What the page cannot move

Six rungs of this ladder measure what a page does to a field: its slope, its aspect, the direction water runs, the shape of a measured surface, the quarter a hillshade is lit from, the sign of its curvature. Every one of them moves, and several of them move enough to change what a reader concludes.

That is a long enough run to invite the wrong general conclusion — that nothing about a field survives being drawn. Something does, and what it is has a one-line proof and a sharp boundary.

The argument, in two lines #

A reader of the page forms the gradient A⁻ᵀg, where g is the field’s gradient on the ground and A is the projection’s own local frame — the matrix whose columns are the page displacement produced by a metre east and a metre north. That is the quantity water runs downhill on the ground, not on the page is about.

A is invertible wherever the map is a map. So

$$\mathbf{A}^{-\mathsf{T}}g = 0 \iff g = 0,$$

and the critical points of the page are the images of the critical points of the ground. Not approximately, not to first order: the same points.

At such a point the page Hessian is A⁻ᵀHA⁻¹, which is congruent to H. Sylvester’s law of inertia says a congruence preserves the signature of a symmetric matrix, so the number of positive and negative eigenvalues is unchanged and a maximum is a maximum, a saddle is a saddle, a minimum is a minimum, on every projection there is.

What was computed, and how #

The proof is two lines and the measurement is the point, because a proof about what a formula does is not a statement about what code does.

Two searches, sharing nothing.

On the sphere: a grid of starts, Newton on ∇f = 0 in the local east–north frame, using the field’s own analytic gradient and a differenced Hessian. No projection is evaluated anywhere in it.

On the page: the composed function G(x, y) = f(p⁻¹(x, y)), differenced in page coordinates, with Newton run in page coordinates. Every value comes through the projection’s inverse; no spherical gradient is evaluated anywhere in it.

The two lists are then matched by position.

The alternating sum — maxima minus saddles plus minima — is 3 on the ground and 3 on every page, which is the structural form of the same fact and is the quantity a topologist would have quoted instead.

What the page does move #

The invariance stops abruptly, and where it stops is worth measuring because it is not where intuition puts it.

A critical point has a shape: the ratio of its two principal curvatures, which is how elongated the hill or hollow is, and the direction that elongation points. Neither is in the signature, so neither is protected.

Mercator preserves it exactly, and the reason is the same one line of algebra. When A is a similarity — a rotation times a scale, which is precisely what conformal means — A⁻ᵀHA⁻¹ is H divided by the square of that scale, so both eigenvalues are divided by the same number and their ratio is untouched.

So the elongation of a hill joins the list of conformal invariants, beside an angle and a shape at a point, and it is separated sharply from the position and the type, which survive everything.

And the equal-area projection is the worst of the four. That is the ordering an areal audit would invert, and it is the same lesson the two ways a map is wrong draws: a quantity is protected by the property that constrains it, and equal-area constrains a product where this needs a ratio.

The bound, and it is attained #

A⁻¹ divides the two principal directions by a and by b, the indicatrix’s own semi-axes. So the Hessian’s eigenvalue ratio can be moved by at most

$$\left(\frac{a}{b}\right)^2,$$

and by exactly that when the hill’s own axes line up with the indicatrix’s.

Measured over twenty-one critical points on three projections: not one exceeds the bound, and the worst uses 99.99986 per cent of it. A bound that is never approached is compatible with several mechanisms; a bound that is attained to six digits identifies one.

That is what turns this from an inequality into a statement about the cause: the change in a critical point’s shape is the indicatrix, acting on the Hessian, and there is nothing else in it.

What this makes safe, and what it does not #

The practical division is unusually clean.

Safe to read off any map, however drawn: where the peaks are, where the passes are, where the hollows are, how many of each there are, which is higher than which — a value is a value and the projection does not touch it — and therefore the whole ordering of the summits, the prominence of every peak, and the topology of the drainage network as a graph.

Safe only on a conformal map: how elongated a summit is, whether a col is broad or narrow, the aspect ratio of a basin’s floor — every quantity that is a ratio of two curvatures and no quantity that is one of them alone.

Not safe on any map: the area of a basin, the length of a ridge, the slope anywhere, the direction water runs, the shape of a contour — which the contour is right and the reading is wrong prices — and which way a hillshade is lit. Six rungs of this ladder measure those.

The uncomfortable member of the third list is the basin area, because the basin is bounded by divides, a divide is not defined by a local condition, and the divide moves on any page that is not conformal. So a reader can be certain there are exactly three summits and cannot be certain how much ground drains to each.

That figure is the sharpest available statement of where the boundary runs. Both quantities are discrete; both are read off a second derivative. One is the signature of the Hessian at a point where the gradient is zero, and it survives every projection. The other is the sign of a curvature along a stated direction at an ordinary point, and it flips.

The difference is that the first is a property of a matrix under congruence and the second is a property of a matrix applied to a vector, and a congruence preserves a signature while doing anything it likes to a quadratic form’s value in a given direction.

Where the model stops #

The field is stated and smooth. Real terrain is not: a sampled DEM has critical points at the sampling scale, most of them artefacts, and the whole apparatus of Morse theory on discrete data needs a persistence threshold to be usable at all. The invariance survives that — a projection cannot move a discrete extremum either, since it is a comparison of values — but the count becomes a property of the sampling, and the slope of a field that was measured is the rung about what sampling does.

Degenerate points are excluded. A monkey saddle has a vanishing Hessian determinant and no signature to preserve; what survives there is the winding number of the gradient, which is a stronger invariant and is what the flexion ladder’s degeneracy rung uses. Nothing here has one.

And a projection with a singularity is not a map there. A is not invertible at the pole of an azimuthal projection or along an interruption, so the argument does not apply, and a critical point that lands on one is genuinely lost rather than moved.

Why the measurement is worth making at all #

An invariance with a two-line proof invites the question of what a measurement adds, and the answer is specific rather than a matter of principle.

The proof is about a formula and the measurement is about code. Everything on this site is computed, and a projection’s Jacobian in this library is a numerical object with a differencing step and a convergence tolerance in it. The theorem says a critical point cannot move; what a measurement can find is that this implementation moves it — because the inverse projection is a Newton iteration that stops early, because the page differencing step is too large where the scale factor is large, or because a search on a page with a badly conditioned frame converges to a different point.

None of those would announce itself. A critical point search that returned six points instead of seven, or a saddle where a maximum belongs, would look exactly like a field with a different structure. The theorem is what says the number to expect; the measurement is what says the code produces it.

And the boundary is the part that could not have been guessed. That the shape is preserved by conformal maps and not by others is available from the algebra, but that the equal-area member of the library is the worst of four, by a factor of eleven where a compromise projection manages two, is not — and it is the number a reader choosing a projection for terrain analysis would actually want.

The generalisation #

A quantity survives a change of coordinates when its definition does not mention the coordinates.

The gradient vanishes here mentions no basis. The Hessian has two negative eigenvalues mentions no basis. The two eigenvalues are in the ratio three to one mentions a metric, and a projection is not an isometry — which is why no map is faithful — so it is exactly the quantities with a metric in them that go.

This collection has the same division elsewhere with different objects. What survives a change of coordinates makes it about the indicatrix itself: h and k depend on the graticule and a, b, the areal factor and ω do not. A velocity needs a frame and a strain rate does not makes it about the ground’s own motion.

What is new here is a case where the invariant is discrete. The others are numbers that survive; these are a count and a classification, and a count cannot be nearly right. That makes the check unusually strong — three maxima on every projection is a statement that fails loudly or not at all — and it makes the failure of everything else easier to state, because the boundary between the two lists is a boundary between what a topology sees and what a metric sees.

One consequence for how a summit list is made #

The safe list has a practical edge worth drawing out, because it settles a question that comes up whenever terrain is analysed on a national grid.

A summit list — the peaks of a region, with their heights and their prominence — is built by finding maxima and the cols between them. Every quantity in it is on the safe list: the maxima are invariant, the cols are saddles and are invariant, a height is a value and is invariant, and prominence is a difference of two values. So a summit list computed on any projection is the summit list, and the choice of grid cannot be argued about.

That is not true of anything else the same analysis produces. The area a summit dominates, the length of the ridge joining two of them, the steepness of an approach: all three are metric and all three move. A list that mixes the two kinds — as most do, since a peak’s area of dominance is a natural thing to report beside its prominence — is half projection-independent and half not, and nothing in it says which half.

One consequence for this site’s own machinery is worth recording. The critical-point search on the page runs entirely through the projection’s numerical inverse, which is a Newton iteration with a convergence tolerance, and the positions still agree with the sphere’s to 10⁻⁷ degrees. That is a statement about the inverse as much as about the theorem: an inverse that stopped short would move a critical point, and none of the four projections tested does.

Who found it, and when #

That a diffeomorphism preserves critical points and their indices is elementary Morse theory, from Morse’s work in the 1920s and 1930s, and the Euler characteristic relation is older still. Sylvester’s law of inertia is from 1852. None of it is cartographic and all of it applies.

What appears not to be written down is the cartographic reading, which is that the terrain analysis a reader can trust off an arbitrary projection is exactly the topological part: the peaks, the passes, the pits, their number and their ordering. That is a usefully large amount, it includes the whole of what a summit list or a drainage graph contains, and it stands in sharp contrast to the metric quantities — slope, aspect, area, curvature — that six previous rungs found to be untrustworthy.

The one reading a reader can trust off any sheet #

The contrast at the end of that account is the ladder’s summary and deserves stating in the form a practitioner can use, because it is unusually clean for a result of this kind.

Everything topological survives any projection. How many peaks, how many passes, how many pits; which peak is higher than which; which passes connect which basins; the drainage graph’s structure. None of that moves when the sheet changes, because a projection is a diffeomorphism and a diffeomorphism carries critical points to critical points with their indices intact.

Everything metric moves. Slope, aspect, curvature, area, the direction of illumination, the spacing of contours, the length of a divide. Six previous rungs measured how much, and the answers ranged from a few per cent to a factor of fourteen.

The line between them is exactly the line between what a homeomorphism preserves and what it does not, which is a statement a reader can apply without computing anything: ask whether the quantity would survive the map being drawn on a rubber sheet and then stretched. A count survives; a length does not.

The rule also says where the boundary is subtle. A prominence value is a height difference and is therefore metric, but a prominence ordering is decided by which pass connects which peaks, which is topological — so the list of summits ranked by prominence is stable under reprojection while the numbers beside them are not. A reader taking the ordering is safe and a reader taking the values is not, from the same table.

And the surviving list is larger than it sounds. A summit list, a prominence ordering, a catchment graph, a count of basins — these are the outputs of a great deal of terrain analysis, and every one of them is safe on any projection whatever. The reader who takes only that from this ladder has taken the part with no caveats attached.

A further consequence is worth naming for anybody assembling such a list from data. Because the counts are invariant, a disagreement between two summit lists derived from the same elevation model on two different projections is not a projection effect at all — it is a resampling effect, introduced when the model was warped from one grid to another, and it should be attributed there. The projection cannot create or destroy a peak; the interpolation that redrew the raster can.

Where the ladder goes next #

Seven rungs price the reading of a field on a page and now the reading that cannot be spoiled. The quantity none of them reaches is the one that sits between the two lists: a ridge, which is neither a critical point nor a metric quantity but a curve defined by a condition on the second derivative along one direction. It has no coordinate-free definition anybody agrees on, several competing ones give different curves on the same terrain, and every one of them moves with the page.