Distortion has a direction

An indicatrix is drawn as an ellipse, quoted as two numbers, and reported as though the ellipse had no orientation. It has one, it is computable, and on most projections it points somewhere unexpected.

Two numbers, and the third nobody prints #

Naming a projection’s property is not measuring it, and the site has been careful from the start about the difference between the pair $(h, k)$ and the pair $(a, b)$. The first is the scale along the meridian and the scale along the parallel, and both depend on the graticule rather than on the map. The second is the pair of principal scale factors, the semi-axes of the indicatrix, and they survive a change of coordinates.

That distinction is standard once stated, and it leaves something out. The principal scales are the two extreme values of the scale factor over all directions at a point, so each of them belongs to a direction — the direction the ground is stretched most, and the direction perpendicular to it, which is stretched least. Those directions are as much a property of the map as the numbers are, and they are almost never given.

The reason is probably that the classical presentation reaches $a$ and $b$ through $h$, $k$ and the angle $\theta$ between the projected meridian and parallel, by way of Tissot’s formulae:

$$a + b = \sqrt{h^2 + k^2 + 2hk\sin\theta}, \qquad a - b = \sqrt{h^2 + k^2 - 2hk\sin\theta}$$

Those give the lengths and stop. Getting the direction as well means treating the whole thing as what it is.

The calculation is a singular value decomposition #

At a point, a projection sends a small displacement on the ground to a small displacement on the page, linearly. Write that linear map as a matrix $A$ whose columns are the page displacement produced by one metre east and one metre north:

$$A = \begin{pmatrix} \partial x/\partial E & \partial x/\partial N \\ \partial y/\partial E & \partial y/\partial N \end{pmatrix}$$

The east and north derivatives come from the projection’s four partial derivatives divided by the metric factors — $N\cos\varphi$ for a step in longitude, $M$ for a step in latitude — so nothing new is needed to build it.

The singular value decomposition of $A$ answers the whole question at once. Its two singular values are $a$ and $b$, and its right singular vectors are the two ground directions that come out perpendicular on the page. Forming $A^{\mathsf T}A$ and taking its eigenvectors is the shortest route, and it is what the library does.

Two independent routes to $a$ and $b$ are then available — Tissot’s formulae and the decomposition — and they must agree, which is worth having because a direction obtained from a decomposition that got the lengths wrong would be worth nothing.

What the arithmetic tried to teach #

The eigenvector of a symmetric $2\times2$ matrix has two textbook expressions, $[f, \lambda - e]$ and $[\lambda - g, f]$, which are algebraically identical and numerically are not. Each collapses when its second component vanishes, and that happens precisely when the major axis lies along one of the two basis directions.

That is not an edge case in cartography. It is the normal case: every cylindrical projection and every conic keeps its meridians and parallels perpendicular, so the major axis lies along the parallel almost everywhere on most maps in use. The first implementation used one expression, and an Albers conic promptly reported its major axis $5\times10^{-3}$ degrees off the parallel it lies exactly along. Taking whichever expression has the larger denominator fixes it, and the residual across the aligned cases is $7\times10^{-10}$ degrees.

Where the direction is the graticule, and where it is not #

The rule is simple to state and it explains the whole pattern.

If the graticule stays orthogonal on the map, the graticule is the pair of principal directions. Perpendicular directions that stay perpendicular are the extremes, so for cylindrical projections, conics, and azimuthal projections in normal aspect, the scale along the meridian really is a principal scale and the classical $h$ and $k$ are $a$ and $b$ in some order.

If it does not, they are not. The pseudocylindricals bend their meridians while keeping their parallels straight, so meridian and parallel meet at an angle $\theta \ne 90°$, and the principal directions are two other directions bisecting the awkwardness.

The sinusoidal projection at 60° east, 45° north is a compact example. There $h = 1.2443$ and $k = 1.0000$ — the parallel is at true scale, which is the projection’s whole design — while $a = 1.4366$ and $b = 0.6961$. The largest stretch is 44% and the meridian only registers 24% of it, because the meridian is not the stretched direction: the major axis lies 34.8° west of north.

Along a parallel the swing is systematic. On the sinusoidal at 45° north the departure from north runs 0° on the central meridian, 39.8° at 30° out, 34.8° at 60°, 30.5° at 90°, and 21.4° near the antimeridian: it rises steeply from zero, peaks in the near half of the sheet, and then declines as the growing elongation of the indicatrix begins to dominate its tilt.

The forty-five degrees is a bound, not a threshold #

The assertion requires a skewed projection to put its major axis up to 45.0° away from the graticule, and the sinusoidal reaches 44.6. That is not a coincidence of the library; it is the largest the swing can ever be, and the condition for reaching it is one line.

Write the ground’s east and north directions as the basis. The map sends east along the parallel with scale k and north along the meridian with scale h, meeting on the page at an angle θ. Then

$$A^{\mathsf T}A = \begin{pmatrix} k^2 & hk\cos\theta \\ hk\cos\theta & h^2\end{pmatrix}, \qquad \tan 2\psi = \frac{2hk\cos\theta}{k^2 - h^2},$$

with ψ the major axis’s angle from east on the ground. As the numerator grows relative to the denominator, 2ψ approaches a right angle and ψ approaches 45° — and it reaches it exactly when h = k, where the denominator vanishes.

So the swing is bounded by 45° from whichever graticule direction is nearer, always, on any projection, and it is attained precisely where the scales along the meridian and along the parallel are equal. That is a real and reachable condition rather than a degenerate one: it says the whole of the map’s distortion at that point is shear, with no difference between the two graticule scales for the direction to prefer.

The sinusoidal’s worked point makes the arithmetic visible. There h = 1.2443 and k = 1.0000, and the graticule’s own angle follows from hk sin θ = ab = 1, giving sin θ = 0.8037 and θ = 53.5° — the meridian and parallel meet at fifty-three degrees rather than ninety. Substituting into Tissot’s two formulae returns a + b = 2.1327 and a − b = 0.7405, which is the 1.4366 and 0.6961 the essay quotes, by a second route.

The two graticule scales are within a quarter of each other there and the swing is 34.8°; where they cross, the swing is 45° and the two principal directions bisect the graticule’s own angle exactly. A projection whose h and k cross somewhere therefore has a point on that curve at which the principal directions are as far from the graticule as any map can put them, and finding it is a matter of solving h = k rather than of searching.

The one case with no answer #

A conformal projection has no principal direction anywhere.

That is not a gap in the measurement. Conformal means $a = b$, which means the indicatrix is a circle, which means every direction is stretched by the same amount and there is no direction of maximum stretch to report. Asking a circle which way it points is not a question.

The measured anisotropy across three conformal projections at twenty-one points each is below $2\times10^{-8}$, which is the noise floor of the derivatives rather than a real ellipse. Any figure that draws a direction field over a conformal projection is drawing its own rounding error, and the machinery here reports the point as circular rather than returning the direction the arithmetic happened to favour.

The same caution applies at isolated points of a non-conformal map. An equal-area conic is exactly true in both directions at its standard parallels, so the indicatrix there is a circle and the direction is again undefined — and a sweep that ignores this reports the Albers conic as 41° off the graticule at the one latitude where it cannot be off anything. Points whose indicatrix is round to within 1% are excluded from both the figure and the assertion for that reason.

What the direction is for #

Three uses, in increasing order of how often anyone thinks of them.

Reading a map. A country drawn near the corner of a pseudocylindrical world map is not simply enlarged or simply squashed; it is sheared along a diagonal, and the diagonal is what the direction field gives. That is why land masses at the outer edges of a Mollweide look tilted rather than merely stretched — a distortion of orientation that no pair of numbers conveys.

Placing a region. Choosing the aspect of a projection to suit a region is easier when the pattern being rotated is visible as a field. A long thin country wants its long axis along the direction of least stretch, and the direction of least stretch is perpendicular to the strokes drawn here. That is the same reasoning the worst point applies to magnitudes, applied to orientation instead.

Detecting a mistake. A direction field is a strong check on a forward map. An error in a sign or in a metric factor changes the numbers by an amount that can be argued about, and changes the field’s pattern into something obviously wrong — a fan that fails to close on the central meridian, or a set of strokes that ignore a symmetry the projection has. The two figures in the fleet that carried no tick labels at all went unnoticed for a long time because the symptom was absence; a wrong pattern is much louder than a wrong number.

The projection that has it both ways #

The American polyconic is the sharpest case in the library, because it was built to make a promise about directions and keeps it only on one line.

Its construction gives every parallel its own tangent cone, so every parallel is true to scale and so is the central meridian. That is a statement about the scale along two families of curves, which is a statement about $h$ and $k$ — and it says nothing whatever about $a$ and $b$, because away from the central meridian the polyconic’s graticule is not orthogonal.

So the polyconic’s true parallels are true along the parallel, which is not the direction the ground is stretched most, which is why a ruler laid along a polyconic parallel reads correctly while a shape drawn near it does not. The promise and the distortion are about different directions, and only a field makes that visible.

The third failure, given an axis #

Scale distortion is a third way a map can be wrong, independent of the angular and areal failures: a map can preserve every angle and every area ratio in a region and still draw everything at the wrong size. The direction adds the corresponding refinement to the first failure.

Angular deformation $\omega$ says how far the worst-treated angle at a point is bent. It does not say which angle that is, and the answer is fixed by the principal directions: the two directions bent most are the two bisectors of the principal axes, at 45° to them, and they are bent towards the major axis. So a field of principal directions is also a map of which pairs of bearings a reader should distrust — a question the two ways a map is wrong poses and answers only in aggregate.

For a navigator that is the operative form. The number $\omega = 12°$ is a warning; the statement that bearings near north-east are the ones running 12° out, and bearings along the principal axes are exactly right, is instructions.

Why it is not in the literature #

A guess, and it is worth stating as one.

The indicatrix is almost always drawn as a decoration rather than computed, and a decorative ellipse cannot carry an orientation because nobody worked one out. Where indicatrices are computed, the output is usually tabulated as $a$, $b$ and the maximum angular deformation $\omega$, which are the three numbers the classical criteria need — and the criteria integrate over a region, at which point a direction averages to nothing useful unless the averaging is done as a tensor.

That last point is real rather than a complaint. A mean direction over a region is not the direction of the mean distortion, and adding up angles that live on a circle is a way of getting an answer that means nothing. The field is a local object and is honest as one.

The invariance, restated #

The invariants essay makes its case by swapping the output axes: $h$ and $k$ change, $a$, $b$, the areal factor and $\omega$ do not. The direction belongs to the second group with one qualification: it rotates with the page.

Swap the axes and the major axis on the page swaps with them; but the ground direction it corresponds to — the bearing, measured from north on the sphere — is unchanged. That is why the library returns both, a bearing on the ground and an angle on the page, and why the two are worth keeping distinct. The bearing is a property of the projection. The page angle is a property of the projection and of how the sheet was laid out, which makes it the right thing for a figure to draw and the wrong thing for a table to quote.

The same information as a field of ellipses, for comparison, at twelve points rather than a hundred and eighty.

What was computed here #

The linear map from the local east–north frame to the page was assembled at every sample point from the projection’s four partial derivatives, taken by Richardson-extrapolated central differences and divided by the ellipsoid’s or sphere’s metric factors. The eigenvectors of $A^{\mathsf T}A$ give the principal directions and its eigenvalues their squared scale factors, which are checked against the $a$ and $b$ that Tissot’s formulae produce by a different route.

Three claims are asserted on every build. A projection with an orthogonal graticule puts its major axis along a graticule line, worst case $7\times10^{-10}$ degrees over five projections at four points each. A projection with a skewed graticule puts it up to 45.0° away, over three projections at 525 points each, excluding points whose indicatrix is round to within 2%. And a conformal projection has no direction at all, with a measured anisotropy below $2\times10^{-8}$ at sixty-three points.

What the pictures cannot show #

A stroke has no sense, only an orientation, so the field cannot distinguish a direction from its reverse — correctly, since the major axis of an ellipse has no preferred end.

The strokes also carry the elongation only crudely, in their length, and saturate at a ratio of three. A field that encoded both the direction and the magnitude faithfully would be a field of ellipses, which is the figure this one exists to replace.

Who found it, and when #

Tissot published the indicatrix in 1881, and the principal directions are in his construction from the beginning — the ellipse is defined as the image of a circle, and an image of a circle has axes. What has drifted out of the presentation over the following century is the reporting: modern software emits $a$, $b$ and $\omega$, and the direction survives only in the drawing, where nobody measures it.

The singular value decomposition arrived independently in the 1870s, from Beltrami and Jordan, and is now the natural way to say what Tissot’s construction is. The two subjects took a century to be described in the same words.

Where this goes next #

Distortion has a size, a place and now a direction. The remaining question is how far the whole apparatus can be avoided: over how small a patch of the Earth is every one of these quantities small enough to ignore, and what fixes the answer.