable, and measured here.

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.

The families

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.

Every mixture of Equirectangular at 50.46° and Aitoff. Airy's criterion over the whole sphere, for every weighted average of the Winkel tripel's own pair. Each mixture is normalised to its own best constant scale first, so the comparison is about shape rather than size. The curve has its minimum at w = 0.45, where the criterion is 0.3037 against 0.4114 and 0.4290 at the two ends. That is a projection nobody constructed beating both projections somebody did, by 26 per cent.

What each projection optimises

The average of two projections

The Winkel tripel is literally the arithmetic mean of two other projections, and this site's implementation of it agrees with that mean to the last bit. Averaging beats both ingredients by 26 per cent — and it preserves conformality exactly, destroys equal-area completely, and can turn eighteen per cent of the world inside out without either distortion measure saying so.

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.

The families

The azimuthal family is one function

Five azimuthal projections are taught as five viewpoints — from the centre, from the far pole, from infinity. They are one projection with five choices of how far out to draw a point at angular distance ρ, and each named property is a differential equation in that one function: f′ = f/sin ρ integrates to the stereographic projection and f f′ = sin ρ to Lambert's, from nothing but true scale at the centre.

The whole library