About

This is a growing collection of illustrated essays about map projections. Each takes a single idea and draws it until the argument is visible — and the distortion in every figure is computed from the projection's own derivatives rather than described in words.

No projection is called conformal until it has been measured

Almost every illustration of a map projection asserts the projection's property rather than checking it. A map is captioned conformal or equal-area because that is what the projection is called. Tissot indicatrices are drawn as decorative circles, scattered and unlabelled, with no numbers attached; the reader is told there is distortion and never shown how much.

Every one of those quantities is exactly computable. From the four partial derivatives of the projection — how x and y each change with longitude and with latitude — the machinery derives the scale factor along the meridian, the scale factor along the parallel, the angle between them, and from those the two principal scale factors. Everything worth knowing follows:

• Conformal means the two principal scales are equal at every point, so the maximum angular deformation is zero.
• Equal-area means their product is one at every point.

So assertConformal samples several hundred points and requires the angular deformation at all of them to be zero to arithmetic noise, and assertEqualArea requires the areal factor to be one. A projection that fails is not drawn with the caption it claims, and the build stops.

The assertions have to reject, and the gate proves it

A test that accepted everything would pass every other check on this site while making its whole premise void. So the gate requires Mercator to fail the area test, Mollweide to fail the angle test, and — the case that makes the point best — Web Mercator to fail the property named in its own title.

Web Mercator carries almost every map on the internet. It applies spherical Mercator formulae to ellipsoidal latitudes, which means it is not conformal. The machinery here found that without being told to look for it, and the essays report by how much and at which latitudes.

The impossibility is a theorem, not a limitation

No map of the sphere is faithful. This is not a shortcoming awaiting a cleverer cartographer, and it is worth being precise about why.

Gaussian curvature is intrinsic: it can be computed from measurements made entirely within a surface, without reference to any surrounding space. That is Gauss's Theorema Egregium, and its consequence is that curvature is preserved by any isometry. A sphere of radius R has curvature 1/R² everywhere; a plane has zero. So no isometry between them exists, and every flat map distorts.

The same computation shows why a cylinder and a cone can be unrolled flat without distortion — both have zero Gaussian curvature. It also shows why calling a projection cylindrical does not make it faithful: the sphere is not a cylinder, so wrapping paper round it is not an isometry of anything.

The trade-off is two lines of algebra

Conformal means the principal scales are equal. Equal-area means their product is one. Both together force both scales to equal one, which is an isometry, which the theorem above forbids.

So no projection is both, and this is not a matter of nobody having found one yet. The site demonstrates it computationally as well: every projection in the library is run through both assertions, and nothing passes both.

What else is checked

• Every projection round-trips. Forward then inverse must return the starting coordinates. A sign error or a swapped argument survives visual inspection and dies here.
• The derivatives agree by two routes. Where a projection has a known analytic scale factor, the numerically differentiated one is checked against it — two independent routes to the same number, which is the only reason to trust either.
• The invariants are invariant. The principal scales, areal factor and angular deformation must not change when the sphere is re-parameterised, which is the property that makes them descriptions of the map rather than of the coordinates.
• Great circles really are shorter. The geodesic distance between two points is compared against the rhumb line between the same two, and the rhumb never wins.
• Areas are computed, not quoted. Where an essay says Greenland is a certain fraction of Africa, that number came from spherical excess over real coastline polygons at build time.

Every figure here is itself a projection

The page is flat. There is no way to show a sphere on it except by projecting the sphere, so every picture on this site — including the ones that look like globes — is an instance of the phenomenon under discussion. An orthographic view of a hemisphere is a projection with its own distortion, and naming it as such is more honest than presenting it as the undistorted truth.

This is not an apology to be made once and forgotten. It is the theorem applied to the site itself, and the figures state which projection they are drawn in for the same reason a laboratory states its instrument.

Where the models stop

The Earth is not a sphere. Most of this site treats it as one, because the spherical case carries every idea and the ellipsoidal corrections obscure them. Where the distinction matters — and for Web Mercator it is the entire point — the essays say so and the machinery uses the ellipsoid.

Distortion at a point is not distortion of a region. The indicatrix describes what happens to an infinitesimally small circle. A country is not infinitesimally small, and summarising a whole map by a single number requires choosing a region and a weighting, both of which are judgements rather than measurements.

No projection is best. Every one minimises something, and comparing two projections without naming the purpose is comparing them against nothing. The essays name the purpose.

On being wrong

Corrections are welcome and will be made. A map with a wrong caption looks exactly like a map with a right caption, which is the entire argument for computing the caption.