Degrees of freedom — where it appears
Named by 31 essays across 7 fields — each of them below, with the objects they name alongside it.
The condition does not always decide the map
Write a family as a shape with an unknown function in it and every classical property becomes a differential equation. In three families the equation has one solution and the named projection is what comes back. In the fourth it has a whole function of solutions, which is why that family has forty members and the others have three.
The third parameter, run
An aspect has three numbers and this site has been searching two of them, with a note admitting it. Searching all three is worth up to 2.1 times — and the obvious way to do it, starting from the two-parameter answer and letting the third move, finds a fraction of that or nothing at all.
Every equal-area map is every other one
Take Mollweide and slide every row of the page sideways by an amount that depends on the row. The result satisfies the equal-area condition to 6 × 10⁻¹², exactly as well as Mollweide does, and it is a ruin — the angular deformation at one ordinary point has gone from 11° to 60°. Equal-area is one equation, and one equation leaves a whole function free.
A coordinate is the output of a solve
Six essays measure a tape, close a traverse, spread a misclosure and reduce a chain. The coordinate that comes out of the far end is the solution of a least-squares problem, and the problem has a decision in it that is not a measurement: what to hold fixed. Change it and every coordinate moves by centimetres while not one residual moves at all.
The nearest map to an impossible request
Rung seven found that not every distortion can be asked for. It never asked what happens when one is asked for anyway — and the answer has a shape: the achievable fields are the solutions of an elliptic equation, a request is a point off that set, and the nearest point leaves a residual whose floor is the curvature rather than the size of the ask.
The difference of two coordinates
Three essays give a single coordinate a width. Every practical use of one is a difference of two — a distance, a bearing, a movement, an area — and the width of a difference is not the two widths combined, because the errors are not independent. Far from its datum a one-leg baseline is six times more certain than the positions it joins.
The best flat picture is not a map
A set of dots whose separations are as nearly right as separations can be made beats every projection in this collection — by a factor of thirteen on four world cities, and by eight per cent on sixteen. The collapse between those two numbers is not about cartography. It is that a picture of n places has 2n − 3 free numbers and n(n − 1)/2 distances to spend them on.
The nearest equal-area map to an impossible request
Every number in the measurement before this one is inside the conformal achievable set, because Liouville's is the conformal condition. The equal-area set is one equation on two functions and never refuses: the same four requests are met to nine parts in a billion, and charged for in angle instead.
The network's answer is decided before it is measured
An adjustment is usually judged by what it does with observations. Every quantity a specification is written about — the error ellipses, the redundancy numbers, the smallest detectable blunder — is a function of the geometry and the weights alone, and does not contain an observed value anywhere. Four layouts of five stations with the same ten distances differ by a factor of 20.9 in their worst coordinate.
Five distances of six, and never more
A flat picture of n places has 2n − 3 free numbers and n(n−1)/2 distances, so it can be exactly right about 2n − 3 of them and no more — five of six for four places, twenty-nine of a hundred and twenty for sixteen, and 3.98 per cent for a hundred. The bound is reached by a construction with no fitting in it, and an azimuthal equidistant map spends only n − 1.
A family is a function, not a list
The equal-area pseudocylindricals are the solutions of one equation in two unknown functions, so one function is free and the named members are points in a space of them. Writing that function as six numbers and searching over them finds a map at 25.03° of mean angular deformation against Eckert IV's 28.08 — and two numbers are already enough to beat every map the family has a name for.
Five bearings of twelve, and only eight ways to draw them
A flat picture of four places can hold five of their twelve one-way bearings exactly, the same count as distances, and the five are found by solving equations rather than searching. The equations cannot tell a line from its reverse. Of the 192 ways to choose five, eight can be drawn with every arrow pointing forwards, all eight leave London and New York unheld, and on sixteen cities no choice made at random can be drawn at all.
The residual reports the error the fix was immune to
A fourth sight gives a four-line fit something three lines never had: a leftover. The leftover is a real measurement of the sights, and it is useless exactly where it is needed. One number computed from the four azimuths alone — before any star is observed — says how much of a common error reaches the residual, and it is 1 when the bodies stand at the quarters of the compass, where such an error moves the fix by nothing at all, and 0.004 when they are bunched within sixty degrees, where it moves the fix by two kilometres.
The pair that cannot be held is not the one the geometry names
Four world cities have exactly one pair of the six that no drawable picture holds, and the obvious candidate — the pair whose bearing there and bearing back disagree most — is not it. That quantity predicts something else exactly: whether a held pair's arrow has a forced direction. It separates the two groups with a clean gap and nothing in between. And at five places there is no forbidden pair at all, so the four-place case was a statement about how little room five constraints leave rather than about the map.
What the extra unknown costs where nothing can see it
Four lines of position can estimate a common error in every sight as well as the position. The estimate is honest — three kilometres put in comes back as 2.8 — and where the residual is blind, its scatter is plus or minus thirty-one, and the fix that carries it is scattered fourteen kilometres from the ship against the biased fix's two and a half. Letting the residual choose between the two is worse than either, because a test with no power is a one-in-twenty lottery on the worse answer.
A mixed picture holds one more, and what it is holding is north
A flat picture of n places can hold 2n − 3 distances exactly, and it can hold 2n − 3 bearings exactly, and it is tempting to read one budget spent twice. It is not one budget. A distance cannot see which way the page is turned and a bearing cannot see how large it is drawn, so a picture constrained by both kinds is blind to one freedom instead of two — and holds 2n − 2. For four cities that is six of the eighteen available where either kind alone can hold five, and nothing whatever can hold seven: nought of 31,824 choices.
An equal-area strip removes the corner and charges nothing for it
Goode's seam matches three quantities and breaks the fourth: every meridian but the central one turns a corner there. A third map spliced between the two removes it — four conditions on one function, satisfied uniquely by a cubic — and the expected price does not arrive. The spliced composite carries less angular deformation across the seam than the plain join, not more: 34.7° against 37.7° at sixty degrees of longitude. What it does leave is the same defect one derivative up, a jump in curvature falling as the reciprocal of the strip's width.
A criterion worth using is one whose answer is not unique
Forty-two starts on one region and one tolerance give forty-two different maps, keeping from 36.98 per cent of the area to 67.50. The two starts anybody would actually use — Chebyshev's map and the mean-square map — end 27.1 points apart, which is more than the criterion buys over either of them. And the spread is widest at exactly the tolerance where the criterion is worth using at all, because both facts are the same fact: an objective with a flat top and a sharp edge has somewhere to hide answers.
Two numbers about the paper buy nothing about the places
A picture of n places has four freedoms, and a constraint holds only what its own blindness leaves: distances reach 2n − 3, bearings 2n − 3, the two together 2n − 2. Position is reached by neither, because nothing about how places stand to one another says where on the sheet to put them. A stated position does reach it, and two of them take the budget to 2n exactly. What the count conceals is that the extra two buy nothing: a largest picture holds six place constraints with the pins and six without, the same 7,596 of them, holding the same pairs at the same rates. The constraint system is a direct sum, and a pin is a different currency.
Only Eckert II's family can be the strip, and its meridians are straight
The cubic that smooths Goode's seam is four conditions on the length of a parallel, and equal area turns them into four conditions on how the parallels are spaced. The sinusoidal's spacing is flat, so a strip can leave it only where its own spacing is momentarily flat — and of eleven named equal-area maps, only Eckert II's spacing ever turns over. One member of its family meets all four conditions exactly, over seventeen degrees centred four hundredths of a degree from Goode's join, with every meridian straight. The cubic over the same span is the same map to four parts in a hundred thousand.
A second pin is a measurement of the places
Counted exactly, a picture of places is a direct sum: the distances and bearings spend its shape and two pins spend where it sits on the sheet. Given tolerances, the sum survives in one form and fails in another. Pins at one place put none of their error into the shape at any tolerance, and wherever the pins are, their average takes none of the distances' error. But every place pinned after the first is a statement about the geography, and held exact it makes the shape worse once its error passes about two thirds of the shape's own.
Two islands are three numbers, and the proof does not survive them
A sea with one island at its centre has one conformal invariant and a forced best map. Put two islands in it and there are three invariants, readable in closed form because stereographic projection carries circles to circles — and the forcing is gone. The theorem that picked the map needed the region to be unchanged by rotation, not to have one hole, and a single harmonic term two hundredths in size beats the stereographic by 0.19 per cent the moment an island moves off centre.
Three numbers are all a local survey can check of its control
Adjust a local network on its own lines, then fit it onto its three control stations, and the misfit at the control is the only place the control's error can show. It is exactly the excess the held adjustment's variance factor already carried, counted on three degrees of freedom instead of fifteen, and it catches a disturbed monument 79 times in a hundred where the held report catches it 49. Give the fit a scale and it hides 98 per cent of that excess, and lends the control's error to every length.
A second clause decides the half that was already decided
A specification written as a band admits forty-two maps keeping between 37.0 and 67.5 per cent of a region. Adding a tie-break — among those, take the one whose worst departure is least — is a linear programme, and it works: it makes each of the forty-two definite. It also merges three neighbours at the bottom into one, leaves 29.1 of the 30.5 points of spread in area kept, and names a single map only at the tolerance where every map keeps everything and the first clause has stopped asking for anything at all.
Naming the control station that moved takes a fifth
A local survey tied to distant control can tell that a monument has moved, and with three control stations it cannot say which. Each station gives the local lines one number they know well — its distance — and a shift of the whole control uses up two of them, so three stations leave one and four leave two. Five evenly spaced name a station moved five centimetres 83 times in a hundred; three name it 20. Better lines cannot stand in for the missing stations, and every station added also exposes more of the control's own published error.
One cut belongs to the island that would cost most alone
A sea with two islands pays a few per cent for being drawn whole by an angle-keeping map, and one cut buys back part of it. Which island to cut is a design choice that leaves the ground exactly where it was, and it is not decided by size: across twenty arrangements of a large island and a small one, the cut worth most goes to the island that would cost the sea most if it were alone in it — in eighteen of twenty, the other two near-ties — which is usually the more central one. A cut joining the two islands buys almost nothing, and the better cut is nearly always the longer.
A residual cannot count the pieces of a map
A compiled map does not say how many sheets went into it, and trying every count and keeping the best fit cannot tell. Seven pieces always fit better than two, because they contain them. A criterion that charges ln N for each number a piece spends recovers the true count in every trial, for one copied stretch and for two, until the digitising error reaches a thousandth of the map. AIC charges two and takes every piece it is offered. And an eased copy defeats the right criterion the other way: it is not a projection, so BIC buys pieces to explain it, three seams, then four, then six.
The second cut goes across the sea
With two islands and one cut, the island that would cost the sea most alone names the cut. With three islands and two cuts, the obvious rule — cut the two dearest — names the best pair in only 23 of 40 seas, and loses up to 23 points of what three cuts would buy. It fails when the two dearest sit on the same side of the sea, where their cuts do each other's work. The best pair keeps the dearest island in 37 of 40; its partner is the island across the centre from it, which is right in 32 of 40 and never costs more than 12 points. A cut joining two islands is never worth a second cut to the sea.
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.
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.
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.
Named alongside it
The objects these essays reach for when they reach for this one.
VerificationConstraintLeast-squaresConformalityResidualCovarianceEmbeddingPurposeSeamEqual-areaEstimatorInvariant