Cartographic Projection
every map is wrong, and by exactly how much
Depth

Ladders — page 2

A field says what an essay is about. A ladder says what else there is to say about it — the distinct arguments that stand against one idea, from the one that introduces it to the one that assumes all the others.
Everywhere within 3,000 km of London, drawn in Lambert cylindrical. The set of places exactly 3,000 kilometres from London by the shortest route, projected point by point. On the ground it is a circle — every point of it is the same distance from the centre, in every direction. On this page the longest radius from the drawn centre is 4.17 times the shortest, so a reader with a ruler measures two different distances for one ground distance depending on which way the ruler points. The two extreme radii are drawn.

Reach

  1. 1 A circle of a distance is not a circle
  2. 2 Nearest of many is a partition
  3. 3 A corridor has a width the page cannot keep
  4. 4 A reach set with a cost that depends on direction
  5. 5 The set that can be reached is not the set that can reach
  6. +4 more
9 rungs · paths Four cities, met — the triangular, x first construction. Every cell drawn here holds the same area of ground and a different area of page: the map has been constructed so that its areal scale factor is the density it was handed, cell by cell, over a contrast of 25.99 to one. The residual against that target is 5.1e-6, measured from the map's own derivatives rather than from the construction. What it cost is the shape: the redistribution alone reaches 139.1° of angular deformation and averages 70.4°, on top of whatever the equal-area projection under it was already doing.

Cartogram

  1. 1 A map drawn to a density it was handed
  2. 2 Every density can be met and none is free
  3. 3 A map that meets its target can fold
  4. 4 Everything else on the page pays for the areas
  5. 5 The cheapest map that meets its areas
  6. +3 more
8 rungs · distortion A curve built to have dimension 1.2619. The generator replaces every segment with four of equal length at headings 0, +60.0°, −60.0° and 0. Closing the displacement fixes the length ratio at 0.33335, and four copies at that ratio give a dimension of exactly log 4 / log(2 + 2 cos θ) = 1.2619. Nothing here is measured yet: this is the construction the measurement will be checked against. Drawn at depth 5, which is 1024 segments, with the second-level shape shown faint beneath it.

Generalise

  1. 1 A line has a length only at a scale
  2. 2 Simplification does not commute with the projection
  3. 3 A tolerance is a promise about the picture
  4. 4 A boundary that two features share
  5. 5 A thousand features are wrong in the same direction
  6. +3 more
8 rungs · applied Tissot's ellipse and the one that governs gradients, at 20°E 48°N. The solid ellipse is the image of a small circle — Tissot's indicatrix, semi-axes a and b. The dashed one is the image of a unit gradient, whose semi-axes are 1/b and 1/a because a gradient transforms by the inverse transpose of the Jacobian rather than by the Jacobian. Its long axis therefore lies where the indicatrix's short one does. On Mercator both are circles, so a gradient's direction survives; on Lambert cylindrical the two ellipses are the same shape turned through a right angle, so the worst direction for a gradient is the best direction for a shape.

Gradient

  1. 1 A slope is not a shape
  2. 2 The contour is right and the reading is wrong
  3. 3 Water runs downhill on the ground, not on the page
  4. 4 The slope of a field that was measured
  5. 5 The light comes from a page direction
  6. +3 more
8 rungs · distortion How each projection escapes being one to one. Eighteen projections, and the theorem allows no fourth column. A map of the whole sphere either leaves ground undrawn, or draws one place as a curve, or is cut so that one ground curve appears twice — and 11 of the eighteen do more than one of those. The three columns are a solid angle, a count of places and a page length, which is why they are three columns rather than one score: there is no rate at which a hemisphere converts into a pole.

Topology

  1. 1 No map of the whole sphere is one to one
  2. 2 North cannot be up everywhere
  3. 3 Two opposite places on the same spot
  4. 4 How many times, not whether
  5. 5 Four colours, and what a cut cannot do to them
  6. +2 more
7 rungs · impossibility The areal factor over 0° to 60° north, and the points it is measured at. Mercator's areal factor shaded over 0° to 60° north, from 1.0001 to 3.8473, with the 12 × 12 grid of cell centres a regional measurement uses drawn on top of it. The cross marks where the quantity is actually largest, found by a search that is allowed to leave the sample; the ring marks the largest value the sample contains. The grid's answer is 3.4639 and the real one is 4.0000, short by 13.40% — and the reason is visible in the picture, because no cell centre is ever on an edge.

Sampling

  1. 1 The worst point is not on the grid
  2. 2 A mean that does not exist can still be printed
  3. 3 There is no equal-area lattice on a sphere
  4. 4 A refinement that stops moving
  5. 5 Which of these numbers are the sampler's
  6. +1 more
6 rungs · distortion One sentence, three readings of it. "a straight line from the initial point on the Rio Grande to a point on the Colorado" — Treaty of Mesilla, 1853. Every curve here answers to those words. the geodesic, the rhumb line, straight on the sheet, drawn between the same two monuments on a conformal conic fitted to the segment itself. The widest pair, the geodesic against the rhumb line, are 7.53 kilometres apart at their worst and enclose 3,925 km². The shading is that ground. It is not an artefact of the drawing: the same figure of the 141st meridian shows one line, because on a meridian every one of these readings is the same curve.

Boundary

  1. 1 One sentence, and the ground between its readings
  2. 2 The line a commission can actually run
  3. 3 A meridian boundary moves when its datum does
  4. 4 An equidistance line belongs to a surface
  5. 5 A tripoint defined three times
5 rungs · practice The four places, and every distance between them. London, New York, Tokyo, Sydney on a Mollweide projection, with every great circle between a pair drawn. The six ground distances run from 5,570 km to 16,994 km, and they are the whole of the input to every figure in this ladder — no coordinate, no projection and no coastline enters any of them. The arcs are drawn only to say which pair each number belongs to; on this page they are curves, and on the ground they are the shortest routes.

Embedding

  1. 1 Four cities that cannot be drawn to scale
  2. 2 How wrong a flat picture has to be
  3. 3 The best flat picture is not a map
  4. 4 Five distances of six, and never more
  5. 5 The escape is not a dimension
5 rungs · impossibility a transform boundary: what the ground is doing to itself. The same 72 places as the velocity field, with the strain rate computed from the motion's own four partial derivatives rather than from its size. Each cross carries two principal rates: the long stroke is the greater extension, the barred one is shortening. The largest second invariant anywhere here is 241.0 nanostrain/yr, and the arithmetic is the same arithmetic that reads a projection's indicatrix. Drawn in Azimuthal equidistant centred on the window.

Strain

  1. 1 The ground has an indicatrix too
  2. 2 A velocity needs a frame and a strain rate does not
  3. 3 A grid stops fitting the ground it was laid on
  4. 4 The strain a map adds to the ground's
  5. 5 A vertical rate needs a height system
5 rungs · datums The same field, the same statistic, two pages. A single northern concentration — v = 10 + 90 exp(−d²/(25°)²), d the distance from 20°E 55°N — drawn twice, with the shading rule and the data identical. The ground's own area-weighted mean is 14.009. A reader weighting by the area actually on the page takes 17.490 off Mercator, which is +24.85%, and 14.009 off Gall–Peters, which is +0.00%. Neither map has misdrawn a single cell.

Thematic

  1. 1 A choropleth is read by area
  2. 2 A symbol has a size on the page and an area on the ground
  3. 3 A dot map's density is partly the projection's
  4. 4 The class breaks were computed on the page
4 rungs · distortion

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