Tissot's indicatrix

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.

Six numbers, of which four matter #

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

$h$ is the scale factor along the meridian: how much the map stretches a step northward. $k$ 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.

$a$ and $b$ 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.

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

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

$$\sin\frac{\omega}{2} = \frac{a-b}{a+b}$$

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.

Which angle is the one that is wrong #

$\omega$ is a maximum over all pairs of directions, and the definition does not say which pair achieves it. It is worth extracting, because the answer is not the one most readers guess.

A direction making an angle $\alpha$ with the major principal axis comes out on the map at $\alpha'$, where $\tan\alpha' = (b/a)\tan\alpha$. The single direction whose bearing is misrepresented most is at $\tan\alpha = \sqrt{a/b}$, and the deviation there is exactly $\omega/2$.

Gall–Peters at the equator makes this concrete. There $a = \sqrt2$ along the meridian and $b = 1/\sqrt2$ along the parallel, so the worst-served bearing sits at $\arctan\sqrt2 = 54.7°$ from north — roughly north-east. A course of 054.7° on the ground is drawn as 035.3°, an error of 19.5°. The angle between that bearing and its mirror image about the meridian therefore changes by twice that, 38.9°, which is $\omega$.

The directions along the axes themselves — due north and due east — come out exactly right. The distortion is worst in between, which is why a map can look plausible along its graticule and be badly wrong across it.

The two definitions fall out immediately #

Conformal means angles are preserved, which means $\omega = 0$, which means $a = b$, 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 $a \cdot b = 1$ 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.

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 $\omega$ 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 $a\,b$ 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.

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.

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 $ab$ times the original, and the indicatrix stops predicting it.

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 $h$, $k$ and the angle between them, and computes $a$ and $b$ 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 scale distortion nobody counts #

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

Even a conformal projection, with $a = b$ everywhere, has $a$ 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.

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.

An indicatrix is an ellipse and an ellipse has an orientation. Drawing only the orientations, over the whole map, shows a pattern that a field of ellipses hides.

Where h and k stop being the principal scales #

Most treatments of distortion give $h$ and $k$ — the scale along the meridian and the scale along the parallel — and stop there, and a table of the two is the usual published summary of what a projection does. It is exact for a smaller class of projections than the habit suggests.

$h$ and $k$ are scales measured along the graticule. $a$ and $b$ are the largest and smallest scales in any direction. The two pairs coincide only when the projected graticule is orthogonal, because only then does one of the graticule directions happen to be the direction of greatest stretch. The consistency relation quoted above says exactly how much is lost when it is not:

$$a\,b = h\,k\,\sin\theta'$$

with $\theta'$ the angle between the projected meridian and the projected parallel. At $\theta' = 90°$ the two products agree and the tabulated pair is the principal pair. Below that they part, and the arithmetic is unforgiving in a quiet way: at 75° the factor is 0.966, which is under four per cent and invisible in a table rounded to two places; at 60° it is 0.866; at 45° it is 0.707, and a summary built from $h$ and $k$ is then overstating the area the map covers by nearly half.

The direction field above is the same fact seen from the other side. A principal axis leaves the graticule only where the graticule has stopped being orthogonal, so a stroke that swings 45° away from the meridian is a point where $\theta'$ has moved far from a right angle and where the meridian scale is not the largest scale available.

Which projections are safe is then a question with a clean answer rather than a judgement. Every cylindrical projection in normal aspect has an orthogonal graticule by construction, and so does every conic and every azimuthal one in its own polar aspect: for those, $h$ and $k$ are $a$ and $b$, and the published table is the principal pair under another name. Every pseudocylindrical away from its central meridian is not, because its parallels stay horizontal while its meridians lean. Neither is Hammer, nor any oblique aspect of anything, because rotating the sphere under a projection does not carry the graticule’s orthogonality with it.

That is why this site computes $a$ and $b$ and reports those, and treats $h$ and $k$ as intermediate quantities that happen to be easy to differentiate. The pair a reader wants is the pair that does not depend on which lines somebody chose to draw.

An indicatrix is a statement about a point, and every quantity above is a property of one place. Two points a thousand kilometres apart are related by something no collection of point measurements can describe, because the relation between them is an integral along a path and the ellipse knows nothing about paths.

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 $3\times10^{-11}$.

The derivation follows the standard route: $h$ and $k$ 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 $a$ and $b$ from those by the usual half-sum and half-difference. The consistency relation $a\,b = h\,k\sin\theta'$ 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 $h$ and $k$ 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: $a$ and $b$ are its singular values, and the principal directions are its singular vectors. Tissot did not have that vocabulary and did not need it.

Why the construction outlived its vocabulary #

The observation that the indicatrix is the singular value decomposition of the Jacobian in all but name is worth more than a footnote, because it explains both the construction’s durability and the shape of its limitations.

Tissot found the right object without the language for it. The two semi-axes are the singular values, the principal directions are the singular vectors, and the whole apparatus is what linear algebra would later call the polar decomposition’s stretch half. Arriving at it geometrically, from a circle and its image, was harder than deriving it algebraically and produced exactly the same thing.

Which is why it has never needed revising. A construction that happens to be a canonical decomposition of a matrix is not a convention that a later generation might improve on; it is the complete answer to what does a linear map do to a neighbourhood, and there is nothing left over.

And it is why the limitations are where they are. Everything the indicatrix omits, it omits because a stretch tensor omits it: the rotation, which the polar decomposition puts in the other factor and which the fourth number recovers; and everything second-order, because a Jacobian is a first derivative and no decomposition of it can hold what it does not contain.

So the honest summary is that the instrument is complete for its order and silent beyond it. That is a much better position than most instruments are in, and it is worth saying because incomplete and silent about a different question are easily confused — the first invites a patch, and the second invites a second instrument.

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.