The indicatrix is a limit

Every serious treatment of map distortion says the same sentence: Tissot’s indicatrix is the image of an infinitesimal circle. Then it draws one about a centimetre across.

The word carries the whole of the construction. The indicatrix is the projection’s derivative — the linear map that takes a metre of ground into a displacement on the page — and a derivative describes a function exactly at one point and approximately near it. How near is a measurement nobody appears to make.

The quantity, and why it is a ratio #

The obvious measurement is the distance between the true image of a point and where the linear map put it. That number is in page units, which are arbitrary — a projection returns radians, a plotting library multiplies by whatever fits the column — so it says nothing on its own.

What means something is the departure relative to the size of the ellipse being drawn. The indicatrix at a point has semi-axes $a$ and $b$; a circle of geodesic radius $\rho$ is drawn as an ellipse of mean semi-axis $\tfrac{1}{2}(a+b)\rho$; and dividing the worst departure by that gives a pure number, which is the fraction of the picture the ellipse gets wrong.

Nothing is fitted anywhere in this. The predicted ellipse is not an ellipse chosen to match the image — it is the Jacobian the site differentiates for every other quantity it computes, applied to a ground displacement of $\rho$ in each direction in turn.

The law, and it is not the one that was expected #

The absolute departure is second order in the radius. That is what “the indicatrix is the derivative” means and it is what any account of the construction implies.

The relative departure is first order, because a quadratic divided by the ellipse’s own linear size leaves something linear. Both exponents are fitted here rather than asserted, and each is required to reject the other’s law:

Four times the radius is sixteen times the absolute departure and four times the relative one. The fitted exponents are 2.016 and 1.016 across five projections, and a linear law fits the first no better than a quadratic fits the second.

The practical consequence is worth stating plainly, because the intuition runs the other way. Halving the region halves the error the indicatrix makes about it. A second-order effect would be quartered, and a reader who expects second order will believe an indicatrix twice as far as it deserves.

What the ellipse is, before it is asked how far it reaches #

The two numbers the indicatrix carries are the semi-axes, and they are computed at a point by a decomposition with no circle in it at all: build the matrix that takes a metre east and a metre north into a page displacement, and its singular values are $a$ and $b$. That is the object Tissot’s indicatrix draws and what survives a change of coordinates shows is a property of the map rather than of the graticule.

The distinction matters because the two quantities behave differently across a map. The semi-axes at 40° on Mercator are both 1.305, and they would be the same numbers whatever radius the drawing used. The honesty of the ellipse at that point is a third number, and it is not derivable from the first two.

The rate has a closed form, on the projection where the algebra is short #

The coefficient of that first-order law is not a free constant. Expanding Mercator’s northing about a point gives a second derivative of $\sec\varphi\tan\varphi$ against a first derivative of $\sec\varphi$, and the ratio of the two is the whole of it: the relative departure of a circle of radius $\rho$ is

$$\frac{1}{2}\,\rho\,\tan\varphi$$

with nothing left over. Measured against that prediction at 15°, 30°, 45° and 60°, the worst discrepancy is 0.14 per cent, which is the residual third-order term.

That formula says what the coefficient is in general, even where the algebra is long: it is the logarithmic rate at which the projection’s own scale factor changes with position. A projection whose scale is nearly constant over a region has an indicatrix that stays honest across it; a projection whose scale is changing quickly has one that does not, regardless of how large the distortion itself is.

The two are independent, and separating them is the point of the measurement. Mercator at 40° has an areal factor of 1.70, which is enormous, and an indicatrix good to a per cent out to 148 kilometres. The gnomonic projection 20° from its centre has a smaller areal factor and an indicatrix good to 46.

How large a circle each indicatrix actually describes #

Inverting the question gives the number a reader can use: at a stated tolerance, how big may the circle be? Bisection rather than the first-order law, because the law is what the other figure is testing.

At one part in a thousand — a tolerance a survey might recognise — every one of these numbers divides by ten, because the law is first order. The gnomonic’s ellipse is then a fair description over five kilometres, and Albers’ over eighty-five.

The dependence on latitude follows the same closed form, and on a cylindrical projection it is severe: $\tan\varphi$ is zero at the equator and unbounded at the pole.

The image of a circle is in the wrong place as well as the wrong shape #

The departure measured above is of shape. There is a second measurement in the same picture and it is the one that connects this essay to the operations field.

The image of a circle is not centred on the image of the circle’s centre. The second-order term is not symmetric about the point, so the centre of area of the drawn ring sits away from the projected centre — by 0.084 per cent of the ellipse at 13 kilometres and 4.2 per cent at 637, which is the same first-order law with a slightly smaller coefficient.

That is exactly the defect a centroid belongs to a plane measures at country scale, arriving here at the scale of an indicatrix and from the opposite direction. A centroid computed in a projected plane is not the centroid of the ground it came from, and the reason is visible in a picture six pixels across.

Where the limit does not exist at all #

The whole account above assumes the projection has a second derivative. One member of the site’s library does not.

The Robinson projection is defined by a table of numbers at five-degree intervals — it is not an approximation to a formula, because there is no formula — and interpolating it linearly makes the first derivative a step function. At a table entry the derivative jumps, and the second derivative is a spike.

The consequence is not that the departure is large. It is that the departure does not shrink:

A hundredfold reduction in the circle leaves the error where it was. Tissot’s construction describes the limit of a shrinking circle, and at a table entry on Robinson there is no limit to describe: the ratio the definition takes a limit of simply does not converge.

Two degrees away, between entries, the projection is a polynomial and behaves ordinarily — 0.005 per cent at 640 metres, 0.045 at 6.4 kilometres, 0.453 at 64. Perfectly first order. So this is a statement about the table and its interpolation, not about the shape Robinson drew.

The site’s own implementation comments that a cubic interpolation would be closer to Robinson’s intent and that the difference is “well below the width of a drawn line”. For the drawn map that is true. For this quantity it is not true at all, and the measurement is the only thing that would have said so.

What this does to a published figure #

Almost every indicatrix in print is drawn at a size chosen so the ellipses are visible — a few degrees of arc, sometimes ten or fifteen, scattered across a world map at twenty-degree intervals.

At 4° of radius on Mercator at mid-latitude the ellipse is right to about a fifth of a per cent, which is far better than the drawing’s line width and entirely fine. At 16° it is wrong by around four per cent, which is a visible discrepancy in a figure whose whole content is a shape. And at the sizes some textbook figures use — circles large enough to touch each other — the ellipse is describing something the projection does not draw.

This site’s own indicatrices are drawn at a fixed size in page units rather than in ground units, which is a different decision and has to be said out loud: Tissot’s indicatrix draws each ellipse from the measured semi-axes at a point, at a size chosen for legibility, and does not claim that a circle of that ground radius maps to it. The numbers are the content and the ellipse is their picture.

The same failure, with a triangle instead of a circle #

A circle is not the only finite object an infinitesimal statement gets applied to. The other one is a triangle, and it fails in a way that is easier to see and harder to dismiss.

That is conformal does not mean the angles are right and it is the same mistake as reading a finite circle off an indicatrix: an infinitesimal property, exercised at a size where the infinitesimal has stopped applying. The two essays measure the two shapes, and the pattern they share is worth naming — a pointwise property is a promise about a neighbourhood whose size is never quoted.

The relation to flatness, which is the same question asked once further out #

How small is flat enough asks what error a plane makes about a patch of sphere, and finds it quadratic in the patch’s size. This asks what error the derivative of a projection makes about a patch, and finds it linear once the ellipse’s own size is divided out.

They are the same expansion truncated at different places. The flat-Earth error is the second-order term of the sphere, with no first-order term available to divide by; the indicatrix error is the second-order term of the projection, relative to the first-order term the indicatrix already carries. A quantity measured against a leading term that is itself proportional to the size loses one power.

The pair is worth holding together because the two numbers are so different in practice. A plane is good to one part in a million over ten kilometres; an indicatrix is good to one part in a thousand over fifteen. The projection’s own variation is a far cruder thing than the curvature of the Earth.

Where the rate is largest, and why it is not where the distortion is #

The rate is a derivative of the scale factor, so it is largest where the scale is changing fastest — which on a cylindrical projection means near the poles, and on an azimuthal one means at the rim.

A projection can therefore be very distorted and very well described, or nearly undistorted and badly described. The Albers conic at 40° is close to true scale and has the honest indicatrix of the eight tested; the gnomonic 20° from its centre is moderately distorted and has the worst. Nothing about the size of the distortion predicts either.

The measurement’s own controls #

Three, and each catches a different way this could have been a fact about the machinery rather than about the projections.

The circles are geodesic circles on the sphere, generated from the destination formula at a fixed distance and a sweep of azimuths, not from a small square in longitude and latitude. A ring at constant $\Delta\varphi$ and $\Delta\lambda$ is not a circle on the ground at all, and using one would measure the graticule’s own anisotropy and report it as the indicatrix’s error.

The predicted ellipse is the site’s own Jacobian, Richardson-extrapolated from central differences, the same object what survives a change of coordinates shows the invariants are built from. If it were a separately fitted ellipse the measurement would be of the fit.

Both exponents are required, and each rejects the other. A check that only demanded the relative departure be small would pass on a projection that was linear everywhere; one that only demanded second-order absolute growth would pass while the useful statement — what happens to the drawn ellipse — went unmade.

What the varying radius asks of a sampling scheme #

The radius over which an indicatrix honestly describes its neighbourhood is a function of position, and that turns the usual way a regional criterion is computed into something that needs justifying rather than assuming.

A regional score is a sum over a lattice. Sample the distortion at a few hundred or a few thousand points, weight them, and total. The lattice spacing is chosen for cost, and nothing in the criterion’s statement says what it should be.

The honesty radius says what it should be. Each sample describes a patch of the size over which its own linearisation holds, and the samples should tile the region rather than overlap or leave gaps — so the spacing wants to be comparable to the local radius. Where the radius is large the sampling can be coarse and the extra samples buy nothing; where it is small the sampling must be fine or the sum is of descriptions that do not reach each other.

And the radius varies by orders of magnitude across a world map, so a uniform lattice is necessarily wrong somewhere. It is wasteful in the well-behaved middle of a projection and insufficient near the places the projection is doing something — which are the places that decide the score.

The adaptive version is not difficult. The radius comes from the same second derivative the criterion’s own machinery computes; scaling the local spacing by it is one more pass, and the result is a sum whose terms describe disjoint patches of ground rather than an arbitrary set of points.

The check, for anybody unwilling to change their sampling, is the ordinary one: refine the lattice and see whether the regional score moves. A score that is stable under refinement is summing patches that already tile the region. A score that drifts is being computed at a spacing the projection does not permit, and the drift says in which direction.

It also gives a reader a way to judge a published regional comparison they cannot rerun: ask what the sampling was, and whether it was uniform. A uniform lattice on a world map is a declaration that the criterion was evaluated at a spacing chosen for convenience.

None of this is an objection to the criteria themselves. It is a statement about the resolution at which they are meaningful, and the point of computing the radius is that the resolution stops being a matter of taste.

Where the ladder goes next #

The indicatrix has now been asked what it is (an ellipse with two axes), what it says (two independent failures), which parts of it survive a change of coordinates, where its worst point is, which way it points, and how large a circle it describes.

What remains on this ladder is the step from a point to a region, which distortion over a region begins and does not finish: integrating a pointwise quantity over a country requires a weighting, and the weighting is where measurement stops and judgement starts. This essay adds a constraint to that integration nobody states — the integrand is only a description of the ground over a radius that is itself a function of position, so a regional criterion sampled on a coarse grid is summing descriptions of overlapping patches of varying honesty.

That is not a defect in the criteria. It is a statement about the sampling density they need, and it now has a number attached to it.