Two indicatrices do not make a third

Ten rungs of this ladder have treated the indicatrix as the report on one map. It is also, unavoidably, the report on a chain of them: the moment a file is read off one projection and written onto another, the object being drawn is a composition, and it has its own indicatrix.

The natural expectation is that the two known reports determine the third. They do not. Composing two maps whose shape errors are 31.8° and 35.4° gives a composite with 20.8°; composing two whose errors are 38.9° and 32.5° gives one with 55.5°. Neither is an average, neither is a sum, and no rule involving only those four numbers can produce both.

What a reprojection is, locally #

A projection sends a ground point to a page point. Its local behaviour is a two-by-two matrix — the page displacement per metre east and per metre north — and everything Tissot’s construction reports is the singular value decomposition of that matrix: the two principal scale factors are its singular values, the areal factor is its determinant, and the fourth number is the rotation in its polar decomposition.

Reprojecting from map $A$ to map $B$ is: undo $A$, then do $B$. Locally that is the matrix product $A_B A_A^{-1}$, and the composite’s indicatrix is the singular value decomposition of that.

The determinant part composes exactly as expected, because determinants multiply: the composite’s areal factor is $s_B / s_A$, and two equal-area maps therefore give a composite of exactly one. That much is safe and it is the only part that is.

The singular values do not multiply. They obey inequalities, and the inequalities are wide.

The four numbers, and which of them compose #

It is worth being precise about which parts of the report survive composition, because two of the four do and two do not, and the two that do are the two the subject spends least time worrying about.

The first row is why the two ways a map is wrong splits the way it does: area is a determinant, determinants are multiplicative, and a quantity that composes by multiplication is a quantity a chain of operations can be reasoned about one step at a time. The last row is why shape is different. Nothing in the pair $(a, b)$ of either parent constrains the composite’s pair beyond an inequality, and the inequality is not tight on either side.

That asymmetry is the whole rung compressed into a table, and it has a consequence for how a chain should be audited. An areal error can be tracked additively in logarithms all the way down a pipeline. A shape error cannot be tracked at all without keeping the orientation of every step.

The bound, and how wide it is #

Write $c = a/b$ for a map’s condition number — the ratio of its principal scale factors, which is the ellipse’s elongation and the thing $\omega$ is a monotone function of. For a product of matrices,

$$\left|\log c_A - \log c_B\right| \;\le\; \log c_{AB} \;\le\; \log c_A + \log c_B,$$

which is the multiplicative form of Weyl’s inequality and holds at every point of every pair. Both ends are attained. So the composite of two maps whose elongations are each a factor of two can be anything from perfectly isotropic to a factor of four, and knowing both parents’ indicatrices completely fails to narrow it.

Across six pairs the composite lands at 19, 59, 71, 73 and 9 per cent of the way up its own range. The spread is not noise and it is not a property of the projections’ families: Mollweide and Hammer are both equal-area and land at 19 per cent; the sinusoidal and Gall–Peters are both equal-area and land at 59.

The one number that decides it #

The missing quantity is an angle, and it is the same one this ladder’s previous rung was about.

The angle between the two maps’ major principal axes predicts where in the interval the composite lands, with a correlation of 0.993 on Miller-to-Eckert IV and between 0.90 and 0.96 on the other pairs. That angle is not reported by either indicatrix. An ellipse drawn on a map carries its two semi-axes and their orientation on the page, and the two pages here are different pages — so the quantity needed is the relative orientation of two principal frames neither of which the drawn field distinguishes from a rotation of itself.

The composite is therefore not unpredictable. It is exactly predictable from one number nobody records.

The situation is closely parallel to whether the ellipses point the same way, which found that the direction of a map’s stretch varies over a region in ways the field of ellipse shapes does not show, and to distortion has a direction, which is the same observation at a point. Both of those are about one map. This is the first place on the ladder where the direction is not a refinement of the description but the whole of the answer: two maps with identical indicatrix fields and different principal directions compose to different maps, and no amount of care with $a$, $b$ and $\omega$ will separate them.

The law #

The relation is not merely a correlation. Two-by-two algebra gives it in closed form, and it is short.

Let $u$ and $v$ be the two parents’ log condition numbers and $\theta$ the angle between their major principal axes. Then the composite’s log condition number $w$ satisfies

$$\cosh w \;=\; \cos^2\theta \,\cosh(v - u) \;+\; \sin^2\theta \,\cosh(v + u).$$

At $\theta = 0$ this is $w = |v - u|$ and at $\theta = 90°$ it is $w = v + u$, which is the bound above recovered as the two ends of one curve. Everything between is a cosine squared, which is why the empirical relation looks so nearly linear in $\theta$ over the middle of its range.

The check that matters is not that the formula can be derived but that it agrees with the object it claims to describe. The composite is computed twice — once by forming $A_B A_A^{-1}$ and taking its singular values, and once from the closed form using only $u$, $v$ and $\theta$ — and the largest disagreement across every pair and every sample point is 2.1 × 10⁻⁸, which is what the Jacobian’s Richardson-extrapolated step leaves behind.

The two controls #

Two special cases behave exactly as the naive expectation says, and they are the two cases the subject is built around — which is a large part of why the general behaviour has gone unremarked.

Two conformal maps compose to a conformal map. Both have $c = 1$, so both ends of the bound are zero and there is nothing for the angle to decide. Mercator followed by the stereographic returns an angular deformation of 0.000° at every point sampled. That is the theorem that makes conformal maps a group exercised rather than cited.

Two equal-area maps compose to an equal-area map. Determinants multiply and both are one, so the composite’s areal factor is one to eleven digits — even when, as in the hero figure, its shape error is worse than either parent’s.

There is a third case worth having because it is the practically useful one. Reprojecting off a conformal map is free. A conformal map contributes $u = 0$, so the law collapses to $w = v$ whatever the angle: Mercator to Mollweide has the composite’s mean angular deformation equal to Mollweide’s own, 31.841338436° against 31.841338436°. A pipeline that stores in a conformal projection and renders into whatever is wanted pays nothing for the storage step, which is a real design consequence of an identity that looks like a curiosity.

Where it goes the other way #

The case that is hardest to anticipate is the one where composing helps.

Over eighty per cent of the sphere the Mollweide-to-Hammer reprojection has a smaller shape error than either map it joins. The two are both equal-area pseudocylindrical-family maps with a similar meridian curve, so their principal directions agree closely — a mean separation of 14° — and the second map’s stretch is largely along the first’s, where it partly cancels.

The comparison that makes it sharp is the sinusoidal to Gall–Peters, which is the same statement about two maps that are also both equal-area. Their mean separation is 44°, the composite lands at 59 per cent of its range instead of 19, and it beats both parents at no point on the sphere at all. Same families, same conserved quantity, opposite outcome, and the only quantity that distinguishes them is the angle.

The share is worth reading as a distribution rather than as a headline. Mollweide to Hammer beats both parents on 80 per cent of the sphere; the equirectangular to Robinson on 21 per cent; Miller to Eckert IV on 3; the sinusoidal to Gall–Peters on none. Three of those four run in the order of their mean separations — 14°, 54° and 60° against shares of 80, 21 and 3 per cent — and the fourth breaks it. The sinusoidal-to-Gall–Peters pair has a mean angle of 44°, smaller than two of the others, and the worst outcome of all four, because its two parents are both strongly anisotropic, so the range its angle is choosing a position within is much wider. The angle says where in the range; the parents say how wide the range is; the outcome needs both.

There is a design reading of this that the anchor has not had before. Choosing an intermediate projection for a pipeline is choosing a $\theta$ against every downstream map, and a map whose principal directions run close to the destination’s is a good intermediate almost regardless of its own distortion — which is why the two members of one pseudocylindrical family compose so gently. An average of ellipses is not an ellipse makes the neighbouring point about summarising a field; this is the same non-linearity met on the way in rather than on the way out.

What was computed, and how #

Every number here comes from the same local frame the rest of this ladder uses: the projection differentiated by central differences with a Richardson step, converted to page displacement per metre on the ground, and decomposed. Nothing is re-derived and no projection is treated specially.

The composite is formed by inverting the first map’s frame and multiplying, which requires that the frame be non-singular — so points where either map’s Jacobian dies are dropped rather than reported, and that is why the samples stop at 84° of latitude.

The angle between principal axes is taken from the stretch factors of each polar decomposition rather than from the rotations, which matters and was got wrong first. A principal axis is a direction without a sign, so the difference between two of them lives in a half-turn and folds into $[0°, 90°]$. Folding further, into $[0°, 45°]$, is the natural-looking thing to do and it destroys exactly the distinction being measured: aligned and perpendicular become the same number, and the correlation with the composite falls from +0.84 to −0.48. The wrong version fits well enough to look like a result.

The assertions require five things separately: that two conformal parents give a conformal composite, that two equal-area parents give an equal-area one, that no composite anywhere escapes the multiplicative bound, that the six pairs land in materially different parts of their ranges — otherwise the bound explains nothing — and that some composite somewhere is gentler than both its parents, since a rule that reprojection only ever adds distortion is what everybody already assumes.

What it means for a pipeline #

A chain of transformations does not close is the datum version of this, and it is about accumulated error. This is about accumulated distortion, which behaves differently: error accumulates monotonically and distortion does not.

So the practical statement is not “minimise the number of reprojections”. It is that a two-step route can be gentler than either of its steps, and that whether it is depends on the relative orientation of two stretch fields — which means a pipeline’s intermediate projection is a real design choice with a computable answer, rather than a formality to be minimised away.

The rule that does survive is narrower and more useful than the folklore one. Store in a conformal projection and any single reprojection out of it costs nothing in shape; store in an equal-area one and area is preserved down the whole chain whatever else happens. Each of those is a genuine invariant of a pipeline and each is the exact half of the trade the operation decides the coordinate system is about. What is not available is a pipeline that keeps both, for the reason no map is faithful gives, and composing maps does not create a loophole in it: the composite of a conformal and an equal-area map is neither, and it is neither at every point where the first map was doing anything at all.

Reprojecting a raster invents values prices the interpolation half of the same operation. The two are independent: interpolation error depends on the resampling kernel and the local scale ratio, and the geometric distortion measured here would be there in a vector pipeline where nothing is interpolated at all.

Where the model stops #

This is a statement about one point. The composite’s indicatrix is a first-order object exactly as its parents’ are, and everything the indicatrix is a limit says about a finite region applies again, now to a map whose second derivative nothing here has computed.

The angle is not itself an invariant of the pair. It depends on both maps’ aspects: rotating one of them about the sphere changes every principal direction and therefore changes the composite, while changing neither parent’s own indicatrix anywhere. That is the sharpest possible statement of what the ellipse fails to carry, and it also means the number quoted for a pair is a number for a pair in a stated aspect.

Six pairs is a sample. The correlations between 0.90 and 0.99 are measured on six reprojections between maps that are all cylindrical or pseudocylindrical in the usual aspect. A pair involving an azimuthal projection, whose principal directions run radially, would have a very different distribution of $\theta$, and the law would still hold because it is an identity.

The generalisation #

The rule is that a composition of linear maps is decided by their relative orientation, and an invariant description throws that away.

Every quantity this collection insists on is invariant: $a$, $b$, the areal factor and $\omega$ survive a change of coordinates, which is what makes them the right things to report about a single map. Invariance is exactly what makes them insufficient here, because the relative orientation of two frames is not invariant under rotating one of them — it is the thing that changes when one of them is rotated.

That is not an argument for reporting less-invariant quantities. It is an argument for reporting one more: the pair $(a, b, \gamma)$ with the rotation included is a complete local description, and rung ten already showed the rotation is the grid convergence a surveyor cannot work without. This rung is a second use for the same missing number, arriving from an entirely different question, which is the usual sign that the omission is structural rather than an oversight.

Who found it, and when #

The matrix facts are old. Weyl’s multiplicative inequalities date from 1949, the polar decomposition from Cauchy, and the closed form above is a page of two-by-two algebra that any of the people who wrote about map distortion in the nineteenth century could have done.

They did not, and the reason looks structural rather than accidental. Reprojection as a routine operation is a computational habit: before it, a map was made once from the sphere, and composing two projections was a thing nobody had cause to do outside the theory of conformal mappings, where both factors are conformal and the composite is trivially conformal. The interesting case — a compromise followed by a compromise — became common only when it became one command.

The literature that does treat compositions treats them as sources of numerical error: floating-point loss, coordinate round-trip drift, the accumulated slop of a pipeline. That is a real subject and it is a different one. The geometric statement is that the composite has its own first-order distortion, that this distortion has a two-term formula, and that reading it requires a number the standard picture of map distortion does not draw.

Where the ladder goes next #

Eleven rungs have measured what an indicatrix reports and, twice now, what it does not. Both omissions were about orientation — the rotation on one map, the relative rotation between two. The instrument is otherwise complete for what it claims to describe, which is a map at a point, and the next question the ladder has to face is what happens when there is no point: a map with a boundary, where the object being reported on stops.