The fourth number the ellipse does not carry

Nine rungs of this ladder have taken the indicatrix apart as an instrument — what it measures, what it is a limit of, where it fails to exist, how it is drawn, how it is sampled. Not one of them has asked the simplest structural question about it.

A Jacobian has four independent entries. An indicatrix reports three.

Two principal scale factors and one orientation. That is a complete description of a symmetric matrix, and the map from the ground frame to the page is not symmetric.

The split #

Every 2 × 2 matrix with positive determinant factors uniquely as

$$A = R(\gamma)\,S,$$

with $R$ a rotation and $S$ symmetric and positive-definite. That is the polar decomposition, and applying it to the map from the local ground frame — one metre east, one metre north — to the page separates the two things a projection does at a point: it turns the frame, and then it stretches it.

$S$ is the stretch. Its eigenvalues are the two principal scale factors and its eigenvectors are the principal directions, which is to say: the indicatrix is $S$. Not a summary of it, not a projection of it — it is the same object drawn as an ellipse.

That identification is checked rather than asserted. For six projections at six points, $R\cdot S$ reconstructs the Jacobian to a residual of $10^{-16}$, and $S$’s eigenvalues agree to seven figures with the $a$ and $b$ that the classical route through h, k and θ produces from entirely different arithmetic.

$R$ is the rest. One number, and nothing in the classical description carries it.

It is worth being clear that $R$ is not a residual or an error term. The decomposition is exact and unique, the two factors are independent, and a projection with a large rotation is not thereby a worse projection — a map that turned every frame through ninety degrees and stretched nothing would be a perfectly faithful map printed sideways. What $R$ is, is the part of the derivative that a description built to be independent of the page cannot see.

Which rotation #

There is more than one angle that could reasonably be called “the rotation”, and they are not the same. That is worth separating carefully, because the fact that they coincide in the case everybody thinks about is why nobody has had to.

The polar rotation $\gamma$ is the honest fourth number: the rigid turn in $A = R S$.

The convergence is where the image of north points on the page, which is the angle a surveyor applies to turn a grid bearing into an azimuth.

And the image of east points somewhere too, and on a sheared map that somewhere is not perpendicular to the image of north.

For a conformal projection $A$ is a scalar times a rotation, so the stretch is isotropic, it turns nothing, and all three angles are the same. Off conformality the stretch turns the two axes by different amounts and they separate.

The zeros in that second figure are the control and they come in two kinds. Mercator’s is conformality. The plate carrée’s and the Miller’s are something else: their principal directions lie along the meridian and the parallel everywhere, so the frame is stretched but never sheared, and a stretch along the frame’s own axes turns nothing. Neither reason has anything to do with the map being good, which is the usual shape of a zero in this collection.

The fourth number has a name already #

The transverse Mercator is conformal, so its three angles coincide, and it is the projection every national grid is built on. Its rotation is therefore exactly the quantity grid north is not north is about — the meridian convergence, $\gamma \approx \Delta\lambda\sin\varphi$ to first order, printed in the margin of every topographic sheet and applied by every surveyor setting out a bearing.

That check crosses two libraries that share no arithmetic. projection.js differentiates a spherical transverse Mercator numerically and decomposes the result; ellipsoid.js computes the convergence from the survey formula with its own transverse Mercator series. They agree to 5 × 10⁻¹⁰ degrees, which is two nanoseconds of arc and is the numerical differentiation’s own floor.

So the number the indicatrix discards is the one a surveyor cannot work without, and the collection had computed both for a hundred and ninety essays without ever comparing them.

What it does to the instrument #

The consequence for the indicatrix as an instrument is the one the previous rung was building toward, and it is a fourth unstated choice on top of that rung’s two.

A drawn indicatrix field carries the sizes and the orientations of the ellipses. It carries no arrows, so it does not say which way north went, and there is nothing in the picture from which a reader could recover it. Two maps whose ellipse fields are identical to twelve decimal places can differ by 37° in what they do to a bearing — the hero figure is that pair — and a reader comparing the two fields would call them the same map.

That is not an argument that the indicatrix should be replaced. It is a much narrower claim: the indicatrix is a complete description of what a projection does to shapes and an incomplete description of what it does to directions, and the two are routinely conflated because a shape has an orientation and it is easy to think that is the same thing. It is not: whether the ellipses point the same way measures the orientation of the principal axes, which is inside $S$, and none of that rung’s numbers changes when the sheet is turned.

Counting the numbers #

It is worth doing the accounting explicitly, because the arithmetic is the argument.

A projection’s derivative at a point is a linear map from a two-dimensional tangent space to the page. In the ground’s own orthonormal frame — east and north, both in metres — that is a 2 × 2 matrix: four numbers.

The classical description supplies:

Three. Every other quantity this collection computes is a function of those: the areal factor is $ab$, the angular deformation is $2\arcsin\frac{a-b}{a+b}$, the Airy and Kavrayskiy numbers are averages of functions of $a$ and $b$, and the second derivative is a different tensor entirely.

So the missing number is one, it is a rotation, and it is not recoverable from any of the others — because $(a, b, \theta)$ determines $S$ exactly and $S$ says nothing about $R$.

The one place it is nearly recoverable is worth stating so the claim is not overstated. A projection is a gradient map, so its Jacobian field satisfies an integrability condition, and that condition ties $\gamma$’s derivatives to $S$’s. Given the whole indicatrix field over a connected region, $\gamma$ is therefore determined up to a single additive constant — which is exactly the constant the turned sheet changes. The field determines the rotation up to the orientation of the page, and no more, which is the precise sense in which the hero figure is the whole of the ambiguity.

What was computed, and how #

The polar decomposition is closed form for a 2 × 2 matrix and no iteration is used: $\sqrt{A^{\mathsf{T}}A}$ has the explicit form $(C + \sqrt{\det C}\,I)/(\sqrt{\lambda_1}+\sqrt{\lambda_2})$, and $R = A S^{-1}$ follows.

Three assertions carry the rung and they reject in three different directions.

The split must be a split. $R\cdot S$ must return $A$ to $10^{-9}$ relative, and $S$’s eigenvalues must be the classical $a$ and $b$. A sign error anywhere in the decomposition breaks the first; a confusion between $A$ and $A^{\mathsf{T}}$ breaks the second while leaving the first intact, which is why both are needed.

The rotations must agree exactly when the map is conformal and separate when it is not. Three conformal projections must return a gap below $10^{-6}$ degrees and some non-conformal one must exceed half a degree. A version that had computed the same angle twice would pass the first half and fail the second.

And turning the page must move $R$ and not $S$. Composing a fixed 37° rotation onto a projection’s output changes the rotation by exactly 37° at every point and leaves the indicatrix identical to $2.5\times10^{-12}$. That is the exhibit, stated as a check, and a decomposition that had leaked the rotation into the stretch would fail it.

What it changes about reading a distortion figure #

Three practical consequences, and the third is the one that matters for how this collection draws things.

A field of ellipses cannot be used to plot a bearing. It says how a small shape is deformed and it does not say which way anything ends up pointing. That sounds obvious stated baldly and is not obvious when looking at a picture, because an ellipse has an axis and an axis looks like a direction.

Two projections whose indicatrix fields differ only slightly can differ enormously in convergence. Two projections that cannot be told apart is about a different kind of indistinguishability — two maps that put the control points in the same places — and this is a third kind: identical deformation and different orientation. A comparison made on distortion alone will not separate them, and a survey will.

And a projection can be improved on one measure by turning it. Nothing about a page rotation changes any distortion figure in this collection, so an aspect search that scores by any of them cannot see it — which is exactly the finding measure.js records when it says that the third aspect parameter is stationary at some optima and not others. The reason is here: the third parameter turns the page, page rotations live entirely in $R$, and every criterion in the search is a function of $S$. Where the third parameter does matter, it is doing something other than turning the page.

Where the model stops #

A page rotation is a trivial exhibit and it is deliberately so. Nobody is confused about whether a turned sheet is a different map; the exhibit’s job is to isolate the fourth number, not to be surprising. The non-trivial content is that $\gamma$ and the convergence separate by tens of degrees on ordinary projections, and that the classical description carries neither.

The rotation is not an invariant. It depends on the page’s own orientation, exactly as the exhibit shows, so it is not the kind of quantity what survives a change of coordinates admits. Its variation over a map is invariant, which is why the convergence between two points is a real quantity while the convergence at one point is a convention about where the grid’s north is.

And a rotation is not a distortion. A map that turned every frame by ninety degrees and stretched nothing would be a perfectly faithful map printed sideways. That is precisely why the indicatrix discards it and why doing so was the right decision for the question Tissot was asking. It is the wrong decision for the question a surveyor asks, and this collection has been printing indicatrix fields on pages that also discuss convergence without noticing that the first has nothing to say about the second.

The generalisation #

The rule is about the difference between describing a map and describing what it does.

An invariant description throws things away on purpose, and the things it throws away are somebody’s subject. The indicatrix is invariant under a change of the page’s coordinates, and it achieves that by discarding the one part of the Jacobian that a change of page coordinates moves. That is the correct construction for comparing projections and it is the wrong construction for using one.

This collection has met the same structure before from the other side. Only the invariants describe the map is the rule that made h and k second-class quantities here, because they depend on the parameterisation of the sphere; the rotation is second-class for the mirror reason, because it depends on the parameterisation of the page. Both are still measurements. A quantity being a convention does not make it unmeasurable, and the whole practice of survey is the arithmetic of conventions.

The practical form is one sentence: a description that is invariant under a group has nothing to say about anything the group moves, and if somebody has to work in a fixed frame then the group is exactly what they cannot ignore.

Who found it, and when #

The polar decomposition is standard linear algebra and its use in continuum mechanics — where $R$ is the rigid rotation of a deformed body and $S$ is the stretch tensor — is a century old. The identification of Tissot’s indicatrix with the right stretch tensor is made explicitly in the differential-geometry treatments of cartography and is not made at all in the applied ones.

Tissot published the indicatrix in 1881 and constructed it as the image of an infinitesimal circle, which is a construction that cannot see a rotation: a circle is invariant under one. The construction and the omission are the same fact.

Meridian convergence is older than the indicatrix and belongs to a different literature. It appears in survey manuals, in grid specifications and in the margin diagram of every topographic sheet, and it appears in no treatment of map distortion whatsoever — because distortion is measured by the indicatrix, and the indicatrix does not contain it. The two quantities have been in separate chapters since before either was named, and this rung is the arithmetic that puts them in the same one.

Which uses need the fourth number #

The two literatures have been separate since before either quantity was named, and the split is not arbitrary. It follows exactly the split between what a map is looked at for and what it is used for.

Every use that compares shapes needs the three the indicatrix carries. Is this country drawn too large, is this shape stretched, is this angle between two coastlines right — all of them are questions about the stretch, and a rotation of the whole neighbourhood changes none of them. That is the whole of the distortion literature’s subject, and the indicatrix is complete for it.

Every use that carries a direction needs the fourth. A bearing taken off the sheet, a heading plotted with a protractor, a grid azimuth reduced to a geodetic one, an alignment set out on the ground: each of them relates a direction on the page to a direction on the Earth, and that relation is precisely the rotation the indicatrix discards.

Which is why the quantity lives in the survey manuals. Surveying and navigation are the two trades that use a map to point at something, and both of them discovered the rotation independently, gave it a name, printed it in the margin, and built corrections around it. The distortion literature never needed it because it never pointed at anything.

So the omission is a feature of the instrument rather than a mistake in it, and the correct statement is narrower than the indicatrix is incomplete. It is complete for the class of questions it was built for. It is silent on a second class, that class has its own century-old apparatus, and the two have never been written down as parts of one four-number description — which is what this rung does and is the whole of what it does.

Where the ladder goes next #

Ten rungs have audited the indicatrix as an object, as a picture, as a sampled field and now as a decomposition. What remains untouched is the assumption underneath all of them: that the thing being differentiated is a map of a fixed body. The ground has an indicatrix too, and the ground moves — so at any place with a strain rate there are two tensors and one page, and whether their principal directions have anything to do with each other is a question neither anchor has asked.