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

Ladders

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.
Angular deformation against latitude, four projections. The same quantity for mercator, gallPeters, winkelTripel, 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. +16 more
21 rungs · choosing 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 rungs · 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 rungs · distortion 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. +10 more
15 rungs · 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. +10 more
15 rungs · 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 rungs · practice 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 rungs · 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 rungs · datums 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. +8 more
13 rungs · wrong 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 rungs · datums 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. +8 more
13 rungs · paths 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. +8 more
13 rungs · practice 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 rungs · applied 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 rungs · applied 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. +7 more
12 rungs · families 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. +6 more
11 rungs · choosing 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. +6 more
11 rungs · families 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 rungs · distortion 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. +4 more
9 rungs · wrong 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 rungs · distortion

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