Tissot — the ladder · Cartographic Projection

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.

rung 1 · distortion

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.

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.

rung 2 · distortion

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.

rung 2 · distortion

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 six named here are Mercator, Web Mercator, Gall–Peters, Mollweide, Equirectangular, Stereographic.

Measuring instead of naming

A projection is called conformal because that is its name. Running the definition as a computation over several hundred points takes about twenty lines, catches a projection the whole internet uses, and is almost never done.

rung 2 · distortion

The least distortion possible over a 20° region. Scale factor along a radius of the cap, each projection normalised to unit scale at the centre. Chebyshev's criterion names the projection whose scale is constant on the boundary as the conformal map of least scale variation, and for a cap that is the stereographic projection centred on it — reaching exactly sec²(ρ/2) = 1.0311 at the rim, marked. Every other conformal projection drawn here rises past that line before it gets there. This is the only figure on the site showing an optimum rather than a comparison.

Scale distortion is the third failure

A map's stated scale is its scale at one place. Everywhere else it differs, by a factor that no conformal projection can avoid and that neither of the two usual distortion measures reports. It is the failure everybody uses and nobody counts.

rung 3 · distortion

The same six projections, ranked over Europe and Chile. Each column orders the projections by Kavrayskiy's regional criterion — the root-mean-square departure of the two principal scales from unity, integrated over the region with the area element. The lines cross, which is the point: Lambert conformal conic leads over Europe and comes fifth over Chile. A table of projections ordered by distortion is a table about somebody's region.

Distortion over a region

An indicatrix describes a point and every practical question is about a country. Going from one to the other means integrating, integrating means choosing a weighting, and the weighting is the step that turns a measurement into somebody's opinion.

rung 3 · distortion

The image of a circle on Mercator, against its own indicatrix. A circle of three radii on the ground at 30°, 40°, projected exactly — the solid curve — against the ellipse the indicatrix predicts for it, dashed. The filled dot is the image of the circle's centre and the hollow one is the centre of area of what was actually drawn, which is not the same point. The departure runs from 3.14 per cent at 4° to 15.80 per cent at 16°, so it grows in proportion to the radius rather than to its square: halving the circle halves the relative error and does not quarter it.

The indicatrix is a limit

Tissot's ellipse describes an infinitesimal circle, and every published one is drawn finite. The error of the description falls as the radius rather than as its square — halving the circle halves the lie — and on the Robinson projection at a table entry it does not fall at all: sixty metres and six kilometres are both 1.17 per cent wrong.

rung 3 · distortion

The scale factor across Europe, from the middle outwards. Each band is the range the larger principal scale factor takes on the ring at that distance from the centre of Europe: 0 is the middle, 1 the frontier. two of the three projections drawn are conformal, and every one of those reaches its maximum at the right-hand edge, because the logarithm of a conformal map's scale factor is subharmonic. None of them reaches a maximum inside the region.

Where the worst point is

The largest scale error on a conformal map of a country is always on the frontier, never inside it, whatever the country's shape and whichever conformal projection was chosen. It is a theorem rather than a tendency, and it is the reason Chebyshev's criterion works.

rung 4 · distortion

Which way Sinusoidal stretches the ground. The major axis of Tissot's indicatrix at 180 points, drawn as a stroke rather than an ellipse because the argument is the orientation. Longer strokes mark more elongated indicatrices. The direction departs from the graticule by up to 44.4°, so the scale "along the meridian" is not the scale along the direction that is actually stretched most.

Distortion has a direction

Tissot's indicatrix is an ellipse, an ellipse has an orientation, and the orientation is never reported. On most projections it is not along the meridian, on a conformal one it does not exist at all, and both facts are computable from the same four derivatives as everything else.

rung 4 · distortion

The circle a reprojection is about to turn into an ellipse. A circle drawn on the Mollweide plane, and its image under the map that carries that plane to Mercator — normalised so each pair has the same area, because the whole map can be rescaled and the shape is what is being shown. The dashed circle is what an undistorted reprojection would leave. This is Tissot's construction with a plane in place of the sphere, and it is the right picture for a reprojection because both ends are pictures. Worst angular deformation over the sampled points: 81.2°. Drawn on the source plane, in Mollweide.

A projection between two projections

Every distortion measured on this site so far compares a map with the sphere. The operation a machine actually performs compares a map with another map — and that map has its own two principal scales, its own areal factor and its own angular deformation, none of which is the difference of the two it was built from.

rung 5 · distortion

The indicatrix is a limit, and here it is being taken. The axis ratio of the image of a circle of radius ρ, measured from the picture rather than computed from the derivatives, against the ratio the derivatives give. It converges, and it converges as ρ² — each halving of the radius quarters the departure. At the smallest radius drawn the two agree to 7.6e-4 relative. This is what "the indicatrix is a limit" means when it is working: there is a number for the ratio to settle on, and it settles.

The indicatrix at a point that has none

Ten essays on this ladder take the derivative for granted. Rung three proved the indicatrix is a limit and every figure since has assumed the limit exists. There are points on the maps this collection draws where it does not, and they are not exotic: the horizon of a gnomonic map, and every corner of every polyhedral net.

rung 6 · distortion

Four averages of one region's indicatrices — Robinson over the whole sphere. The faint ellipses are the indicatrix at ninety sampled places; the four drawn over them are four candidate averages of exactly that set. The arithmetic mean of the tensors gives semi-axes 1.1621 and 1.0182; the log-Euclidean mean gives 1.0053 and 0.9608; the Karcher mean gives 1.0042 and 0.9619; and taking the mean of a and the mean of b separately — which is what a published average distortion usually is — gives 1.2335 and 0.8200. Their maximum angular deformations are 7.57°, 2.60°, 2.47°, 23.23°, against a mean of the pointwise deformations of 20.67°. Five numbers, one set of ellipses.

An average of ellipses is not an ellipse

Rung three integrates distortion over a region and finds the weighting is somebody's opinion. It never asked what was being averaged. An indicatrix is a positive-definite matrix, matrices form a cone rather than a vector space, and the arithmetic average of an equal-area map's indicatrices comes back inflating area by up to 11 per cent.

rung 7 · distortion

How aligned a region's ellipses are, for every projection and every region. The resultant length of the doubled indicatrix orientations, over 12 projections and 9 regions. One means every ellipse in the region points the same way; zero means they are spread evenly and cancel. Three cylindrical rows are 1.00 throughout, and everything else varies down the row and across it — which is the answer to the question the number was first stated without: alignment is a property of the pair. Mercator's row is the exception that is not a measurement, because a conformal projection's indicatrix is a circle and a circle has no orientation.

Whether the ellipses point the same way

The previous rung found that alignment decides whether the average of a region's deformations is above or below the deformation of its average, and stated it at ten projections over one region. Swept over a hundred and eight pairs, the answer is that alignment belongs to the projection 28 per cent, to the region 38, and to neither 33 — and two rows of the table turn out not to be measurements at all.

rung 8 · distortion

Two conventions for one indicatrix field on Mercator. The left panel is the convention nearly every published field uses: every ellipse drawn at the same area on the paper, so the picture carries the shape and throws the size away. The right panel draws the image of the same ground circle at every place, so the drawn area IS the areal scale factor. On Mercator the axis ratio spans 1.0000 and the areal factor spans 14.93, so the left panel shows nothing the other one shows. Neither caption states which is which, on any published map this collection has found.

The ellipses are a sample, drawn at a size somebody chose

Thirteen essays measure with the indicatrix and none audits it as an instrument. A published field has a gauge nobody states and a placement nobody states: on Mercator the standard convention draws twenty-five identical circles while the areal factor runs over a factor of 14.9, and the average a reader takes off any of these fields is between 22 and 64 per cent too high.

rung 9 · distortion

Two maps with the same ellipses. Mollweide, and the same projection with the sheet turned 37° on the desk. Every ellipse is the same size and the same shape as its counterpart, and its angle to its own graticule is the same — the two indicatrix fields agree to 1.8e-12, which is arithmetic noise. The rotation differs by 37° at every point. So the classical description of a map's distortion is exactly blind to the difference between these two, and the difference is the whole of what a surveyor calls convergence.

The fourth number the ellipse does not carry

A Jacobian has four independent entries and an indicatrix reports three. The missing one is the rigid rotation in the polar decomposition A = R·S, the indicatrix is exactly S, and R is what turns north into grid north: for the transverse Mercator it agrees with the survey formula for convergence to 5 × 10⁻¹⁰ degrees, from a different library and a different derivative.

rung 10 · distortion

What a reprojection does, drawn where it does it. The angular deformation of the map that takes Sinusoidal's page to Gall–Peters's — which is what a reprojection is — shaded over the sphere and drawn on Mollweide so that equal ground areas are equal page areas. Both parents are equal-area, so the composite preserves area exactly; its mean angular deformation is 55.5° against 38.9° and 32.2° for the two maps separately. Composing two maps did not average their shape errors; it made a larger one.

Two indicatrices do not make a third

Reprojecting is composing, and the composite's indicatrix is not a function of its parents'. Mollweide followed by Hammer is gentler than either map over eighty per cent of the sphere; the sinusoidal followed by Gall–Peters is worse than both everywhere. What separates them is one angle, and it is the number rung ten showed the ellipse does not carry.

rung 11 · distortion