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

This collection’s opening move is that a projection’s properties are measured rather than named, and the instrument it measures with is Tissot’s indicatrix. Thirteen essays on this ladder use it. None has asked what a published field of them actually shows.

Two choices go into drawing one, neither of them is ever stated, and between them they decide most of what a reader takes away.

The gauge #

An indicatrix is the image of an infinitesimal ground circle, and an infinitesimal circle has to be drawn at a finite size. The indicatrix is a limit prices what the finiteness costs in accuracy. This is a different question: which finite circle.

Two answers are available.

A constant page radius. Every ellipse is drawn with the same area on the paper. The picture then carries the shape — the axis ratio, which is the angular deformation — and carries no information about size at all.

A constant ground circle. Every ellipse is the image of the same circle on the Earth, so its drawn area is the areal scale factor and its shape is still the shape.

Almost every published field is the first. Almost every reader reads it as the second — the same confusion an average of ellipses is not an ellipse turns on, where a picture of a construction is taken for the construction — because a picture of ellipses of visibly different sizes is what the second would look like and the first does not announce itself.

What that does on a conformal projection #

On Mercator the consequence is not an approximation. It is a blank sheet.

A conformal projection has its two principal scale factors equal everywhere, so the axis ratio is exactly one at every point and every ellipse is a circle. Drawn at a constant page radius, they are twenty-five identical circles. The field, on the most-used projection in the world, contains no information whatsoever.

Meanwhile the areal scale factor over the same sample runs over a factor of 14.93.

Those two numbers are the same number, and that is the finding worth carrying out of this rung.

Mercator and the Lambert cylindrical are the conformal and the equal-area members of one family. Both carry a factor of sec²φ of distortion; Mercator puts all of it into area and none into shape, the Lambert cylindrical puts all of it into shape and none into area. They are equally wrong, in the same amount, about different things — which is the trade-off, forced — and the standard indicatrix convention displays one of them and hides the other.

So a reader comparing the two published fields sees a blank sheet and a field of violently flattened ellipses, and concludes that the first projection is better. The measurement says they are the same.

Why nobody draws the other one #

The constant-ground-radius field is honest and is nearly unusable, and the reason is in the same number.

An ellipse drawn at a constant ground radius on Mercator at 75° north has 14.9 times the area of one at the equator, which is 3.9 times the diameter. A field legible at the equator is a collision of ink at the poles; a field legible at the poles is a scatter of dots at the equator. The dynamic range a single page would have to carry is the whole of what is being measured.

That is a real constraint rather than an oversight, and it has a real answer: state the gauge in the caption. One clause — drawn at a constant page radius, so the sizes carry no information — costs nothing and removes the whole misreading. This collection’s own figures carry the projection they are drawn in for exactly that reason, and until this rung they did not carry the gauge.

The equal-area field, for contrast #

That is the control the previous section needs. The gauge is not always destructive: on an equal-area projection it costs nothing at all, because the quantity it suppresses is constant. What decides whether the convention matters is which of the two failures the projection has chosen to take, and the two ways a map is wrong is the essay that names them.

The unpleasant corollary is that the convention is most destructive exactly where a reader most needs help. A conformal map’s areal error is the thing nobody can see by looking at the map, and the instrument built to show what cannot be seen shows nothing.

The indicatrix at its own scale #

A single indicatrix carries both numbers because it is drawn at whatever size the arithmetic gives it. It is only the field that has to choose a gauge, and it has to choose one because twenty-five ellipses at their own sizes do not fit on a page — which makes the gauge a consequence of the page rather than of the mathematics, and therefore exactly the kind of choice that goes unrecorded.

The placement #

The second unstated choice is where the ellipses go, and the usual answer is at graticule intersections.

A graticule is not a uniform sample of a sphere. A row of placements at 75° covers cos 75° = 0.26 of the ground a row at the equator covers, and it carries the same number of ellipses. So a field placed this way over-represents the high latitudes by up to a factor of four — and the high latitudes are exactly where every projection in the library is worst.

A reader averaging the field by eye is therefore integrating against the wrong measure.

And on Mercator’s areal factor, where the effect is largest because the quantity grows fastest with latitude: the placements give a mean of 5.45 where the ground gives 3.04, an overstatement of 79 per cent.

Every one of those biases is upward, and it has to be: the placement over-weights the poles and the poles are where the distortion is. An instrument whose error always has the same sign is not noise.

What the two errors do together #

The two choices are independent and they compound in a specific direction.

The gauge decides which quantity is visible, and distortion has a direction besides. The placement decides how much of it a reader thinks there is. So on an equal-area projection — where the gauge lets the shape through — the placement then inflates the reader’s estimate of it by up to 64 per cent, and the projection looks worse than it is. On a conformal projection the gauge hides the areal error entirely and the placement is inflating a number the reader cannot see.

The instrument therefore flatters conformal projections twice: once by showing nothing, and once by making the alternative look worse than it is. Mercator’s reputation is a subject this ladder has touched from the other side, and this is a mechanism nobody in that argument has raised.

What this collection does about it #

Three changes, and the first two are cheap.

Every field here states its gauge, the way every figure states the projection it is drawn in. The figures above name which convention they are drawn under, in the caption, beside the projection.

Every average here is area-weighted. Distortion over a region has integrated against cos φ from the start, so the numbers this site prints are the true mean column above rather than the eyeballed one — and the difference between the two columns is what a reader of an ordinary published field would have got instead.

And the placement stays on the graticule. That is a deliberate choice rather than an oversight: an equal-area placement would put the ellipses somewhere a reader cannot relate to the graticule, and the graticule is how anybody navigates a world map. What changes is that the caption says the sample is not uniform.

What a better field would be #

Three designs, and none of them is what anybody draws.

Two fields side by side, one per gauge, which is what the figures in this essay do. It costs a second panel and it is the only presentation in which both quantities are visible at once. The objection is space, and the objection is real for a printed atlas and is not real for anything else.

One field with the ellipses at a common ground radius, on a page whose latitude range is narrow enough that the dynamic range fits. A field over one country rather than over the world has an areal factor spanning perhaps 1.2, and at that range the honest gauge is perfectly legible. The reason the dishonest gauge became standard is the world map, and the standard then spread to maps that did not need it.

Or the ellipses drawn at a constant page radius with the areal factor as a shading underneath. Two channels, two quantities, one field — which is the presentation the modern distortion literature does use, and which is not usually described as a repair to the indicatrix because nobody had stated that the indicatrix needed one.

The third is the best of them and it makes the point of this rung plainly: the field of ellipses is not the instrument. It is one channel of a two-channel instrument, and it has been shipped alone for a hundred and forty years.

What the field is for, and what it is not #

An indicatrix field is usually presented as a summary of a projection’s distortion, and it is a poor one — which is worth saying plainly, because this collection uses the construction constantly and uses it for something else.

As a summary it is beaten by two numbers: the mean and the maximum of the angular deformation over the region, which is what distortion over a region computes and what every quantitative comparison here is built on. Those numbers have no gauge and no sampling scheme; they are integrals.

What a field is for is locating: seeing where on the map the distortion is, and in which direction it acts. That is a question a pair of numbers cannot answer and a picture can, and it is why the construction survives.

The distinction matters because the two uses want different conventions. A field read for location wants the ellipses legible and evenly spread, which is what the standard convention gives. A field read for magnitude wants them at a common ground radius, which is unreadable. The convention is right for the job the field is good at and is read as though it were doing the job the field is bad at.

Where the model stops #

The sample here runs to 75° of latitude, which is where the library’s own worst behaviour is cut off. Extending it to 85° makes every number in the tables larger and changes none of the comparisons, because both means move together — but the ratio between them grows, so the biases quoted are lower bounds on what a field drawn to the pole would carry.

The conic projections are left out of the gauge table and the reason is arithmetic rather than taste: a conic’s apex is inside the sampled band at these latitudes, so its areal factor spans four million and would be the whole of any range the table printed.

And “what a reader concludes” is modelled as an unweighted average, which is a guess about perception rather than a measurement of one. A reader’s eye is probably drawn to the largest ellipses rather than averaging them, which would make the overstatement worse rather than better; nothing here measures that.

The generalisation #

An instrument has a gauge and a sampling scheme, and a picture made with it inherits both. That is not a statement about cartography and it is the reason this rung exists on a ladder about measurement.

The pattern recurs wherever a field is drawn rather than tabulated. A wind-barb map has a barb length nobody states; a strain-rate map normalises to its own maximum, so a plate interior and a plate boundary produce the same picture; a heat map has a colour scale. In each case the reader is being handed the shape of a field and a scale that was chosen for legibility, and in each case the choice is invisible in the output.

The remedy is the same in each case and it is one line of caption. What makes it worth an essay is that the failure is silent: nothing about a blank Mercator indicatrix field looks wrong.

Who found it, and when #

Tissot published the construction in 1881, and the practice of drawing the ellipses at a common page size is as old as the practice of drawing them at all — it is what makes a printed field legible, and every atlas that carries one does it.

The criticism has been made before in pieces. Cartographic texts warn that indicatrices exaggerate, and area-weighted distortion measures have been standard in the quantitative literature since the 1970s. What this rung adds is the arithmetic on a library: the sizes of the two biases, the sign of the second, and the identity between Mercator’s areal range and the Lambert cylindrical’s shape range that makes the first one’s cost exact.

The lattice is a third bias, and it runs the same way #

Two biases are measured above — the gauge and the sample. There is a third in the same instrument, and it is in where the ellipses are put rather than how big they are drawn.

A field is laid out on a graticule. Ellipses at every fifteen or thirty degrees of latitude and longitude is what every printed indicatrix field does, because it is what a graticule offers and because the result looks orderly.

A graticule lattice is not an even sample of the ground. Equal steps in longitude are $R\cos\varphi$ apart on the ground, so the number of samples per unit area rises as $\sec\varphi$: a factor of two at 60°, nearly six at 80°. The polar regions get the same number of ellipses as the equatorial ones while holding a fraction of the surface.

And the poles are where the distortion is worst, on every cylindrical projection in the library and most of the others. So the sample over-weights precisely the region whose ellipses are largest and strangest, which is the same direction as the gauge bias and the same direction as the placement bias the essay already measures.

Three biases, all pointing the same way, is not a coincidence. Each of them comes from making the picture legible and orderly, and legibility on a distorted sheet means giving space to the places the sheet has stretched. The instrument is drawn in page coordinates, and every decision taken in page coordinates inherits the distortion it was built to display.

It is also the one that would change the published pictures least, since the ellipses would move rather than change size, and a reader who did not know the lattice had changed would see much the same field with fewer ellipses crowded into the corners.

The fix for this one is the cheapest of the three: sample on an equal-area lattice rather than a graticule. The ellipses no longer line up in rows, which is why nobody does it, and the field then reports what a randomly chosen piece of ground experiences rather than what a randomly chosen graticule intersection does.

Where the ladder goes next #

The instrument has been audited for its gauge and its sample. What has not been audited is its order: an indicatrix is a first derivative, and a field of them is a first-order description of a map that has a second derivative — which is the whole of the neighbouring anchor, and which the field is silent about by construction.