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

Series — page 2

A field says what an essay is about. A series follows one idea essay by essay — from the question that introduces it to the one that assumes all the others.
The same address length, a tenth of the area. Every cell of a lon/lat quadtree at level 4 carries an identifier of the same length. The heavy curve is each cell's area as a fraction of the largest, against its latitude: a polar cell is 10.2 times smaller than an equatorial one. The light curve is the inverse of the cell's aspect ratio, which falls from 1.00 near the equator to 0.10 at the top — the cells stop being anything like square long before they stop being usable.

Cells

  1. 1 An address is an area
  2. 2 Hexagons cannot tile the sphere
  3. 3 A cell system trades area for shape
  4. 4 A query is a disc, and a disc is not a cell
  5. 5 The address is a curve through the sphere
  6. +7 more
12 essays · applied 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. +7 more
12 essays · impossibility A great circle and the ruled line, on Mercator. The shorter of the two routes is the curved one. The great circle between the two marked points is drawn against the straight line a ruler would give between them on Mercator; over 70° of arc the curve departs from the ruled line by 11.9 per cent of the chord. The flexion at the midpoint, which is the rate the image turns per radian of arc, is 0.788; the angular deformation there is 0.00°.

Flexion

  1. 1 Tissot stops at the first derivative
  2. 2 Bending and stretching are one failure
  3. 3 The second derivative has its own ranking
  4. 4 The second derivative over a region
  5. 5 A local model has an order
  6. +5 more
10 essays · distortion Two lines of position replace two circles thousands of kilometres across. A ship at 50° N 20° W observes two bodies at altitudes of 45.60° and 42.69°, which place it on two circles 4,938 and 5,261 km in radius, drawn faint. From an assumed position 150 km to the north-east, each body's azimuth is laid off (dashed) and its intercept marked — 12.2 km and 111 km — and a line of position drawn perpendicular to the azimuth through that point. On Mercator the two lines cross 3.28 km from the ship: each line is only its circle's tangent at the intercept, and by the time the two lines meet they have left their circles.

Position fix

  1. 1 A line of position is Newton's method, but only on a conformal chart
  2. 2 A cocked hat holds the ship one time in four
  3. 3 The residual reports the error the fix was immune to
  4. 4 What the extra unknown costs where nothing can see it
  5. 5 A confidence ellipse is honest only where the sheet keeps angles
  6. +4 more
9 essays · paths What the 5th decimal place of a coordinate is worth on the ground. A latitude written to 5 decimal places steps 1.112 m north for one unit in its last digit, at every latitude, because the meridian does not care where it is measured. A longitude written to the same 5 places steps 1.112 m east on the equator and 0.097 m at 85°, because a degree of longitude is a degree of a circle whose radius is R cos φ. The same written precision means two different distances at the same point, and a different pair at every other.

Precision

  1. 1 A coordinate is a number with a width
  2. 2 An error ellipse is an indicatrix
  3. 3 The average of noisy positions moves
  4. 4 The difference of two coordinates
  5. 5 The error ellipse is not an ellipse
  6. +4 more
9 essays · distortion 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 essays · 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 essays · 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 essays · distortion 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 essays · distortion 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 essays · 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 essays · distortion

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