The length belongs to the journey

The staircase that is not the diagonal diagnoses its own paradox and then says the diagnosis cannot be drawn: “length is a property of the parametrised object rather than of the point set, which is the technically correct diagnosis of everything above and which no picture on this page can show, since a picture only ever draws the point set.”

That sentence deserves an essay of its own, because it is stronger than it looks. It is not that a point set is a lossy record of a curve. It is that the number a reader calls the curve’s length is not a function of the point set at all.

The same picture, three answers #

Take an arc and three ways of walking it. Walk it once. Walk to the far end and return along it. Walk out, back and out again. At every moment all three are at some point of the arc, and between them they visit exactly the same points — the images are identical, and the figure checks that by sampling each map densely and requiring every sample to lie on the first’s image.

The lengths are 1.464, 2.927 and 4.391, in the ratios one, two and three.

There is nothing paradoxical here and that is the point. Each number is the honest total distance travelled by something following that map, and the three maps travel different distances because two of them go over the same ground more than once. What the example refutes is a habit rather than a theorem: the habit of saying the curve’s length while pointing at a picture, when the picture is the image and the length is a property of the map.

So “curve” is two different objects sharing a word. A curve as a set of points has a length only if one is chosen for it; a curve as a map from an interval has one by definition, and the definition is the supremum over inscribed polygons computed along the parameter.

Which of the two the supremum computes #

The definition is a supremum over partitions of the parameter interval, and that is where the dependence enters. Partition $[0,1]$, take the points the map sends the partition to, join them in order, and add up. For the out-and-back map the polygon runs to the far end and comes back along itself, and its length is twice the arc’s — because the partition is of the parameter and the map visits each point twice.

A supremum over partitions of the image would be a different construction and it is not the one anybody uses, for a reason worth stating: an image has no order on it. A polygon needs its corners joined in sequence, and the sequence is what the parameter supplies. Take away the parameter and there is no polygon, only a set of points — and a set of points in the plane has no length, it has an area, which for an arc is nought.

That is the cleanest way to see that the two notions are genuinely different. The image’s own one-dimensional measure is a third quantity, it equals the length when the map is injective, and it equals the image’s length rather than the journey’s when the map doubles back. For the out-and-back map that measure is 1.464 and the length is 2.927, and both numbers are correct answers to questions that sound identical in English.

The definition, rebuilt with the parameter in view #

Before the three maps can be compared it is worth watching the definition pick a partition, because the partition is of the parameter and that is where the whole dependence lives.

The supremum is over partitions $0 = t_0 < t_1 < \cdots < t_m = 1$ of the parameter interval, and the polygon joins $p(t_0), p(t_1), \ldots, p(t_m)$ in that order. Two things about that construction are worth saying out loud. The order is the parameter’s, not the plane’s — the corners are joined in the sequence the map visits them, which for a retracing map means the polygon retraces too. And the corners are points of the map, so a parameter value visited twice contributes twice.

The figure shows what a badly chosen partition does. At three corners the wave’s own zeros are sampled and nothing else, so the polygon is the straight segment along the axis and its length is exactly one — the width of the interval. Refine to seven and the humps are caught. The supremum does not care, because a supremum is over all partitions and the bad ones are simply not the ones attaining it; what a figure has to be careful about is not reporting a bad partition’s answer as the supremum.

Which is the same caution the variation needed one step earlier, and it has the same resolution: refine until the increases stop, and say what refinement was used. Every number in this essay’s figures is a polygon at two thousand or four thousand corners, and the figures say so.

What the integral of the speed is measuring #

The formula everybody learns is $L = \int_0^1 |p'(t)|\,dt$, and it is worth asking which of the two quantities it computes. The answer is the map’s, and the figure is the check: the integral agrees with the polygon length for the out-and-back map, at twice the image’s length.

The reason is in the derivation. A short piece of parameter $dt$ carries the point a distance of about $|p'(t)|\,dt$, and adding those up totals the distance travelled — with no mechanism anywhere for noticing that a piece of ground has been covered before. An integral over the parameter counts what happens at each moment, and going back over old ground is something that happens.

So the formula is not an approximation to the image’s length that happens to be wrong for retracing maps. It is an exact computation of a different quantity, and the quantity it computes is the one the definition defines.

The agreement between the two is also a theorem rather than a restatement, and it needs a hypothesis. The supremum over inscribed polygons equals the integral of the speed when the map is continuously differentiable; for a merely rectifiable map the supremum still exists and the integral may not, since the speed need not be defined anywhere. So the formula is the special case, and the supremum is the definition.

Speed is invisible and length is not #

A map’s length does not determine the map either, and the figure is the smallest example. Trace the arc once at a steady speed and once accelerating; the images match, the lengths match, and the maps are different. The parameter marks are what shows it, and the parameter marks are not part of the picture in any usual sense.

That is the other half of the relationship and it says how much the length knows. The length is a function of the map and not of the image, so it knows more than the image. It is invariant under reparametrisation — changing the schedule without changing the route leaves the total distance alone — so it knows less than the map. Length sits strictly between the two, and what it is exactly is the map up to increasing reparametrisation.

That equivalence class has a canonical member, and this is where the subject tidies itself. Reparametrise so that the point travels at unit speed: then the parameter is the distance travelled, the length is the parameter interval’s own length, and every rectifiable map has exactly one such representative. The arc-length parametrisation exists because the length function is increasing and can be inverted, which is the monotonicity that argument is built on doing one more job — the same device that turns a secant into a tangent by holding both ends on the curve.

What this settles about the staircase #

The staircase paradox now has two diagnoses and it is worth seeing that they are the same one.

The first diagnosis, which the staircase that is not the diagonal gives, is that length depends on direction and the staircases’ directions never converge. The second is that the staircases converge to the arc as point sets and not as maps: give each staircase the obvious parametrisation and the sequence of maps does not converge to any parametrisation of the arc, because at almost every moment the staircase is heading horizontally or vertically while the arc is heading at forty-five degrees.

The two are the same statement because a map’s derivative is its direction and speed. Converging as maps in the sense that preserves length means converging with the derivatives, and the staircases fail exactly there. So the diagnosis is not that the point sets were the wrong object to look at; it is that convergence of point sets is the wrong convergence to use for a quantity attached to maps.

Which makes the paradox an instance of the general rule the earlier essay states: a quantity survives a limit when it depends on no more than the limit controls. Length depends on the map’s derivative; convergence of images controls nothing about derivatives; so length does not survive. Everything else on this page is that sentence with the objects named precisely.

The staircase, for comparison #

Setting the staircases beside the three maps above makes the two failures distinguishable, which is worth doing because they are usually run together.

The three maps have the same image and different lengths. The staircases have different images — each is a jagged path near the arc, not on it — converging to the arc’s image, and lengths that do not converge to the arc’s length. The first shows that length is not a function of the image; the second shows that it is not continuous in the image either. Neither implies the other, and the second is the one that looks like a paradox.

What connects them is that both are settled by looking at the map. For the three maps the answer is that they are three different maps and the length is telling them apart correctly. For the staircases the answer is that they are maps whose derivatives are wrong everywhere, so a quantity depending on the derivative has no reason to converge. Both readings are the same instruction: name the object before measuring it, and the object with a length is the map.

There is a version of the staircase construction that does converge, and it is the inscribed polygon. Its corners are on the curve, its directions converge to the curve’s directions, and its lengths converge to the curve’s length. So the repair is not a subtle one — it is the same construction with the corners moved onto the curve, and moving them is what makes the derivatives right.

Where the distinction bites outside the paradox #

Three places, and in each the confusion is a real error rather than a pedantic one.

Measuring a length from samples. Anything computing a length from sampled points is computing the length of the polygon, which is the length of a map — the one that goes straight between consecutive samples. If the underlying journey retraced, the samples may not record it, and the answer comes out short. If the sampling is noisy, the polygon zigzags and the answer comes out long, and the error does not shrink with more samples.

Curve simplification. Replacing a curve by one with fewer corners — the same trap the staircase sets, one application along is usually specified as a constraint on how far the new curve strays from the old — which is a constraint on the images. That controls the area between them and says nothing about the lengths, which is exactly the staircase’s situation and exactly why simplified curves have lengths that wander.

Closed curves and winding. A map going twice round a circle has twice the length and the same image, and the number of times it goes round is not recoverable from the image either. That count is what the winding number is, it is a property of the map, and it is the discrete analogue of everything above: a quantity that a picture of the image cannot carry.

Two maps, one length, and the class in between #

The relationship between the three objects — image, length, map — is worth writing down as a chain, because each step in it loses something specific.

A map determines its image, by applying it. The image does not determine the map: three of them above share one. So the image is a genuine quotient and the information lost is the schedule and the covering count.

A map determines its length, by the supremum. The length does not determine the map, since a reparametrisation leaves it alone. So length is also a quotient, and what it forgets is exactly the increasing reparametrisations.

The two quotients are not the same and neither refines the other. Two maps with the same image can have different lengths, and two maps with the same length can have different images — a straight segment of length two and a semicircle of length two are the obvious pair. So knowing the image tells nothing about the length and knowing the length tells nothing about the image, and the map is what has both.

The one place they coincide is worth naming because it is the case everybody has in mind. For an injective map — one that never returns to a point it has visited — the length equals the image’s own one-dimensional measure, and then the length of the curve is unambiguous and the habit this essay complains about is harmless. Almost every curve anybody draws is of that kind, which is why the ambiguity survives.

What the pictures cannot show #

The three maps in the hero figure are drawn on top of each other, because they have the same image — so the panels are identical drawings with different numbers under them, which is the honest picture of the claim and a strange one. What distinguishes them is the dots, which mark equal steps of the parameter, and even those coincide for two of the three panels at some points.

The sameness of the images is a computation rather than something visible. The figure samples each map at four thousand points and requires every sample of every map to lie within a thousandth of the first map’s image; that is what “the same image” means in the figure and it is what a drawing of two identical pictures cannot establish.

And no picture shows a speed. A speed is a rate and a drawing is a state, so the accelerating map’s dots crowding at one end is the nearest a static figure comes — and the crowding is a picture of the parameter, which is the thing the whole essay is about not being part of the image.

Still open: what the image alone determines #

The image determines its own one-dimensional measure, which is a number, and it does not determine the length of any map onto it. The question of what a set determines about the maps onto it is a subject with real content, and the one-dimensional case is where it begins.

The sharpest known statement is a characterisation of which sets admit a map of finite length onto them at all: a compact connected set is the image of a rectifiable map exactly when it has finite one-dimensional measure, and then the shortest such map has length between that measure and twice it. The factor of two is attained — a set shaped like a tree has to be traversed twice along every branch — and which sets need the full factor is understood.

What is much less clear is the version with more dimensions. A surface’s area is a property of its parametrisation by the same argument as here, the analogue of the arc-length parametrisation does not exist in general, and the question of which sets are the images of finite-area maps has no answer of the one-dimensional kind. That asymmetry between one dimension and two is the same one the classification of surfaces runs into from the other side.

What the word was hiding #

Two objects share the word curve, they have the same picture, and only one of them has a length. Nearly every confusion here is that ambiguity, and the paradox these essays start with is what it looks like when the ambiguity is left in place.

The repair is not a definition but a question: length of what? A journey has a length. A route has a length if a journey along it is specified, and whether any journey along it has a finite one is a separate question with its own condition. A set of points has a measure, which for an arc is the length of the journey that traverses it once, and which for a set traversed twice is half what the journey travelled.

Once the question is asked the paradox stops being one and the formula stops looking like an approximation. The integral of the speed is exact, it computes what it computes, and what it computes is the distance something travelled — which is the quantity anybody wanted, provided they were asked to say whose journey they meant.