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

The trade-off is two lines

Conformal means the two principal scales are equal. Equal-area means their product is one. Both at once forces both to one, which is an isometry, which the curvature forbids. That is the entire argument.
15 min read 9 figures The trade-off is forcedA theorem, not a limitationMeasured, not named

Every treatment of map projections says that angles and areas cannot both be preserved. Most leave it there, which invites the reading that it is a rule of thumb or an observation about the projections people happen to have invented.

It is neither, and the derivation is two lines.

Every projection in the library, measured against both propertiesMaximum 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.10⁻⁶10⁻⁴10⁻²110010⁻¹¹10⁻⁸10⁻⁵10⁻²1010000maximum angular deformation / degreesmaximum areal errornothing here, everMercatorStereographicWeb MercatorGall–PetersMollweideEquirectangulartolerances: 0.0001° and 0.00000121 projections measured
Fig. 1 Every projection in this site’s library, measured against both properties. Conformal projections lie along the left edge; equal-area ones along the bottom. The shaded corner, where both errors vanish, is empty — and the argument below is why it always will be.

The derivation

Tissot’s indicatrix gives two numbers at every point: the principal scale factors aa and bb, the largest and smallest amounts by which the map stretches the sphere there. Both definitions are statements about them.

Conformal means angles are preserved. A map preserves angles at a point exactly when it stretches equally in every direction, so

a=ba = b

Equal-area means area is preserved. The indicatrix ellipse has area πab\pi ab against the circle’s π\pi, so

ab=1ab = 1

Suppose both. Substituting the first into the second gives a2=1a^2 = 1, so a=b=1a = b = 1, at every point.

A map whose principal scale factors are both exactly one everywhere preserves every distance. That is an isometry. And no isometry from the sphere to the plane exists, because Gaussian curvature is intrinsic and the two surfaces have different amounts of it.

That is the whole proof.

What the argument does and does not say

It says the two properties are incompatible everywhere at once. It does not say they are incompatible anywhere in particular.

A projection can be both conformal and equal-area at isolated points, or along a curve. Every projection with a standard parallel is exact along it — scale is true there in both directions, so a=b=1a = b = 1 locally. What cannot happen is for that to hold across a region of non-zero area.

It also does not say the two failures are related in size. They are independent quantities, and a projection can be badly wrong about one and exactly right about the other, which is precisely what conformal and equal-area projections respectively do.

The same point at 30°, 55° under four projectionsOne circle on the sphere, four projections, four ellipses. A conformal projection keeps it circular and lets the area run; an equal-area projection keeps the area and lets the shape go. Nothing keeps both, and the dashed circle shows what keeping both would look like.Mercatorω 0°area 3.04×Gall–Petersω 24°area 1.00×Mollweideω 21°area 1.00×Winkelω 11°area 1.14×dashed: undistortedscaled to fit
Fig. 2 One circle on the sphere under four projections. Mercator keeps it round — a = b — and the area runs away. Gall–Peters keeps the area — ab = 1 — and the shape is destroyed. Nothing keeps both, and the dashed circle shows what keeping both would look like.

What the empty corner is worth

The site demonstrates the theorem computationally as well as deriving it, and it is worth being clear about what that adds.

The derivation is a proof and needs no support. What the measurement adds is a check on the library: if some projection in it ever passed both tests, the conclusion would not be that the theorem is wrong but that the implementation is. The assertion is therefore a test of the site’s own machinery, framed as a test of geometry.

That framing matters because the same measurement is what caught Web Mercator. A loop that runs every projection against both properties finds anything inconsistent, whether the inconsistency is in a projection’s claim or in the code measuring it, and it is not necessary to know in advance which one is being looked for.

The shape of the compromise

Since both cannot be had, every projection is somewhere on a spectrum, and the spectrum has structure worth knowing.

Angular deformation against latitude, four projectionsThe 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.0°20°40°60°80°MercatorGall–PetersMollweideWinkellatitudeangular deformationalong a meridian
Fig. 3 Angular deformation against latitude for four projections. Mercator lies flat on zero — it has bought that with the areal distortion. The equal-area projections rise steadily. Winkel tripel sits between, accepting some of both.
How much each projection inflates a cell, by latitudeFive patches of the sphere, each 20° by 10°, and the factor by which each projection enlarges them relative to the equatorial one. An equal-area projection sits flat on 1. Mercator reaches 15.4× at 70°, which is the mechanism behind every complaint about the size of Greenland.equator23°45°60°70°1×MercatorEquirectangularMillerGall–Peterslatitude of the cell1× means treated fairlyrelative to the equator
Fig. 4 The same four projections measured the other way: how much each inflates a fixed patch of the sphere as it moves poleward. The equal-area projections are flat on 1 by construction. Mercator reaches fifteenfold at 70°, which is what buying conformality costs.

Read together, those two figures are the trade-off drawn twice. A curve that is flat in the first is steep in the second and the reverse. Nothing is flat in both, and the shaded corner in the hero figure is where a curve flat in both would have to live.

Three ways out that do not work

The impossibility is robust, and it is worth disposing of the obvious escapes.

Use a better projection. No. The argument does not mention any particular projection, only aa and bb.

Use a smaller region. This helps in practice and does not evade the theorem. A small region has little total curvature and so admits small distortion, but the distortion is not zero and cannot be. A street map is fine because the error is below the width of the ink, not because it is absent.

Interrupt the map. Cutting the sphere into gores and laying them out separately is a real technique — Goode’s homolosine does it, and so does every globe gore printed for assembly. It reduces the distortion within each piece by making each piece small, at the cost of tearing the map apart. The pieces still distort, and now the map has gaps.

The last one is the most interesting because it identifies what is actually being traded. A projection is a continuous map of the whole sphere onto a region of the plane, and giving up continuity buys real improvements. That is why interrupted projections exist, and why they are used for thematic maps of land and never for navigation.

What the trade-off is not about

One clarification, because it is the source of a great deal of argument elsewhere.

The theorem says nothing about which properties are worth having. It says both cannot be had at once, and stops. Whether a particular map should keep angles or areas is a question about what the map is for, and no amount of geometry answers it.

The Mercator–Peters argument is conducted almost entirely as though the geometry had a view. It does not. Both projections are exactly what they claim; each sacrifices what the other keeps; and the disagreement is about purposes, which is a legitimate disagreement conducted in the wrong vocabulary.

The one property everything can have

There is a third classical property, and it behaves differently from the other two in a way that is worth understanding.

Equidistance means distances are preserved — but only along particular lines, or from particular points. No projection preserves all distances, since that would be the isometry the theorem forbids.

The azimuthal equidistant projection preserves distances from its centre, in every direction. The equirectangular projection preserves distances along meridians. Several conics preserve them along their standard parallels.

So equidistance is not a global property at all; it is a family of local guarantees, and a projection can offer one of them alongside neither, either, or — importantly — not alongside conformality or equal-area at the same point in the same generality.

That is why the classical taxonomy lists three properties and only two of them appear in the trade-off argument above. The third is a different kind of claim.

Azimuthal equidistantThe graticule of the Azimuthal equidistant projection at 30° of longitude and 15° of latitude. distances from the centre are true — from the centre, and from nowhere else. It is neither conformal nor equal-area.neither conformal nor equal-areadrawn in Azimuthal equidistant
Fig. 5 The azimuthal equidistant projection. Every distance from the centre is exact and no other distance is, which is a genuine and useful guarantee of a quite different shape from the two this essay is about.

Local exactness is always available

The theorem forbids exactness over a region. It permits exactness on a set of measure zero, and every projection in practical use exploits this.

A tangent construction is exact along one line; a secant construction along two. Choosing where those lines fall is the main tuning available once the family and property are fixed, and it is what converts a projection from a mathematical object into a fit-for-purpose map.

The distortion then grows away from the exact lines, and for a region of modest extent it can be kept small everywhere. A UTM zone is six degrees wide and its scale error stays within about a part in a thousand — which is the curvature integral again: a narrow zone carries little total curvature, so little distortion is forced.

What a standard parallel buysThree equal-area cylindrical projections differing only in where they are exact. 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.0°20°40°60°80°exact at equatorexact at 30°exact at 45°latitudeangular deformationall three are equal-area
Fig. 6 Three equal-area projections differing only in where they are exact. Each has zero angular deformation on its own standard parallel — local exactness, permitted by the theorem — and grows away from it in both directions.

The result is stronger than it looks

One more consequence, because it disposes of a hope that recurs.

The argument shows that a projection cannot be conformal and equal-area over a region. It does not assume the projection is smooth in any strong sense, or that it is defined on the whole sphere, or that it belongs to any family. It uses only that the map has a derivative with principal scales at each point.

So the prohibition survives interruption, survives piecewise definition, and survives any amount of cleverness in the construction. A projection assembled from a thousand different formulae on a thousand different patches still has principal scales at each point, and the same two lines apply at each of them.

The only genuine escape is to give up being a map of a region at all — which is what a globe does, and what an interactive rescaling tool does, and neither is a projection.

What a globe does instead

The one object with no distortion at all is worth mentioning, since it is the alternative the theorem leaves open.

A globe is not a projection. It is the sphere, at a smaller scale, and scaling is a similarity rather than an isometry — every distance is reduced by the same factor, so every angle and every area ratio is exact.

Its cost is not distortion but occlusion: half of it is facing away at any moment, and no amount of it can be seen at once. That is a genuine trade and it is a different trade from the one this essay is about.

Which identifies what a projection actually buys. It is not accuracy — a globe has more. It is simultaneity: the whole surface, visible at once, on something flat enough to print, fold, measure and carry. The distortion is the price of seeing everything together.

OrthographicThe graticule of the Orthographic projection at 30° of longitude and 15° of latitude. the view from infinitely far away — a hemisphere at a time, and the one that looks like a globe. It is neither conformal nor equal-area.neither conformal nor equal-areadrawn in Orthographic
Fig. 7 An orthographic projection, which looks like a globe and is not one. It shows a hemisphere, flattens severely toward the rim, and is neither conformal nor equal-area — every figure on this site that resembles a globe is a projection with its own distortion.

Reading the audit plot

The hero figure repays a careful look, because its structure is the theorem’s shape.

Points along the left edge are conformal: angular deformation at the noise floor, areal error running from small to enormous. Points along the bottom are equal-area: areal error at noise, angular deformation from moderate to severe.

Points in the middle are compromises, and they are neither close to either edge nor especially far from both.

The corner is empty. Not sparse — empty, and the two lines at the top of this essay say it will stay that way.

The one red point is Web Mercator, which is not near the left edge despite claiming to be, and that is the site’s opening argument in one mark.

Tissot's indicatrix across StereographicA small circle on the sphere, drawn where the projection puts it. The dashed circle behind each is what an undistorted map would show. On Stereographic every ellipse is a circle — ω never exceeds 1.2e-6°, and the areal factor reaches 46.6.dashed: an undistorted circledrawn in Stereographic
Fig. 8 Stereographic indicatrices. Every one is a circle, so the projection sits on the left edge of the audit plot. They grow without bound toward the antipode, which is how far up the vertical axis it sits.

The proof needs almost nothing

Worth noting what the argument does not assume, because that is what makes it robust.

It does not assume the projection is given by a formula, or that it is defined on the whole sphere, or that it belongs to any family, or that it is constructed geometrically. It needs only that the map has a derivative at each point, so that principal scale factors exist there.

Which means it applies to a projection assembled piecewise from a hundred different rules, to one defined by a lookup table, and to one nobody has thought of. The two lines run at every point independently, and a map is conformal and equal-area over a region only if they run successfully at every point of it.

That generality is why the result is a theorem about the sphere rather than a survey of known projections.

6 projections of the same sphereThe same graticule under stereographic, lambertAzimuthal, mercator, gallPeters, albers, lambertConformalConic. 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.StereographicLambert azimuthal equal-areaMercatorGall–PetersAlbers equal-area conicLambert conformal conicsame sphere, same graticuleno two agree
Fig. 9 Six projections, three conformal and three equal-area, drawn from three different construction families. Every one is exactly what it claims; none is both; and the theorem applies to each identically regardless of how it was built.

What was computed here

The audit runs every projection in the library — twenty-one of them — through both tests, sampling several hundred points across each. The tolerances are set from measurement rather than taste: the genuinely conformal projections measure around 1.5×10−61.5\times10^{-6} degrees of angular deformation, which is the noise floor of the differentiation, and the tolerance sits sixty times above that and nearly four thousand times below the smallest real failure.

Three assertions guard the conclusion. Nothing may pass both tests. The audit must find projections of both kinds, so that a bug making everything fail would be caught. And the tolerance itself must discriminate, with the margins above and below checked on every build.

The equal-area projections measure their areal error between 6×10−126\times10^{-12} and 1.5×10−91.5\times10^{-9}, which is a much cleaner zero than the conformal ones manage — an areal factor is a determinant and does not suffer the cancellation that comparing two nearly equal scale factors does.

What the pictures cannot show

The audit’s axes are logarithmic and span seventeen decades, which is the only way to get a noise floor of 10−1110^{-11} and a failure of 10410^{4} onto one plot. That compresses the interesting region and exaggerates the uninteresting one, and a reader looking for a sense of how much worse one projection is than another should read the numbers rather than the distances.

The empty corner is also a picture of an absence, and an absence looks the same whether it is a theorem or an oversight. The figure cannot distinguish those; the derivation at the top of this essay is what does.

Who found it, and when

The incompatibility was understood well before it was proved. Practical cartographers knew from experience that a projection good for shape was bad for area, and the two families were named and used for centuries on that basis.

Gauss supplied the reason in 1827 with the Theorema Egregium, though he was working on the geodetic survey of Hanover rather than on map projections as such. Tissot gave the local analysis in 1859 and 1881 that turns the general prohibition into a computable quantity at each point.

Lambert had already, in 1772, produced both a conformal conic and an equal-area cylindrical projection in the same publication — and named the properties as the design goals of each. That the same person constructed both, deliberately, half a century before the theorem, is a fair indication of how well the trade-off was understood in practice before anyone could say why it held.

Where this goes next

The curvature underneath the argument is no map is faithful. The tool that makes it measurable is Tissot’s indicatrix. And for what to do about a trade-off that cannot be escaped, which projection is best.

Where this goes next

planeK = 0 — unrolls flatcylinderK = 0 — unrolls flatconeK = 0 — unrolls flatsphereK = 1.00torusK = 1.79pseudosphereK = -1.00K computed at each centretwo routes, one answer No map is faithful Not "no map yet", and not "no map at page size". Gaussian curvature can be computed from inside a surface, a sphere has some and a plane has none, and that closes the question permanently. a = 1.74b = 1.74an infinitesimal circle, projectedmeasured at this pointh1.7434scale along the meridiank1.7434scale along the parallela1.7434larger principal scaleb1.7434smaller principal scalea·b3.0396areal scale factorω0.00°maximum angular deformationdashed: undistorteddrawn in Mercator 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. EquirectangularMercatorMollweideSinusoidalRobinsonWinkel tripelsame sphere, same graticuleno two agree 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.