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

Quadrature — where it appears

Numerical integration of a stated function, which is how this site computes every meridian arc, strip area and isothermal coordinate. Its results are used here as the independent check on a series, so an error in either shows up as disagreement rather than being absorbed.

Named by 13 essays across 4 fields — each of them below, with the objects they name alongside it.

The boundary of the region the series is the map in. The curve where the transverse coordinate reaches 2.918, which is where the terms stop shrinking. It crosses the equator 83.81° from the central meridian and closes towards the poles, because the same longitude is a smaller transverse coordinate at a higher latitude — the boundary is a curve rather than a meridian. The narrow band beside it is a 3° zone, the width national grids actually use, drawn to the same scale: the practical world sits in about a fiftieth of what the series can reach. Drawn in Mollweide.

Where the series stops being the map

The transverse Mercator has no closed form on an ellipsoid, so every national grid computes a truncated series. Asking how many terms it needs has an answer everywhere; asking whether more terms always help has an answer only within 83.81° of the central meridian, and the boundary is computed rather than assumed.

families · Ellipsoid On a triaxial body, latitude depends on longitude. Walk round each body at a constant planetocentric latitude of 45° and watch the direction of the surface normal, which is what the planetographic latitude is. On Mars it does not move at all — that is what having an axis of revolution means. On Vesta it swings by 1.40° and on Phobos it swings by 6.40°. A body without an axis has no latitude that is a function of position alone, and every coordinate on it is a convention with a body-fixed frame attached.

A body that is not an ellipsoid

Vesta's three axes are 286.3, 278.6 and 223.2 kilometres, all different, so it has no axis of revolution — and on such a body the planetographic latitude of a point at 45° planetocentric swings by 1.4° as one walks round it in longitude, and by 6.4° on Phobos. Latitude stops being a function of position.

datums · Bodies A conformal map of Vesta, in the coordinates that make one possible. The body's own isothermal coordinate, used as the map — which is Mercator's construction carried out on a body that has no axis to be Mercator about. The lines are Jacobi's ellipsoidal coordinates, which are the surface's lines of curvature, at 30° and 20° spacing; the two quadratures that turn them into an isothermal pair are one-dimensional. The measured angular deformation away from the marked points is 1.1e-6°, which is this site's noise floor, and the areal factor spans a factor of 162 — conformal, and emphatically not equal-area. The four marks are the umbilics, where the coordinate collapses and the map has nothing to say.

A conformal map of a body with three axes

This site built an equal-area map of a triaxial body and wrote down what it could not do: the conformal one, which needs an isothermal coordinate that a surface with no axis of revolution was said not to have. It has one, Jacobi found it in 1839 for a different reason, and it takes two one-dimensional integrals.

datums · Bodies Two expansions of one integral. The worst error along the whole meridian, against the number of sine terms kept, for the series in the third flattening and the classical series in e². Both are checked against a Simpson's rule on the defining integral, which shares no algebra with either. At one and two terms they are the same number to four digits and the e² series is fractionally ahead; from the third term the n series pulls away, and at four it is 1175 times more accurate — 7.6e-8 metres against 9.0e-5.

Which small quantity the series is in

Every ellipsoidal formula in this collection is a truncated series in the third flattening, inherited from Krüger in 1912 and justified nowhere. Measured against the classical expansion in e², at four sine terms it is 1,175 times more accurate — and at one and two terms it is fractionally worse, which is not what the folklore implies.

wrong · Ellipsoid Four cities, met — the triangular, x first construction. Every cell drawn here holds the same area of ground and a different area of page: the map has been constructed so that its areal scale factor is the density it was handed, cell by cell, over a contrast of 25.99 to one. The residual against that target is 5.1e-6, measured from the map's own derivatives rather than from the construction. What it cost is the shape: the redistribution alone reaches 139.1° of angular deformation and averages 70.4°, on top of whatever the equal-area projection under it was already doing.

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.

distortion · Cartogram Two maps of one density: one costs nothing and one costs forty-four degrees. Both drawings meet the same areal request — an exponential ramp of contrast 79.8 to one, whose logarithm is harmonic — to arithmetic noise. The first is a conformal map written down in closed form, the conformal map that meets an exponential ramp, log-harmonic, whose angular deformation is 2.4e-8 degrees. The second is the triangular construction the previous rungs use, at 43.8° on the same request. A grid of squares is drawn through each: the first keeps every angle and the second does not.

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.

distortion · Cartogram The density every real cartogram is handed. Two islands in an ocean that is not lightly populated but empty: the density is exactly zero over 83.7 per cent of the sphere. Four rungs of this anchor assume a positive density, because the construction divides by it — and the conditional cumulative of a column with no mass in it is zero over zero. Drawn on the equal-area base, so a cell's ink is a density and its area is ground.

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.

distortion · Cartogram The same count, scattered two ways. 22 dots in every cell of a 30-cell covering, drawn on Mercator. The upper panel places them uniformly on the GROUND — uniform in longitude and in the sine of latitude, which is what uniform on a sphere means — and the lower places them uniformly on the PAGE, which is what a drawing routine handed a polygon does. Both panels carry exactly the same number of dots in exactly the same regions, so both are honest as totals. They are different pictures, and a reader reads a dot map by density.

A dot map's density is partly the projection's

A dot map carries the right number of dots in every region whichever way it is drawn, so it is honest as a total under both placements. It cannot be honest as a density under both: ground on a uniform field reads 0.099 of its equatorial density at 72° north on Mercator, and scattering inside the polygon on the page moves 64.3 per cent of a cell's dots into its northern half without one of them leaving the cell.

distortion · Thematic One field, one classifier, two sets of breaks. A field that varies with latitude put into 5 classes by area-weighted quantiles, on Mercator. The upper panel weights each region by its ground area and the lower by the area it occupies on this page, which is what a classifier handed projected geometry does. 60 of 150 regions land in a different class, marked in the lower panel. The data has not changed and neither has the number of classes.

The class breaks were computed on the page

The three rungs below price what a reader does with a finished map. A classifier is software, it runs on the geometry it has, and the geometry it has is projected: a five-class quantile classification of one stated field puts half of the three hundred and eighty-four regions in a different colour on Mercator, and 87.5 per cent of them at nine classes.

distortion · Thematic One of these averages exists. Mercator's area-weighted mean areal factor, and its Kavrayskiy number, against how close the sampled band comes to the pole. The first is artanh(sin Φ)/sin Φ in closed form and has no limit: it passes 7.04 at a tenth of a degree from the pole and keeps going, gaining a fixed amount every time the remaining gap is halved. The second settles by 85° and does not move again. Both are published as summary distortion figures for the same map.

A mean that does not exist can still be printed

Mercator's area-weighted mean areal factor is artanh(sin Φ)/sin Φ, and it has no limit. A sampler asked for it returns the logarithm of its own sample count plus 1.512 — measured slope 1.001 against ln n — so the number is a property of the person who computed it. The Kavrayskiy number for the same map over the same sphere settles at 0.52124 and is a number.

distortion · Sampling Three ways to cover a sphere with 900 points. The three samplers this collection's numbers are computed with, each with about 900 points, drawn on the Mollweide projection so that equal areas on the sphere are equal areas on the page and the crowding is the samplers' rather than the map's. The graticule piles points at the poles; the equal-area rings space them evenly by area and unevenly by distance; the Fibonacci lattice trades a little of each. None of them is equal-area, and no finite set is.

There is no equal-area lattice on a sphere

Every number in this collection is an average over a point set, and the three point sets available all fail to be equal-area in different ways. The equal-area ring sampler this site has used since its early essays is the one whose outermost ring sits half a step inside the rim — which is how a measured scale spread once came in 3.42 parts in a thousand below a proved bound.

distortion · Sampling The doubling ladder is the one sequence that cannot see it. The largest departure of a small circle from its own indicatrix that a grid of n latitudes over 10° to 70° north finds on the Robinson projection. The filled marks are 4, 8, 16, 32, 64 and 128 — the doubling ladder every convergence study runs — and they rise smoothly to about 4.66e-4 with the increments halving, which is what a convergent first-order sequence looks like. The open marks are grids whose samples land on the projection's five-degree table entries. They report 1.28e-2, twenty-seven times higher, and whether a grid does that is decided by whether n is a multiple of four.

A refinement that stops moving

Doubling the sample and watching the answer settle is how every quadrature in every field is checked. On the Robinson projection the doubling ladder — 4, 8, 16, 32, 64, 128 — converges beautifully, with its increments halving at every step, on a limit that is wrong by a factor of twenty-seven. Whether a grid finds the answer is decided by whether n is a multiple of four.

distortion · Sampling One map, two samples, two answers. The same Miller cylindrical projection carrying two sets of sample points of the same size. On the left the points are uniform over the SPHERE — equal ground area between them, which is what every mean in this collection integrates against. On the right they are uniform over the PAGE, which is what a raster, a pixel loop or any figure that walks its own canvas produces. The right-hand set crowds where the map stretches, and the mean angular deformation it returns is 18.0° against the left-hand set's 7.2°.

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°.

distortion · Sampling

Named alongside it

The objects these essays reach for when they reach for this one.

Areal factorClosed formEqual-areaSamplingVerificationAngular deformationDensityEstimatorArea weightingBiasCartogramConvergence

All concepts