A thousand features are wrong in the same direction

A tolerance is a promise about the picture established that a simplification’s area error is not bounded by the tolerance and that its sign is not predictable: a cut corner takes area off, a bridged bay puts it back, and one outline can lose at one tolerance and gain at the next.

That is a statement about one feature, and it is the one a reader takes away. It leads directly to a comfortable conclusion — that the error is a noise, unbiased, and that summing over many features will wash it out.

It will not. Over a population the sign is not unpredictable at all.

Why the rotation test is the right one #

Rotating a shape does not change it. Every rotation has exactly the same area, the same perimeter, the same curvature distribution and the same complexity, so any variation in the measured error is variation in the algorithm’s response rather than in the input.

What changes with rotation is which vertices the recursion happens to select. Douglas–Peucker’s choice of split point depends on distances to a chord, and the chord depends on where the recursion started, so a rotated presentation of the same curve produces a different retained set with the same guarantee.

Twenty-four presentations therefore give twenty-four independent draws from the algorithm’s own variability, on an input with no variability at all. If the error were a noise, that is the experiment that would show it.

The result, and what seventy standard errors means #

Every one of the twenty-four loses area. The mean is −0.46 per cent and the standard deviation across presentations is small enough that the mean sits seventy standard errors below zero.

Seventy is not a marginal significance. It says the quantity being measured is a property of the operation rather than of the presentation, and that no sample size will bring the mean to zero — because the mean is not zero.

That is the practical difference between a bias and a noise, and it is the whole of this rung. A noise divides by the square root of the count. A bias does not divide by anything.

Even with bays available #

The curve here is deliberately not convex. It has five outward lobes and five inward ones, so at every tolerance the algorithm has corners to cut and bays to bridge, and the previous rung’s mechanism for gaining area is present throughout.

It still loses, at every presentation. The two mechanisms are not symmetric: cutting a corner removes a sliver whose size is set by the curvature there, bridging a bay adds one, and the algorithm’s vertex selection is driven by displacement — which is larger on the convex side of a wiggle than on the concave side of the same wiggle, because the chord falls on the outside.

So the sign is genuinely unpredictable for a particular feature at a particular tolerance, and the expectation over presentations is not zero. Both statements are true and only the first is usually made.

What a bias is, against what a noise is #

The distinction is worth putting carefully because it decides what a user can do.

A noise has expectation zero, in the sense an error ellipse describes. Its effect on a sum of $n$ terms grows as $\sqrt{n}$ while the sum itself grows as $n$, so the relative error of an aggregate falls as $1/\sqrt{n}$ and a large dataset is more reliable than a small one. That is the ordinary and reassuring case, and it is what most users assume they are in.

A bias has a non-zero expectation. Its effect on a sum grows as $n$, exactly as the sum does, so the relative error of the aggregate is the same as the relative error of one term and does not improve at all. A dataset of a million polygons is no better protected than a dataset of ten.

The measurement above puts the simplification’s area error in the second category, at seventy standard errors, on an input with no intrinsic variability. Everything a user might do to average it away — more features, larger regions, longer time series — leaves it exactly where it was.

The bias already dominates at one feature #

Seventy standard errors is quoted as a statement about the mean of twenty-four, and it says something sharper when it is turned back into a statement about one.

The standard error of a mean of twenty-four is the per-presentation spread divided by √24, so the spread itself is 0.46 per cent × √24 ÷ 70 = 0.032 per cent. Against a bias of 0.46, that is a signal-to-noise ratio of fourteen for a single presentation of a single feature.

So the bias is not something aggregation reveals. It was always the dominant term: one feature simplified once loses 0.46 per cent of its area, give or take three hundredths, and the give-or-take is a fourteenth of the loss. The rotation experiment’s twenty-four presentations do not make a small effect visible — they demonstrate that the effect is not an accident of the one presentation anybody would otherwise have measured.

That reconciles this rung with the previous one rather than contradicting it. The earlier finding of an unpredictable sign is about varying the tolerance and the shape, both of which this experiment holds fixed. A feature’s error at a given tolerance is reliably negative and reliably about this size; the same feature at a different tolerance, or a different feature at the same one, is where the sign becomes unpredictable — because the two mechanisms, corner-cutting and bay-bridging, are re-weighted by which vertices the recursion happens to reach.

The practical reading is therefore stronger than a bias survives aggregation. A single generalised polygon’s area is already biased low by a knowable amount, and the reason nobody has noticed is not that the effect is small relative to the noise. It is that nobody compares a generalised polygon’s area against the source’s, because the source has been discarded and the comparison is the thing generalisation was performed to avoid needing.

The size is set by the curvature, not the tolerance #

The two errors are the same order in $\varepsilon$, and the area error is not a function of $\varepsilon$.

At a tolerance of 0.01, a circle of radius 1 loses 0.641 per cent of its area and a circle of radius 0.25 loses 2.550 — very nearly four times as much, which is the ratio of the radii. Two boundaries held to identical displacement guarantees lose different fractions of their area, and the ratio is their curvature.

That is why no correction is available from the tolerance alone. Predicting the loss needs the boundary’s curvature distribution, which is exactly what the simplification was performed to discard.

What that does to a total #

The consequence is arithmetic and it is uncomfortable.

Take a dataset of $n$ polygons — each of them a claim about a surface — simplify them all at one tolerance, and sum the areas. Each contributes a loss with a negative expectation, so the total’s error is $n$ times a negative mean rather than $\sqrt{n}$ times a standard deviation. A national land-cover total computed from a generalised dataset is smaller than one computed from the source, and the shortfall grows with the number of features rather than with their sizes.

Worse, the per-feature loss depends on curvature. A dataset of smooth administrative boundaries and a dataset of crinkled natural ones lose different fractions at the same tolerance, so the same tolerance applied to two layers biases them differently — and any comparison between the two layers inherits the difference.

And to a rate #

A density is a count divided by an area. If the denominator is biased low then every rate computed from generalised geometry is biased high, by the same fraction, and the fraction differs by feature shape.

That is the quiet failure, because a rate looks like a ratio of two measured quantities and one of them is a processing artefact. The attribute is a claim about the geometry makes the general version of this argument one anchor over: a value stored per unit area is only meaningful against the area it was computed with, and every operation that changes the area invalidates it. Simplification is such an operation and is almost never treated as one.

The obvious fix does not fix it #

The natural response is to reach for the algorithm whose threshold is an area, and Visvalingam’s rule is a real improvement in several respects. It is not an improvement in this one.

Controlling the area removed at each step is not controlling the total, and on a convex stretch each step still removes rather than adds. Matched at the same retained point count it loses area on every row measured here and wins only half the comparisons on magnitude.

Two rules, one threshold each, neither of them a promise about the quantity users read.

What an area-preserving rule would cost #

It would have to place vertices outside the original boundary, so that chords cut out on one side and in on the other in balanced pairs.

That breaks containment: the simplified feature no longer lies inside the original, so a point that was inside can end up outside and a point-in-polygon test on the generalised geometry can disagree with one on the source. Which side of the line a point is on is the failure the previous rung already priced, and it is the one containment protects against.

So the two properties are in direct conflict and every mainstream algorithm has chosen containment. That is a defensible choice; it is not documented as a choice, it is documented as a tolerance.

Two layers, one tolerance, different biases #

The consequence that is hardest to see in practice is the differential one, and it is worth an example.

Suppose a coastline layer and an administrative-boundary layer — which share their common edge and are stored twice — are both simplified at ten metres, which is a perfectly ordinary thing for a pipeline to do. The coastline is crinkled at every scale; the administrative boundary is largely straight lines and river reaches. At the same tolerance the crinkled layer loses a much larger fraction of its area than the smooth one, because the loss is set by curvature.

Now compute anything that compares the two — the proportion of a district that is coastal, the length of coast per unit area, the fraction of land within a distance of the sea — and the answer carries the difference of two biases that were never measured separately. It is not even a consistent overstatement or understatement; which way it goes depends on which layer appears in the numerator.

The refusal #

The measure has to be able to return zero and it does. A tolerance below the source’s own vertex spacing removes nothing: the retained point count is unchanged, the line is the original line, and the area error is exactly zero to machine precision.

So every non-zero reading above is the tolerance’s doing rather than the instrument’s, and the twenty-four negative values are twenty-four measurements rather than twenty-four artefacts.

What to record instead of a tolerance #

The measurements here are all cheap and two of them belong in a dataset’s metadata.

The retained fraction — vertices kept over vertices in the source — is the honest description of how much was thrown away, and it is what a user needs to judge whether the geometry is fit for their purpose.

The areal bias, computed once over the layer by comparing the simplified total against the source total, is one number and it is a correction. It cannot be derived from the tolerance, and it can be measured directly by whoever performed the simplification, at the moment when both geometries are still in hand.

Nobody records the second, and it is available only then. Once the source is discarded the bias is unrecoverable, and the discarding is usually the point of the exercise.

Why the algorithm cannot be blamed #

It is worth defending Douglas–Peucker here, because the essay reads as an indictment and is not one.

The rule was designed for a specific purpose — reduce the vertex count of a digitised line while keeping the drawn line within a stated distance of the original — and it does that exactly, provably, and cheaply. Every claim it makes is true and no claim it makes is about area.

The failure is one of interface. A parameter called a tolerance, expressed in ground units, reads as a bound on error, and the users of a generalised layer are rarely the people who generalised it. What travels downstream is a number in a metadata field and an unexamined assumption that it bounds everything.

That is a familiar shape in this collection. Conformal does not mean the angles are right is the same failure in a different vocabulary: a precise technical guarantee, correctly stated, read by users as a much broader one because the word carries more in ordinary speech than it does in the theorem.

Where the model stops #

Everything here is measured in the plane, on synthetic boundaries with stated curvature, at tolerances small compared with the features. A real layer has features spanning several orders of magnitude in size, and a fixed tolerance interacts with that spread: the smallest features are simplified nearly out of existence while the largest are barely touched, so the per-feature bias is wildly uneven and its aggregate is dominated by the small end.

What is also omitted is topological cleanup. Real generalisation pipelines run a simplification and then repair the damage — remove slivers, snap shared boundaries, drop degenerate rings — and each repair moves area again, in directions this measurement does not model.

The comparison that is safe #

Not everything is lost, and the thing that survives is worth naming because it is what generalised data is good for.

A shape comparison between two features in the same layer, simplified at the same tolerance, is largely protected: both lost area for the same reason and by similar fractions if their curvatures are similar, so the ratio is better than either term. That is the ordinary cartographic use — this district is about twice that one — and it is not badly served.

What is not protected is any comparison that leaves the layer: against a total from another source, against a count, against a length, against the same layer at another generalisation level. Every one of those puts a biased number next to an unbiased one, and the bias does not announce itself.

So the rule is the one this field keeps arriving at from different directions. Generalised geometry is a picture, and a picture supports comparisons within itself and not measurements taken out of it — which is what a tolerance promises, read as a statement about what the data is for rather than about what the algorithm does.

Who found it, and when #

The observation that generalised areas shrink is old and is usually reported as an empirical regularity in the cartographic generalisation literature. What is rarely done is the significance test: reporting a mean shrinkage without a spread leaves open the reading that it is a tendency rather than a bias, and a tendency is something an aggregate might survive.

The rotation experiment is the cheap way to close that, and it is cheap because the input is stated rather than sampled. Twenty-four presentations of one shape, one tolerance, one number out — and the number is seventy standard errors from the value the comfortable reading requires.

Where the ladder goes next #

This rung and the three before it ask what happens to a feature that is kept. Which features are kept at all is a different problem with its own law, and the law turns out to be a theorem about a distribution rather than the rule of thumb it is always quoted as.