The thread: Computed, not quoted — page 15
What a machine does with it
A label belongs to no tile
Nine rungs price the tile as a piece of geometry. A label is not geometry — it is a page object placed by collision against other page objects, and collision is a global relation while a tile is rendered alone. Cut a page into tiles and 10.3 per cent of the labels are placed differently; the buffer that closes the gap is half a tile at moderate density and a whole one when it is crowded.
What is taught wrongly
A second tracing tells a drifting hand from a copied stretch
A hand tracing a coast drifts rather than jitters, and a drifting error is a smooth departure no projection explains — so the criterion that counted copied stretches correctly under independent noise counts four or five seams where two were made, and three or four on a coast with no copy at all. Undo the drift before fitting and every count comes back, for no copy, one and two, in every tracing. That needs the drift's correlation length from outside the coast, known to within a factor of three on the short side and much less on the long, and a second tracing of the same coast supplies it: the difference between two tracings holds nothing but the two errors, and read that way the count is right in 22 tracings of 24, and never too high.
What each projection optimises
Three maps make a plane, and the band's best map lies outside their triangle
Every conformal map whose ln k is a fixed blend of three solved maps — the p = 1 map, the least worst map and the mean-square map — is itself a conformal map, so three solved maps are a plane of them. Inside their triangle the third map recovers most of what the line between the first two gave up on thin regions: 0.63 points short of the best exponent on the thinnest ellipse, where the line was 1.48 short. But the band's best map in the plane is not between the three. On 172 of 182 regions and bands it lies outside the triangle, and it keeps up to twenty points more than any exponent; started from it, a search over all twenty-five coefficients adds about a point on four regions of five and seven on a hexagon. Its price is a worst departure up to nearly five times the least worst.
Grids, and what a survey does
An agency can map where its published covariance is optimistic
The local surveys tied to a regional network each estimate, badly, how far the network's published covariance understates the truth, and the agency that receives them can pool what they say. When the understatement is confined to one part of the region, pooling it into one number dilutes it: a patch covering 15 per cent of the region with errors four times their published size reads as 1.30 for the whole. Counted place by place, the same surveys find the patch — two agencies in three find it with 400 surveys and nearly every one with 800, flagging almost nothing outside it. Fed back as one number the scale removes little of the false alarms inside the patch and costs nearly a third of the test's power outside; fed back as a map it cuts the false alarms inside from 20 per cent to 3 and costs little outside.
Paths and directions
A layer is a meeting with the bank, not a corner turned
A reach set drawn in layers — rays from the centre, then from every place a route can leave the bank — ends in the exact answer, and how deep it must go depends on what a layer is. Counted by corners turned, the depth belongs to the polygon the bank is drawn as: five turns on a bank drawn every 400 metres, eighteen on one drawn every 50. Counted by meetings with the bank, it is three at every resolution. One meeting anywhere leaves 1.6 per cent of a ten-kilometre walk unreached, where rays from the bridges alone leave 6.4; two leave 0.47; three leave nothing. A longer walk needs more — up to six meetings at twenty kilometres across thirty-six rivers — and the bend's steepness, which prices the first layer, ranks the depth at 0.24.
The impossibility
A sea with five islands is seldom worth five cuts
With three islands in a sea, the second cut went across the sea from the first, and the number of cuts worth spending looked like the number of directions the islands point. With four and five islands that survives in part and breaks in part. Choosing the dearest island and then, each time, the island farthest in bearing from every island already cut names the best pair of cuts in 28 seas of 40, where the two dearest name it in 16 — and it fails at three cuts of five, because the best three cuts often do not contain the best two. The dearest island is no longer reliably among the best pair. And 39 seas of 40 are worth fewer cuts than they have islands: the count follows the widest gap in the islands' bearings more closely than it follows how many islands there are.
The families
A smooth curve beats the water's best join from its first term
On the land, a smooth equal-area curve with no join needed three terms before it beat Goode's composite rejoined at the land's best latitude, and on the land's own lobes it needs three again. On the water a single term beats the water's best composite, a second and third add more, and the best curve is 0.54 degrees of mean angular deformation better than the composite joined at seventy-five and 1.6 better than Goode's. It is nothing like the composite in shape: its parallels never close to less than seven eighths of the equator's spacing, so the map the water wants is the sinusoidal reshaped by a few per cent at every latitude, not the sinusoidal with a cap. The lobes do not move under it. And where a separate join for each hemisphere was worth three hundredths of a degree, a separate curve for each is worth a quarter of a degree.
Measuring distortion
The light comes from a page direction
A hillshade's illumination azimuth is declared from the top of the sheet, and the top of the sheet is grid north. On a conic the light therefore swings 52.6° across eighty degrees of longitude, and 72.8 per cent of the sheet is shaded differently from what the legend claims.
What the numbers refer to
A ray from the centre hits the surface twice
Eleven rungs map bodies that are lumpy, triaxial and turning at a drifting rate, and every one assumes the surface is star-shaped about the centre — which is what makes a longitude and a latitude a coordinate at all. A contact binary is not: on a stated body with a neck a third of a lobe wide, 10.9 per cent of the sky has no single radius, and the shape model everybody publishes fills the neck in and adds 1.67 per cent of the volume.
What a machine does with it
Which features survive is not a sample
How many features a scale can carry is answered and treats the population as a number. Which ones survive is a different question: keeping one feature in ten carries 99.99 per cent of the total length and inflates the median feature by a factor of 95, and the shape of the size distribution survives both exactly.
What is taught wrongly
An easing is a seam with a width, and a drifting hand hides the width
A copy eased into a compiled coast is not a projection, so a count that explains the coast in rigid pieces buys four, five or six seams for one copy. Give each seam a width — the copy blended into the host over that width, the same linear fit at every width, the width charged as one more number — and the count is two seams at every easing up to 17.5 per cent of the coast, and the width it reports is the one drawn, wherever the copy departs from the host by many times the digitising error. Where it departs by seven, the width is a guess. On a tracing made by a drifting hand the easing disappears: whitened for the drift, the rigid count already calls two seams in every tracing, because an easing as long as the drift is a stretch of drift, and its width can no longer be read until it is twice the drift's length.
Paths and directions
A season learns the stretch from where the streak was, not from the floors
An ensemble whose members spread four fifths as widely as the forecast's real error covers the diversion floor in 94 crossings of a hundred instead of 95, and the repair is to stretch every member away from the centre by a factor learned from past crossings. Learned the way the rank histogram teaches it — from where the truth's floor fell among the members' floors — a season of a hundred crossings lands within a tenth of the right stretch one time in three, and a season of eight hundred two times in three. Learned from where the streak turned out to be, which the analysis after each crossing supplies whether the streak reached the gap or not, a hundred crossings land within a tenth 85 times in a hundred and two hundred almost always. The floor is the quantity the plan is about; it is the wrong quantity to learn the spread from.
What each projection optimises
A cap on the worst departure keeps the thin regions' gain and gives back the square's
The conformal map that keeps the most of a region inside a band can sit far out in the plane of three solved maps and let its edges go to nearly five times the least worst departure. Capped at twice the least worst, the plane still matches or beats the best exponent on 69 region-and-band pairs of 91: on the thin ellipses it keeps the whole of its gain, because that gain sits in a small pocket just past the exponents' own worst departure and arrives all at once; on the square and the bays it gives most of it back, because there the gain is a ramp that runs out to four times the least worst. Under any one cap the plane still beats the exponents that cap admits, by up to eighteen points — and it can name maps that no weighting of area against worst departure would ever choose.
The impossibility
Gravity is worth four loops a station, and the background still hides a deep root
Levelling loops alone place a range's compensating root at ten kilometres three times in four from 1,600 loops, and do not place one at sixty from 6,400. Gravity measured on the ground reads the root's attraction directly, and a station is worth about four loops for a shallow root and two for one at thirty kilometres. It does not place the deep root either: 1,600 stations put a sixty-kilometre root within a factor of 1.5 of its depth in fewer than one survey in five. With the background field taken away, a hundred stations place it every time. What hides the deep root is not the reading but the geoid's own mass that the range does not explain, and either half of its spectrum is enough to do it. Fitted for depth, gravity reads the deep root as too heavy, where the loops read it as too light.
Grids, and what a survey does
Copied out to sea, a coast keeps its length and loses its bays
Before 1930 a territorial sea was drawn by following the coast in all its sinuosities, and the phrase fits two constructions. Moved twelve miles seaward, a stated coast of five bays gives up 240 square kilometres against the envelope of arcs, most of it over the narrowest bay; moved along its own normals, it ties a loop over every bay. On a fractal coast charted at finer detail the copy grows by a quarter at every level, to 480 kilometres, while the envelope settles at 133.0.
What a machine does with it
Two routes to one scale
A national series is cascaded — the million is derived from the quarter-million, which was derived from the fifty — and the folklore is that the errors accumulate. They do not: Douglas–Peucker and Visvalingam both cascade to the same line the direct route produces, bit for bit, because both output a sublevel set of a per-vertex number. What breaks it is anything else in the chain, and a moving average puts the product nine tolerances away.
Measuring distortion
The curvature of a field is not the curvature of its picture
Whether a place is a spur or a hollow is the sign of a second derivative, and every automatic terrain classification is built on it. Read off an equal-area grid, that sign is wrong on 13.25 per cent of a sheet — and conformality reduces it to 0.30 per cent without removing it.
What is taught wrongly
Averaging tracings reads the easing, until the hand repeats itself
One tracing by a drifting hand cannot show how far a copy was eased, because an easing as long as the drift is a stretch of drift. Average several tracings and the easing stays while the drift shrinks: two tracings read the easing's width at the seam where the copy departs strongly, sixteen read it at the seam where it barely departs, and as the drift falls the rigid count starts buying pieces for the easing again while the eased count stays at two. All of that assumes each tracing drifts afresh. A hand that repeats part of its drift in every tracing leaves that part in the mean, where it sets a floor no number of tracings goes below; and a digitiser who learns the hand from the spread between tracings measures only the part that averages away, and charges the mean too little, so that both counts buy every piece they are offered.
Paths and directions
A revised forecast is relearned in as many crossings as the learner remembers
An ensemble's stretch is a property of the forecast system, and forecast systems are revised. When a revision narrows the members from four fifths of the real spread to three fifths, a learner that weights every crossing since the start of the winter equally still holds 1.44 for a new stretch of 1.67 three hundred crossings later, and leaves the plan under the floor in 93.8 crossings of a hundred instead of 95. A learner that forgets regains ninety-five per cent coverage after about as many crossings as its memory — 31, 57, 113 and 206 for memories of 25, 50, 100 and 200 — and pays for forgetting only when nothing has changed. A memory of fifty costs nothing measurable over a winter. And a bias the whole ensemble shares, which the stretch alone reads as extra spread, is found to a kilometre by a learner that tracks the mean departure as well.
Measuring distortion
The cheapest map that meets its areas
An earlier essay bracketed a cartogram's least cost between a construction charging eighty degrees and a bound valid only for symmetric densities, and recorded the gap as a shortfall. One request settles it: a density of contrast eighty whose least cost is exactly zero, met by a map written down in closed form, while the standard construction charges 43.8° for it.
What a machine does with it
A tolerance in map units is not a tolerance
A snapping tolerance is a number, and the number is in whatever units the file is in. Five map units on Web Mercator is 4.97 metres of ground at the equator and 0.87 at eighty degrees — so a rule that merges two features three metres apart merges them everywhere below 52.8° north and refuses everywhere above it, in one pass, over one dataset, with nothing recording where the boundary is.
What the numbers refer to
The height a coordinate does not carry
Ten rungs treat a datum shift as a map from one pair of angles to another. It is not one: the transformation runs through Cartesian coordinates, so the answer depends on the height — by 7.7 to 121 millimetres per kilometre depending on the datum and the place, which puts Lhasa 432 millimetres from where a height of zero would have said.
What is taught wrongly
The spread between hands charges the mean, and an office's habit hides in the sheet
One hand's tracings cannot see the part of its drift it repeats, so a mean of sixteen by one hand is charged for a third of the error it carries. Sixteen tracings by four hands carry half the error of sixteen by one, and the spread between the hands' means, divided by the number of hands, charges that mean correctly to within a fifth on every sheet from three hands on — without ever estimating how much of a hand is habit. The same spread reads a hand's share of habit from two hands of two. What it cannot read is a habit every hand in the office shares: tracings of a sheet with one are tracings of a different sheet without it, and the charge they give falls further short the more hands and tracings are added.
What is taught wrongly
A known coast charges an office's habit, and a bending one takes it out
An office whose hands all share a habit leaves it in every tracing of every sheet, where the spread between hands cannot see it. One control sheet — a coast known independently, traced the office's usual way — reads the habit's size and restores the charge, if the habit is a drift. If the habit answers to the coast's own curvature, a size read on a rounder coast charges a thinner one at little more than half, but the same control read differently does better than charge the habit: its departures, set against its curvature, give a slope that is the office's own on every coast, and subtracting it leaves about one per cent of the habit in the mean. The slope also tells the two kinds of office apart. A coast that hardly bends teaches neither, and corrects worse than nothing.