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

Areal scale — where it appears

The factor by which a projection multiplies areas at a point, equal to the determinant of its Jacobian and to the product of the two principal scale factors. An equal-area projection holds it at one everywhere, which is one equation and leaves everything about shape free.

Named by 9 essays across 5 fields — each of them below, with the objects they name alongside it.

The indicatrix at 45°, 55° on Mercator. A circle on the sphere becomes this ellipse on the map. Its semi-axes are the two principal scale factors, a = 1.743 and b = 1.743; their product is the areal factor 3.040; and the maximum angular deformation is 0.00°. h and k are shown too, and depend on the coordinates rather than on the map.

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.

distortion · Tissot Angular deformation against latitude, four projections. The same quantity for mercator, gallPeters, mollweide, winkelTripel, 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.

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.

distortion · Tissot Four ways to build the same equal-area projection. The cylindrical equal-area projection's own areal scale factor, measured from its Jacobian against the metric of the body it is drawn for. On a sphere with the spherical formula it is one everywhere, which is the control. Feed the same formula the geodetic latitude a coordinate actually carries and measure against the ellipsoid it refers to, and it is 1.00674 on the equator and 0.99332 at 88° — a spread of 1.34 per cent, on a projection whose entire purpose is that there is no spread. Rescaling to make the totals agree does not repair it. The authalic northing q/2 does, exactly.

Equal-area on the wrong body

The site's headline is that Web Mercator puts geodetic latitudes into a spherical conformal formula and stops being conformal. The same sentence is true with "equal-area" in it and nobody says it: the areal factor is 1.00674 at the equator, 0.99332 at 88°, and averages to almost exactly one — so every check that adds up areas passes while every cell is wrong.

wrong · Audit The area of one 20° × 10° cell at 50–60° north, seven ways. The cell has an exact area — R²Δλ(sin φ₂ − sin φ₁), 1,416,580 square kilometres — so every other row is a measurement of the method rather than of the ground. The equal-area projection returns it to 1.000000 and the spherical polygon formula to 1.000000, which is three routes agreeing — and the same cell integrated on the ELLIPSOID comes out 0.45 per cent away from all three, because the sphere is a model. Taking the shoelace in Mercator gives 3.06 times too much, and treating degrees as a length gives 1.75 times — about sec φ at the cell's middle, which is where that error comes from.

Computing an area needs a surface

A shoelace over a ring of coordinates returns a number whatever the coordinates are. For one twenty-by-ten-degree cell it returns 1.75 times the true area in degrees, 3.06 in a conformal plane, and exactly the closed form in an equal-area one — and the closed form itself is 0.45 per cent out, because the sphere is a model too.

applied · Dataset A 1-hectare parcel across British National Grid, at 52°N. The departure of a plan's dimensions from the ground's, across the width of the zone, as a length and as an area. The area curve is the length curve doubled — an areal scale factor is the square of a linear one, and squaring a small departure doubles it — so a parcel whose sides are each 399 parts per million short on the central meridian is 797 short in area. On a 1-hectare parcel that is 8.0 square metres at the central meridian and 8.0 at 0.00° out.

An area on the grid is not an area on the ground

A grid's scale factor is a property of lengths and what a surveyor sells is an area. Squaring a departure doubles it, so a hectare drawn on the British grid at its central meridian has 10,008 square metres of ground under it — and correcting an area with the line factor instead of its square leaves half the error behind.

practice · Grid Where the middle of a 20° × 20° region is, in five planes. The region is drawn in longitude and latitude — which is itself a projection, and one of the ones being compared. Each filled mark is the shoelace centroid computed in one projected plane and inverted back to the ground; the hollow mark is the centre of area on the sphere, by integration. They spread over 273 kilometres. The equal-area member is 66 kilometres out, because a centroid is a first moment and preserving area says nothing about where the area sits.

A centroid belongs to a plane

Every renderer labels a region at its centroid, and every centroid is a shoelace over coordinates as stored — which is a statement about the plane they are in. Six planes put the middle of one 20° × 20° region up to 273 kilometres apart, the equal-area member is 66 kilometres out, and the disagreement falls as the square of the region's size.

applied · Dataset The error of a spherical formula is twice the flattening. The worst angular deformation of the spherical Mercator formulae applied to each body's own latitudes, against that body's flattening, on logarithmic axes. The points are measured by differencing the projection; the line is 2f radians, which is not fitted. The two agree to a third of a per cent for Mercury, the Earth and Mars, and depart by 3.3 and 5.1 per cent for Jupiter and Saturn, whose flattenings are too large for the first term to be the whole story. The site's headline number — 0.3848° on Earth — is an instance of this law rather than a fact about Web Mercator.

The same projection on a different body

A projection's distortion does not depend on the body's size at all — only on its flattening — so Web Mercator's 0.3848° of angular deformation is not a fact about Web Mercator. It is 2f radians, and the same mistake made on Mars measures 0.6765°, on Jupiter 7.68°, and on a body scaled ten times larger than Mars the identical number to twelve figures.

datums · Bodies A conformal map solved rather than written down. The same patch of parameters on two bodies, mapped to the plane by solving the discrete Cauchy–Riemann equations — one complex equation per triangle, 1568 triangles, least squares, conjugate gradients, and no formula for either surface. Left: a sphere, where the answer is known in closed form and is not used. Right: a body with a bump on it, which has no isothermal coordinate and therefore no closed form at all. The parameter lines cross at right angles in both, to a median of 0.60° and 1.25° of angular deformation.

A conformal map of a body that is not a quadric

Jacobi's ellipsoidal coordinates give a triaxial body a conformal map by two quadratures, and this collection wrote down what that argument uses: the surface has to be a quadric. A real body is not. Solving the discrete Cauchy–Riemann equations instead — one complex equation per triangle, two thousand triangles, conjugate gradients — gives a conformal map of a bumped body to a median of 1.10°, converging at first order in the mesh, with the areal factor spreading by 3.07 and refusing to converge at all.

datums · Bodies The same patch, pinned six ways. Two vertices have to be held or the conformal energy has a similarity's worth of null space. Which two turns out to decide two of the three numbers reported. The median angular deformation is the same to 8 per cent across all six — that is the map. The areal spread runs from 2.68 to 7.44, so the 3.07 reported for this body was a statement about its corners. And pinning two adjacent vertices, which fixes the similarity through a very short lever, ruins the worst point without touching the median: a badly conditioned constraint pays for its scale in one corner.

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.

datums · Bodies

Named alongside it

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

ConformalityToleranceVerificationAngular deformationClosed formEllipsoidEqual-areaNumerical integrationScale factorAuthalic latitudeConventionJacobian

All concepts