An easing is a seam with a width, and a drifting hand hides the width

A copy eased into place loses its seams before its source described how a compiler joins a copied stretch of coast to the sheet around it. A rigid copy dropped into the gap leaves a corner at each end. A compiler with tracing paper does not leave corners: they pin the copy’s two ends and ease it into its neighbours, so the copied coast blends into the host’s over some distance either side of each seam. A residual cannot count the pieces of a map then counted the pieces of a compiled coast with the Bayesian information criterion, charging each piece for the numbers it spends, and found the count right for rigid copies and wrong for eased ones. An eased stretch is no projection of anything, so the criterion paid for it in pieces.

A second tracing tells a drifting hand from a copied stretch dealt with the other smooth departure on a traced coast, the slow drift of a digitiser’s hand, and ended with a proposal for easing. Model it. Let a piece be a projection placed by a similarity and blended into its neighbours over a stated length, charge the length as one more number, and see whether the criterion then counts an eased copy as one copy and reports how far it was eased — and whether what it reports is a draughtsman’s easing or whatever the criterion needed.

The proposal works, under conditions that turn out to be specific, and it fails in a way that answers the question the earlier essay was really asking.

A seam with a width is still a linear fit #

The rigid count explains a coast of 360 traced points as a sequence of stretches. Each stretch is some candidate projection of the coast’s own latitudes and longitudes, placed on the page by a similarity — a complex scale a and a complex offset b, four numbers — and the seams between stretches are the places the count chooses. Each seam is a fifth number. The criterion is the residual, in units of the digitising error, plus the logarithm of the number of observations for every number spent.

An eased seam replaces the step from one piece to the next with a blend. Near a seam, the drawn point is a weighted mixture of the two neighbouring pieces’ predictions, the weight running from 0 to 1 over a width h either side of the seam on the curve a compiler’s easing follows — a smoothstep, the same curve the eased copies here were drawn with. Write $m_k(t)$ for piece k’s weight at point t. The model is

$$w(t) = \sum_k m_k(t)\,\bigl(a_k z_k(t) + b_k\bigr),$$

and once the seams and their widths are fixed, it is linear in every $a_k$ and $b_k$. The fit at any stated set of seams and widths is a least-squares solve as exact as the rigid one, and a rigid seam is simply a seam of width zero. What has to be searched is the seams, their widths and the candidate behind each piece. The search here starts from the rigid count’s exact optimum at each count, moves each seam and its width together over a window and thirty stated widths from a step to a third of the coast, re-chooses each piece’s projection among nine candidates, and repeats until nothing moves — from five starting widths, because widening one seam alone can make a fit worse while widening both makes it much better.

The model is the construction. On the eased coast with no digitising error, the true projections at the true seams with the true width leave a residual of 4 × 10⁻¹⁵ of the map, where rigid seams at the same places leave 4 × 10⁻⁵. Because a rigid seam is a seam of width zero, the eased model contains the rigid one at every count, and its residual is never larger. That is why the width has to be charged for. Each seam in the eased model costs six numbers where a rigid seam cost five, whether the width the fit finds is zero or not.

One copy, counted as one copy #

The coast is the earlier essays’ compilation: Japan’s outline on a conformal conic sheet, with the stretch from 55 to 85 per cent of the way round copied from a sinusoidal sheet and placed by the similarity that pins its two ends to the host’s. The copy is 108 points of 360. Here it is eased over a stated share of the coast either side of each seam, from nothing to a quarter, and traced to a ten-thousandth of the map’s width.

The rigid count does what the earlier essay found. A rigid copy is two seams. Eased over 2 or 3.5 per cent of the coast it is still two, because the easing is short enough that the departure it leaves is inside the digitising error. From 5 per cent the count rises — four seams, then five at 8 per cent, then six, the most it was offered, from 10 per cent on.

The eased count calls two seams at every easing up to 17.5 per cent. It fails at 20 per cent, where it calls four, and the failure has a reason that the model makes visible. The copy is 108 points long, and an easing of more than 54 points either side of each seam — 15 per cent of the coast — means the two easings run into each other. Past that point the copied stretch is nowhere drawn as itself: every point of it is partly host. The eased model, which blends each seam on its own and holds full weight on the copy between them, describes that coast less and less well, and by 20 per cent it is cheaper to describe it in pieces again.

One choice in the measurement has to be stated, because the rigid count never needed it. The count walks a closed coast as a line, from wherever the walk starts, and an easing that runs past the start is split across the two ends of the walk where no stretch of a line can hold it. The rigid count’s walk started at the coast’s northernmost point, 54 points after the copy ends. Here the walk starts opposite the copy, leaving 126 points of host coast either side of it. Started at the north, from 15 per cent of easing the count named the host coast after the copy as sinusoidal: the easing had run off the end of the walk and taken the last of the host with it, which is a failure of the walk rather than of the model.

The width that comes back #

The eased count does more than count. Each seam it finds comes with a width, and the earlier essay’s second question was whether that width is the compiler’s. At the first seam it is. From an easing of 13 points either side to 63 the reported half-width is within three points of the drawn one at every easing, and exact at 18 and 54. Only the narrowest easing, 7 points, is read short, at 4.

The second seam is a different story at small easings and the same story at large ones. From 29 points up the second seam’s width comes back within three points, exactly as the first does. Below that it comes back short: 2 for 7, 6 for 13, 5 for 18. The fit is not confused about where the second seam is or how many seams there are; it simply finds a narrow easing there fits as well as the drawn one.

A seam where the copy barely departs has no width to read #

What an easing changes is how fast the drawn coast moves from the host’s line to the copy’s. The amount it can change is the distance between the two lines over the easing, and that distance is set by the copy’s own shape. The copy is placed by a similarity that pins its two ends exactly on the host’s coast, so at the seam itself copy and host agree and the easing does nothing; it matters only as far as the copy departs from the host within the easing’s reach.

At the first seam the sinusoidal copy departs from the conformal conic host quickly. Over the 18 points either side of the seam its distance from the host averages 27.6 times the digitising error, and the residual rises steeply either side of the drawn width: only 18 itself is within one unit of the best. At the second seam the copy runs close to the host for longer, and the same average is 6.9 times the error. The residual barely moves as the width changes there, so widths from 4 to 8 lie within one unit of the best, and the drawn width, 18, sits 5.1 units up — a fluctuation of the digitising error, not a feature of the coast, has picked a narrower easing.

That is the honest form of a reported width. A width is measured against a departure, and where the copy and the host nearly coincide over the easing, there is nothing for the width to be measured against. The same is true of a rigid seam’s position, which where the control points are and its successors found everywhere in this subject: a parameter is only as visible as the difference it makes.

Six tracings #

One tracing can mislead in either direction, so the 5 per cent copy was traced six times with independent digitising errors of the same size. The rigid count overcounts in every one, by one seam or two. The eased count calls two seams in every one, and the first seam’s width comes back as 18 in every one, against 18 drawn.

The second seam’s width scatters, from 5 to 23, with a middle of 16. The first tracing’s 5 was the low tail, not the rule. Read across tracings, the second seam’s width is an estimate with a spread of about five points either side of the truth, and the first seam’s is exact. That is what the answer is a set asked of every number this method produces: not a value but a value and a width, and here the two seams of one copy carry widths of very different precision, for a reason the geometry states in advance.

The draughtsman’s curve hardly matters #

The eased model blends along one curve, and a compiler does not use a curve: they ease by eye. The test is to draw the copy with an easing of a different shape and read it with the model’s own. A raised cosine is nearly the same curve as the smoothstep, and it is read identically — two seams in every tracing, widths 18 of 18 and 34 of 36. A straight ramp is a different shape, with a corner at each end of the easing where the smoothstep is flat, and the rigid count notices the corners: at 5 per cent it calls four or five seams for the ramp where it called three or four for the smoothstep. The eased count still calls two seams in seven of the eight ramp tracings, and once calls four.

What changes is the width it reports. A ramp of half-width 18 is read as a smoothstep of half-width 20, and a ramp of 36 as one of 38 to 43. The smoothstep that best imitates a ramp is a little wider than the ramp, because it reaches its full weight gently. So the reported width is the width of the equivalent smoothstep, which is within about a fifth of the draughtsman’s for the shapes tried. A width reported this way is recognisable as an easing — 5 per cent of the coast, not 0.5 or 30 — and not a measurement of the curve the hand followed.

A drifting hand and an easing are the same kind of departure #

The earlier essay’s proposal had a premise. With the drift undone, it said, the one smooth departure left on a traced coast is the compiler’s own, and a model that contains easing could then read it. The measurement says the premise is false.

Traced with the drift that essay measured — an error three ten-thousandths of the map in size, each point’s error mostly the last one’s over a correlation length of 18 points — and counted without allowing for the drift, the eased copy is overcounted by both models. The rigid count calls four to six seams; the eased count calls four, the most it was offered, in seven tracings of eight. Drift is a smooth departure, and a model with eased seams can spend them on drift as readily as a rigid model spends pieces.

Whitened at the drift’s true length, as the earlier essay showed how to do, the rigid count calls two seams in all eight tracings. It no longer needs the eased model. And the eased model, whitened the same way, cannot see the easing: it calls two seams in seven tracings and one in the eighth, and the widths it reports at the strong seam are 0, 0, 1, 0, 0, 6 and 20, against 18 drawn.

Both facts have one cause. Whitening divides the weight of each departure by how much like drift it looks. A drift of correlation length 18 points produces smooth departures about 18 points long, and an easing of half-width 18 is a smooth departure of exactly that length. The whitened criterion therefore discounts the easing as heavily as it discounts the drift: the misfit that made the rigid count buy pieces is now cheap, so it buys none, and the difference between an eased seam and a rigid one is no longer worth the number the width costs. A drifting hand does not hide the copy — the seams and the count survive — but it hides the easing completely.

An easing shows through the drift once it is longer than the drift #

If whitening discounts departures by how much they look like drift, an easing much longer than the drift should look less like it. It does. Eased over twice the drift’s length, 36 points either side, the strong seam’s width comes back as 34 in two tracings of four, 48 in the third and 67 in the fourth. Over three times the drift’s length, 54 points, it comes back as 54, 60, 67 and 67. The weak seam, where over an 18-point easing the copy departs from the host by a quarter as much, reads anything from 0 to 82 at every width.

So the answer to the earlier essay’s question has three parts. Easing and drift cannot be told apart by one coast when they have the same length, and the reason is not a failing of the count: they are the same kind of departure, and the criterion that allows for one has allowed for the other. An easing several times longer than the drift can be told apart, at a seam where the copy departs strongly. And every count in this essay came out right once the drift was whitened, eased or not, so on a drifting tracing the easing is a property the count can ignore — which is the useful half of the answer, since the count was what the method was for.

What was held fixed #

The eased model is the construction. With no digitising error, the true projections at the true seams and the true width must fit to a millionth of the map, and rigid seams at the same places must not. They fit to 4 × 10⁻¹⁵ and leave 4 × 10⁻⁵.

The eased model contains the rigid one. At every count searched and every easing, the eased residual must be no larger than the rigid residual, since a seam of width zero is a rigid seam. It never is.

The finding must be there to fail. On the 5 per cent copy at a ten-thousandth of digitising error, the rigid count must exceed two seams and the eased count must be two. They are four and two.

The search is a search, and is stated as one. The rigid count is solved exactly, by dynamic programming over every place the seams could go. The eased model is fitted by a descent over seams, widths and sources, started from that exact rigid optimum and from five starting widths, and a descent can stall. What is guaranteed is only that the eased fit is never worse than the rigid one at the same count; that it finds the best eased fit is not, and the measured widths agreeing with the drawn ones to within three points is the evidence that it does here.

Where the model stops #

One curve. Every easing is fitted as a smoothstep. A ramp is read as a slightly wider smoothstep, and an easing of a shape far from both — a long tail on one side, say — would be read as whatever smoothstep it most resembles.

Easings that meet. The model blends each seam on its own and holds the copy at full weight between them. A copy eased so far that the two easings meet in its middle is described less and less well, and from 20 per cent it is counted in pieces again.

A walk with a start. The count walks a closed coast as a line, and the walk here starts opposite the copy. An easing that runs past the walk’s start cannot be held by any stretch of the line, and on an unknown coast the right start is not known in advance. A count on a closed walk, with no start at all, would remove the choice and cost a second search.

One sheet, one shrinkage. The sheet moved before it was measured found the paper’s own shrinkage to be a smooth, directional departure, present in every tracing. It is a third smooth departure beside easing and drift, and nothing here models it.

The copy is placed by its ends. The copied stretch is pinned to the host at both ends by a similarity, which is why the departure at each seam starts at zero and why the second seam’s width is hard to read. A compiler who placed the copy by eye over its whole length would leave larger departures at the seams, and wider widths might then be easier to read rather than harder.

Still open: whether several tracings can read an easing the drift hides #

One tracing cannot separate an easing from a drift of the same length. Several tracings might, because the two departures behave differently across tracings. The easing is on the sheet and is in every tracing identically; the drift is in the hand and is different every time. The earlier essay used the difference of two tracings to learn the drift, since the difference holds nothing but the two errors. The mean of several tracings does the opposite: it keeps the easing whole and divides the drift’s variance by the number of tracings.

Whether the mean of four tracings is enough to read an 18-point easing at the strong seam, how many it takes at the weak one, and whether a criterion that counts on the mean while whitening for the drift’s reduced size behaves as the single-tracing criterion does with a quieter hand, are questions one tracing cannot ask. A map with no graticule and the datum hides inside the projection’s parameters are where this subject first found that a coast can say more than its control points do; a stack of tracings would be the first time it said more than itself.