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

Series

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.
Angular deformation against latitude, four projections. The same quantity for Mercator, Gall–Peters, Winkel tripel, Robinson, from the equator to 80°. A conformal projection sits flat on zero in the first of these plots and runs away in the second; an equal-area projection does the reverse. Nothing is flat in both.

Choosing

  1. 1 Every projection minimises something
  2. 1 Which projection is best
  3. 2 Compromise projections
  4. 3 Chebyshev's criterion
  5. 3 Giving up continuity
  6. +21 more
26 essays · choosing Construction against property. Every projection in the library, sorted by the shape it was notionally rolled from and by what it actually preserves. The families do not line up with the properties: cylindrical and pseudocylindrical and pseudoazimuthal and azimuthal and conic each contain projections of more than one kind, which is why the cylindrical–conic–azimuthal taxonomy answers a question nobody has. Three entries are picked out: Mercator, Albers equal-area conic, Orthographic.

Families

  1. 1 Cylinders, cones and planes
  2. 2 What a standard parallel buys
  3. 2 The aspect is a free choice
  4. 3 The projections that gave up being one thing
  5. 4 The conic is the whole family
  6. +20 more
25 essays · families London to Tokyo on Mercator. Two routes. The great circle is 9559 km and is the shortest path on the sphere. The rhumb line holds a single compass bearing the whole way and is 11296 km — 1737 km further, or 18.2 per cent. On Mercator the rhumb line departs from straight by 5.0e-9 of its own length.

Paths

  1. 1 The shortest route is not straight
  2. 2 Why Mercator exists
  3. 2 The gnomonic companion
  4. 3 The great-circle vertex
  5. 3 Geodesics on the ellipsoid, and why they are hard
  6. +19 more
24 essays · paths Six surfaces and their Gaussian curvature. Curvature computed at the centre of each surface, by both available routes — from the way the surface sits in space, and from distances measured inside it alone. The two agree, which is Gauss's theorem. A surface with K = 0 can be unrolled flat without stretching: here the plane, the cylinder and the cone do, and the sphere, the torus and the pseudosphere do not.

Curvature

  1. 1 No map is faithful
  2. 2 What can be unrolled
  3. 2 The trade-off is two lines
  4. 3 Total curvature and the scale rule
  5. 3 Measuring curvature from inside
  6. +16 more
21 essays · impossibility Why there are two latitudes, at 45°. A meridian section of an ellipsoid with the flattening exaggerated 12× — at the true value of 1/298 this outline would be indistinguishable from a circle. Geodetic latitude is the angle the surface normal makes with the equatorial plane; geocentric latitude is the angle the radius makes with it. The two lines meet the plane at different points and the angles differ by 11.55 arcminutes on the real WGS84 ellipsoid, which is about 21 km of ground distance.

Ellipsoid

  1. 1 The Earth is a sphere, and when it is not
  2. 2 Transverse Mercator and the series that computes it
  3. 2 Geodetic against geocentric latitude
  4. 3 UTM and the zone system
  5. 3 Datum shifts dwarf projection errors
  6. +16 more
21 essays · wrong Exact distances from two places, and from nowhere else. The two-point equidistant projection with London and Cape Town as its centres, 87.0° apart. The light circles are drawn in the map at radii of 30°, 60°, 90°, 120° about each centre; every one of them is a true distance circle on the sphere, to 1.8e-14 relative. Between two points that are not centres the drawn distance is wrong by up to 633 per cent.

Condition

  1. 1 A projection written as a condition
  2. 2 Three conditions are one too many
  3. 3 A map that cannot be read backwards
  4. 4 Solving for the map instead of choosing it
  5. 5 A map with no formula
  6. +15 more
20 essays · choosing Every candidate fitted to one map's graticule, ranked by what is left over. The map is drawn in Conformal conic over a region 40° tall centred at 45° north, and the projection is not told to the fit. Each candidate is evaluated at the same 121 graticule crossings, its own parameters are searched, the best plane similarity between its output and the picture is removed, and what remains is drawn as a proportion of the map's own width on a logarithmic scale. Conformal conic fits to 1.3e-10, which is the arithmetic's floor; the next candidate is 3.7e+7 times worse.

Identify

  1. 1 A map does not say what it is
  2. 2 Two projections that cannot be told apart
  3. 3 What a careless copy hides
  4. 4 When the answer is not in the library
  5. 5 A residual has more than one explanation
  6. +14 more
19 essays · wrong Four steps between a tape and a drawing. A slope distance of 76.895 km measured at 3.2° on ground 220 m above sea level, reduced to the British National Grid, with every step drawn on a logarithmic scale in millimetres. The slope reduction is the largest by a wide margin at 119.90 m and is the one everybody applies. The grid reduction is 30.23 m. The fourth bar is not a step at all: it is what using the levelled height where the height above the ellipsoid is wanted costs, with a geoid separation of 48.5 m — 583 mm, which is 1.9% of the grid reduction it sits beside and 29 times the tolerance the job closes to. Reading a correction's importance off its share of the chain is how it gets dropped.

Reduction

  1. 1 What a tape measures
  2. 2 A traverse must close
  3. 3 The two ways to spread a misclosure
  4. 3 Setting out runs the chain backwards
  5. 4 Two grids over the same ground
  6. +14 more
19 essays · practice 53° north, 2° west: four places one pair of numbers could be. The same two numbers, read as three datums and as the pair in the other order. The centre is the WGS84 reading; the marks are where the ground is if the numbers were measured against OSGB36, ED50, NAD27 instead, at 103 m, 138 m, 195 m. Every one of those is a plausible position — near enough to look like a survey difference and far enough to be a different field. Reversing the two numbers instead moves the point 7942 kilometres, and is the mistake nobody worries about because it is usually obvious.

Dataset

  1. 1 A coordinate without its system is not a location
  2. 2 A degree is not a unit of length
  3. 2 Computing an area needs a surface
  4. 3 A straight segment is a claim about a plane
  5. 3 The antimeridian is a cut in the numbers
  6. +12 more
17 essays · applied The indicatrix at 45°, 55° on Mercator. A circle on the sphere becomes this ellipse on the map. Its semi-axes are the two principal scale factors, a = 1.743 and b = 1.743; their product is the areal factor 3.040; and the maximum angular deformation is 0.00°. h and k are shown too, and depend on the coordinates rather than on the map.

Tissot

  1. 1 Tissot's indicatrix
  2. 2 What survives a change of coordinates
  3. 2 The two ways a map is wrong
  4. 2 Measuring instead of naming
  5. 3 Scale distortion is the third failure
  6. +11 more
16 essays · distortion Every projection in the library, measured against both properties. Maximum angular deformation across the bottom, maximum areal error up the side, both on logarithmic scales and both measured over several hundred sample points rather than taken from the projection's description. Conformal projections lie on the left edge, equal-area ones along the bottom, and the corner where both vanish is empty because a projection there would be an isometry of the sphere onto the plane. All 26 are plotted; the four named here are Mercator, Mercator (ellipsoidal), Web Mercator, Gall–Peters.

Audit

  1. 1 Web Mercator is not conformal
  2. 2 Mercator against Peters
  3. 2 The projection that shows true size
  4. 3 The plate carrée, the projection nobody chooses
  5. 4 Conformal does not mean the angles are right
  6. +10 more
15 essays · wrong A false origin moves every number and no geometry. British National Grid, drawn twice at the same scale. On the left the coordinates are measured from the projection's own origin, where the central meridian meets the true origin's parallel, and 52% of the country takes a negative easting or northing — the worst reaching -323 km. On the right the authority's published false origin of 400 km east and -100 km north has been applied, and none of them does. Every distance computed from the two sets of numbers agrees to the last bit a double has left after carrying six figures, and every bearing agrees exactly; the only thing that changed is that no coordinate carries a sign. The margins say the origin was fitted to the land rather than placed arbitrarily below it: 77 km spare in the west and 50 m in the south, on a grid 988 km tall.

Grid

  1. 1 A grid has an origin that is not there
  2. 2 What a grid is made of
  3. 2 The scale factor of a line
  4. 3 The scale factor was chosen
  5. 3 The units are part of the coordinate
  6. +10 more
15 essays · practice The icosahedron's faces, drawn on the sphere. The edges of the icosahedron projected radially onto the sphere, which is the partition of the world a polyhedral map uses: everything inside one spherical polygon is drawn on one flat face. Each face spans 25.8° from its centre to its own boundary, and the gnomonic map onto it reaches 6.6° of angular deformation at the corners. Drawn in an orthographic projection of the embedding, which is a map with its own distortion.

Polyhedral

  1. 1 The globe on a solid
  2. 2 The cut has to go somewhere
  3. 3 What a face can preserve
  4. 4 A conformal map onto a face
  5. 5 More faces, less distortion, more cutting
  6. +10 more
15 essays · families The same coordinate on four datums. One pair of numbers — 2.0° west, 54.5° north — read as a coordinate on OSGB36, ED50, NAD27 and on WGS84, and plotted where each reading puts the mark on the ground. The spread runs to 195 metres. The numbers are identical; only what they refer to differs.

Datum

  1. 1 What a coordinate refers to
  2. 2 The seven parameters, and what each one does
  3. 2 A datum is fitted to a region
  4. 3 Where a fit leaves residuals
  5. 3 Molodensky's shortcut
  6. +9 more
14 essays · datums Three surfaces, and the two heights between them. The ellipsoid, the geoid and the ground, with the two heights a coordinate can carry. Ellipsoidal height h is what a satellite fix returns and is measured from a surface defined by four constants. Orthometric height H is what a level and a staff measure and is referred to the geoid — the equipotential surface that best fits mean sea level. They differ by the separation N, drawn here as 45 metres because that is a stated input rather than a computed one: a geoid model is a data product with a truncation degree in it, and this site computes rather than downloads. The arithmetic h = H + N is exact whatever N is.

Height

  1. 1 Height above what?
  2. 2 The ellipsoid is a level surface
  3. 3 A levelled height is not a distance
  4. 3 The plumb line is not the normal
  5. 3 The ground is not the grid
  6. +9 more
14 essays · datums 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. +9 more
14 essays · paths Two latitudes for one point on Mars. A cross-section of Mars with its flattening exaggerated 6× so the two angles can be told apart. The line to the centre defines the planetocentric latitude and the outward normal defines the planetographic one; at 45° they differ by 0.338° on the real body, which is 20 kilometres along the surface. Both numbers are published for Mars, and a coordinate that does not say which it is is ambiguous by that much.

Bodies

  1. 1 A coordinate on another body
  2. 2 The same projection on a different body
  3. 3 A body that is not an ellipsoid
  4. 4 A map of a body with three axes
  5. 5 A body with no sea level
  6. +8 more
13 essays · datums 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 built on them — 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
  6. +8 more
13 essays · impossibility The tile pyramid, four levels down to quadkey 120. The world as a square, quartered three times. Each level's tile is exactly half the width of its parent, so a tile is four tiles at the next level and never needs resampling to serve one — the property the whole scheme rests on, and one that holds only because the projected world is square. Level 3 has 64 tiles at 19567.9 metres per pixel, which at 51.5° north is 12181.3 metres of ground per pixel rather than the number the scheme publishes.

Screen

  1. 1 A screen map is a pyramid of tiles
  2. 2 The scale of a screen map is not one number
  3. 2 A scale bar is right in one place
  4. 2 Zoom is a ladder
  5. 3 The square costs the poles
  6. +8 more
13 essays · applied 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
  6. +7 more
12 essays · practice

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