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

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.
15 min read 8 figures Measured, not namedWhat survives a change of coordinatesComputed, not quoted

Take a circle on the sphere, small enough that the projection is effectively linear across it. Its image on the map is an ellipse. That ellipse is Tissot’s indicatrix, and everything anyone needs to know about the distortion at that point is in its two axes.

The indicatrix at 45°, 55° on MercatorA 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.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
Fig. 1 The indicatrix at 45° east, 55° north on Mercator, with every quantity named. The dashed circle is what an undistorted map would show. The semi-axes are the two principal scale factors; their product is the areal factor; and their ratio fixes the maximum angular deformation.

Six numbers, of which four matter

Differentiating the projection at a point gives four partial derivatives, and six quantities follow.

hh is the scale factor along the meridian: how much the map stretches a step northward. kk is the scale factor along the parallel. These two are the ones most often quoted, and they are the two that depend on how the sphere was parameterised rather than on the map.

aa and bb are the semi-axes of the ellipse — the largest and smallest scale factors over all directions, called the principal scale factors. They do not depend on the parameterisation.

a⋅ba \cdot b is the areal scale factor: how much the map inflates area at that point.

ω\omega, the maximum angular deformation, is fixed by the ratio through

sin⁡ω2=a−ba+b\sin\frac{\omega}{2} = \frac{a-b}{a+b}

and is the largest amount by which any angle at that point is misrepresented.

The last four are properties of the map. The first two are properties of the map together with the coordinate grid, and confusing the two groups is the commonest error in informal treatments of this subject.

The two definitions fall out immediately

Conformal means angles are preserved, which means ω=0\omega = 0, which means a=ba = b, which means the indicatrix is a circle everywhere. A conformal projection may make the circle any size it likes — that is scale distortion, and it is unavoidable — but it must keep it round.

Equal-area means a⋅b=1a \cdot b = 1 everywhere. The ellipse may be any shape at all, as long as its area matches the circle’s.

That is why the site can test a projection’s claim rather than repeating it: both properties are statements about a computable quantity at every point, and both are checkable by sampling.

Tissot's indicatrix across MercatorA small circle on the sphere, drawn where the projection puts it. The dashed circle behind each is what an undistorted map would show. On Mercator every ellipse is a circle — ω never exceeds 1.0e-6°, and the areal factor reaches 4.0.dashed: an undistorted circledrawn in Mercator
Fig. 2 Indicatrices across Mercator. Every one is a circle, which is what conformal means — the measured angular deformation never exceeds the noise floor of the differentiation. They grow enormously toward the poles, which is the areal distortion that conformality costs.
Tissot's indicatrix across Gall–PetersA small circle on the sphere, drawn where the projection puts it. The dashed circle behind each is what an undistorted map would show. On Gall–Peters ω reaches 39°, and the areal factor reaches 1.0.dashed: an undistorted circledrawn in Gall–Peters
Fig. 3 The same figure on Gall–Peters. Every ellipse has the same area, which is what equal-area means. None of them is a circle away from the standard parallels, and near the equator the distortion is severe — the shapes are stretched vertically by a factor of two.

Those two figures together are the trade-off, and it is forced rather than chosen.

What the usual version leaves out

Search for Tissot’s indicatrix and most of what comes back is a map with circles on it. Not ellipses — circles, all the same size, evenly spaced, unlabelled.

That is not an indicatrix. It is a picture of where indicatrices would go.

The entire content of the construction is how much the ellipse departs from a circle, and dropping the numbers drops the content. A reader looking at an unlabelled figure can see that the shapes near the poles are bigger, which they already knew, and cannot tell whether the areal factor there is 3 or 30. On Mercator at 70° it is about 8.5, and at 80° about 33 — a difference the picture does not carry and the number does.

Every indicatrix on this site is computed by pushing a small circle of directions through the projection and measuring what comes back, and the important ones state their axes, their areal factor and their ω\omega in figures.

The construction is also checked against itself. The ellipse drawn by walking directions round a circle must have the semi-axes that the analytic formulae predict from the Jacobian — two independent routes to the same pair of numbers, and the agreement is asserted before the figure is drawn.

The infinitesimal, and what it costs

The indicatrix describes the projection’s behaviour on an infinitesimally small circle. That is not a technicality to gloss over; it is the source of the tool’s main limitation.

A projection is a smooth map, so at a small enough scale it is a linear map, and a linear map sends circles to ellipses. Everything above follows from that. Push the circle up to a finite size and the linearisation stops being exact — a large circle on the sphere maps to something that is not an ellipse at all, and its area need not equal a ba\,b times the original.

So the indicatrix is a statement about a point, and a map’s overall behaviour is not the sum of its points in any straightforward way. Summarising a whole projection by one number requires choosing a region and a weighting, and both are judgements rather than measurements.

The drawn indicatrices are also enlarged, by a factor stated in each figure. A true indicatrix at the scale of these maps would be smaller than a line width. This is the same compromise MacAdam’s ellipses require in colour science and it is worth noticing whenever a figure of this kind appears: the shapes are true and the sizes are magnified.

Comparing points across projections

The most useful thing the indicatrix does is let one point be compared across maps.

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. 4 One circle on the sphere at 30° east, 55° north, under four projections. Mercator keeps it round and lets the area grow. Gall–Peters keeps the area and squashes the shape. Mollweide keeps the area with a different compromise on shape. Winkel tripel keeps neither exactly and both approximately.

Four ellipses, four different answers, one point on the Earth. That figure is the argument against asking which projection is best without saying what for.

Reading a map’s indicatrices

Once the ellipses carry numbers, a map’s whole distortion pattern can be read off at a glance, and a few signatures are worth learning.

All circles, growing outward means conformal. The size tells the scale error, and the growth rate tells how fast the areal error accumulates.

All the same area, changing shape means equal-area. The elongation direction tells which way shapes are being stretched.

Ellipses leaning means the graticule is not orthogonal on the map — the principal directions are not the coordinate directions, and h and k have stopped describing the distortion.

Circles at one latitude, ellipses elsewhere means a standard parallel, and where the circles are is where the projection is exact.

Tissot's indicatrix across Winkel tripelA small circle on the sphere, drawn where the projection puts it. The dashed circle behind each is what an undistorted map would show. On Winkel tripel ω reaches 43°, and the areal factor reaches 1.2.dashed: an undistorted circledrawn in Winkel tripel
Fig. 5 Winkel tripel’s indicatrices. No circles except near two isolated points, no constant areas, no clean pattern — which is the visual signature of a compromise projection, and is what preserving nothing exactly looks like.

That last figure is a useful negative example. A compromise projection has no clean indicatrix signature because it has no exact property, and the ellipses simply vary moderately in both shape and size everywhere.

The magnification problem

Every indicatrix figure on this site draws its ellipses at many times their true scale, and this is worth stating plainly because it is a real distortion of the evidence.

A true indicatrix is infinitesimal. Drawn at true scale on a world map it would be smaller than a line width, and a figure of twenty invisible dots would be accurate and useless.

The compromise is to draw the shape correctly and the size at a stated magnification. Within one figure the relative sizes are meaningful — a Mercator indicatrix at 60° really is four times the area of one at the equator — and across figures at different magnifications they are not.

The same compromise appears in MacAdam’s discrimination ellipses in colour science, for the same reason and with the same caveat. It is the standard treatment for any figure showing a local derivative as a finite shape.

What a finite region does instead

Push the circle up to a finite size and the linear approximation stops holding. A large circle on the sphere maps to a curve that is not an ellipse, its area is not abab times the original, and the indicatrix stops predicting it.

One 20° × 10° cell at 45°, under four projectionsThe same patch of the sphere, drawn by four projections and scaled to fit. The shapes differ and so do the areas: the figure under each panel is the areal inflation relative to the same cell at the equator, so 1.00 means the projection treats the two fairly.Mercator2.41× areaGall–Peters1.00× areaMollweide1.00× areaEquirectangular1.55× areathe cell at 45°scaled to fit, areas as stated
Fig. 6 A 20°-by-10° cell at 45° under four projections. These are finite regions rather than infinitesimal circles, and their shapes are not the ellipses the indicatrix predicts — the sides curve, and the corners are not where a linear map would put them.

This is the indicatrix’s main limitation and it is structural. A country is a finite region; the indicatrix describes a point; and going from one to the other requires integrating, which introduces a choice of how to weight the interior.

Which is why summarising a projection by one number is a design decision rather than a measurement, and why this site reports pointwise distortions and exact finite-region areas as two separate things rather than blending them.

Two routes to the same ellipse

The construction is implemented twice on this site, and requiring the two to agree is what makes the figures trustworthy.

The analytic route differentiates the projection, assembles hh, kk and the angle between them, and computes aa and bb from the standard half-sum and half-difference formulae.

The geometric route walks a small circle of directions on the sphere, pushes each through the projection, and records the longest and shortest images.

They are independent — one is algebra on four partial derivatives, the other is a loop over 360 directions — and they agree to better than a part in a thousand, which is asserted before the detail figure is drawn.

That is the same discipline applied to the tool itself that the tool applies to the projections: the claim is measured rather than assumed, and by two routes wherever two exist.

The indicatrix at 120°, 40° on SinusoidalA circle on the sphere becomes this ellipse on the map. Its semi-axes are the two principal scale factors, a = 1.879 and b = 0.532; their product is the areal factor 1.000; and the maximum angular deformation is 67.89°. h and k are shown too, and depend on the coordinates rather than on the map.a = 1.88b = 0.53an infinitesimal circle, projectedmeasured at this pointh1.6770scale along the meridiank1.0000scale along the parallela1.8786larger principal scaleb0.5323smaller principal scalea·b1.0000areal scale factorω67.89°maximum angular deformationdashed: undistorteddrawn in Sinusoidal
Fig. 7 The indicatrix on the sinusoidal projection far from the central meridian. The ellipse leans, because the graticule is not orthogonal there — and the geometric construction finds the true principal directions without being told where to look, which is exactly what the analytic h and k cannot do.

The scale distortion nobody counts

A third failure hides inside the indicatrix and is usually left out of the classification.

Even a conformal projection, with a=ba = b everywhere, has aa varying from place to place. That is scale distortion: the map is at a different scale in different places, and it is unavoidable for the same reason everything else here is.

It does not appear in the angular deformation, which depends on the ratio, and it does appear in the areal factor, which depends on the product. So it is not independent of the two quantities this site tracks — but it is worth naming separately, because “the scale of a map” is a phrase people use as though it were a single number and it never is.

A map’s stated scale is its scale at the standard parallel, or its nominal scale, and everywhere else it differs. On Mercator at 60° the true scale is twice the nominal one.

How Stereographic distorts, by latitudeAngular deformation and areal scale plotted against latitude along the meridian at 0°. On Stereographic the angular deformation reaches 0.0° and the areal factor reaches 3.4.-60°-30°0°30°60°angular deformation, to 2°areal factor, to 3.4×latitudetwo independent distortionsalong the 0° meridian
Fig. 8 Stereographic measured along a meridian. The angular deformation is flat on zero — conformal — and the areal factor grows without limit, which is scale distortion accumulating in a projection that has no angular distortion at all.

The construction’s durability is worth a final remark. It is a nineteenth-century idea, it computes what would now be called the singular value decomposition of the Jacobian, and nothing has replaced it in a hundred and fifty years. That is unusual, and the reason is that the question it answers — what does this map do to a small circle here — is exactly the right question, and it has not changed.

What was computed here

Every quantity on this page comes from the projection’s own derivatives, obtained by Richardson-extrapolated central differences and validated against the analytic scale factors for the projections that have them — Mercator, equirectangular, sinusoidal and the Lambert cylindrical all agree to better than 3×10−113\times10^{-11}.

The derivation follows the standard route: hh and kk from the lengths of the two partial-derivative vectors divided by the corresponding distance on the sphere; the areal factor from the Jacobian determinant; the angle between the projected meridian and parallel from the ratio; and aa and bb from those by the usual half-sum and half-difference. The consistency relation a b=h ksin⁡θ′a\,b = h\,k\sin\theta' holds by construction and is a useful thing to check by hand when adding a projection.

One assertion in the library guards the distinction this essay is about. The principal scales, areal factor and angular deformation must not change when the projection’s output axes are swapped, while hh and kk exchange. That is the cleanest available demonstration that the first group describes the map and the second describes the coordinates, and it comes out at exactly zero drift.

What the pictures cannot show

The magnification, already stated. And the fact that these are point measurements: an indicatrix says nothing about what happens to a country, only about what happens at a place.

There is also a limit specific to the comparison figure. Four ellipses drawn side by side have been scaled to fit the panels, so their relative sizes across panels are not meaningful — only the shape within each panel and the numbers underneath. A figure that showed the true relative sizes would be mostly white space with one enormous Mercator ellipse in it, which is honest about the areas and useless about the shapes.

Who found it, and when

Nicolas Auguste Tissot published the construction in 1859 and developed it in his Mémoire sur la représentation des surfaces of 1881. He was a French cartographer, and the problem he was addressing was practical: given several candidate projections for a survey, how does one compare them without arguing about appearances.

His answer was to stop looking at the map as a whole and look at what it does to a point. That move — from the global impression to the local derivative — is what made the comparison quantitative, and it is the reason a nineteenth-century construction is still the standard tool.

The mathematics underneath is older, and is the singular value decomposition of the Jacobian in all but name: aa and bb are its singular values, and the principal directions are its singular vectors. Tissot did not have that vocabulary and did not need it.

Where this goes next

The distinction between the two groups of quantities is what survives a change of coordinates. The two independent failures it measures are the two ways a map is wrong. And the method the whole site is built on is measuring instead of naming.

Where this goes next

a = 1.08b = 0.92an infinitesimal circle, projectedmeasured at this pointh1.0834scale along the meridiank0.9231scale along the parallela1.0834larger principal scaleb0.9231smaller principal scalea·b1.0000areal scale factorω9.16°maximum angular deformationdashed: undistorteddrawn in Gall–Peters 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. 0°20°40°60°80°MercatorGall–PetersMollweideWinkellatitudeangular deformationalong a meridian 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. 10⁻⁶10⁻⁴10⁻²110010⁻¹¹10⁻⁸10⁻⁵10⁻²1010000maximum angular deformation / degreesmaximum areal errornothing here, everMercatorStereographicWeb MercatorGall–PetersMollweideEquirectangulartolerances: 0.0001° and 0.00000121 projections measured 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.