Cartographic Projection
every map is wrong, and by exactly how much
How this site is made

The figure library

Every picture here is generated from code at build time. This page lists the generator families, how many figures each one draws, and how many essays call it.

No figure on this site is a drawing that was made once and saved. Each one is a function: it takes parameters and returns SVG, so the same generator produces the zone at the equator and the zone at 60° north without either being redrawn.

That is the reason the collection can keep growing without the illustrations drifting apart. A generator is written once, checked once, and every essay that calls it inherits the same line weights, the same colour roles, and the same behaviour in dark mode. There are 29 families so far, drawing 253 distinct figures between them across 384 essays.

A family is one object and each of its figures is one question asked of it — the same generator drawing the map, then the same map's distortion, then the quantity summed over a region. That is what keeps the count of generators from growing with the count of essays, and it is the reason an essay here can develop one picture instead of touring several. Each family has a page of its own listing what it draws and every essay that calls it, which is also the blast radius of changing it.

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-figure

9 essays 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-figure

8 essays Tangent-warped cube cells, shaded by area. The tangent-warped cube at level 2, drawn on Mollweide with each cell shaded by its own measured area. The largest cell is 1.21 times the smallest. Every area is computed with the spherical polygon formula from the cell's own boundary, not from the scheme's intentions, and the shading is what the numbers say rather than what the mesh looks like.

cell-figure

5 figures · 13 essays The residual is curvature, and it is cubic. The worst trimetric residual inside a region, against the radius of that region, on logarithmic axes. The fitted slope is 2.99: doubling the region multiplies the residual by eight. That is the signature of a curvature effect on a distance — the relative error of flattening a patch grows as the square of its size, and a distance across the patch grows as its size, so the absolute error grows as the cube. At a radius of 2° the residual is 0.16 km; at 32° it is 636 km.

condition-figure

4 figures · 13 essays 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-figure

78 figures · 31 essays 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-figure

11 figures · 30 essays 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 six named here are Mercator, Web Mercator, Gall–Peters, Mollweide, Equirectangular, Stereographic.

distortion-figure

7 figures · 41 essays Five latitudes that are not the latitude, on WGS84. Each curve is the amount by which one auxiliary latitude falls below the geodetic latitude a coordinate actually means, in arcminutes. All five vanish at the equator and at the poles and peak near 45°, where the geocentric latitude is 11.55 arcminutes below the geodetic one — about 21.4 km on the ground. The curves never cross, which is forced by the algebra rather than by this ellipsoid's particular flattening.

ellipsoid-figure

13 figures · 53 essays 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.

field-figure

8 essays 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-figure

8 figures · 23 essays 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-fit

13 essays Tissot's indicatrix across Mercator. A small circle on the sphere, drawn where the projection puts it. The dashed circle behind each is what an undistorted map would show. On Mercator every ellipse is a circle — ω never exceeds 1.0e-6°, and the areal factor reaches 4.0.

indicatrix-figure

8 figures · 32 essays Five stations, ten distances, three spare. A braced quadrilateral with a centre point. Every distance between the corners and every distance to the centre is observed, 10 in all, each with a standard deviation of 8 mm. Holding one station and one bearing leaves 8 unknown coordinates, so the network has three degrees of freedom: three independent statements the observations make that could be contradicted. Everything the adjustment can tell anybody about the quality of the work comes out of those three.

network-figure

6 essays 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.

operation-figure

14 figures · 27 essays On a triaxial body, latitude depends on longitude. Walk round each body at a constant planetocentric latitude of 45° and watch the direction of the surface normal, which is what the planetographic latitude is. On Mars it does not move at all — that is what having an axis of revolution means. On Vesta it swings by 1.40° and on Phobos it swings by 6.40°. A body without an axis has no latitude that is a function of position alone, and every coordinate on it is a convention with a body-fixed frame attached.

planetary-figure

5 figures · 15 essays The sphere on an icosahedron, unfolded. A polyhedral map: the sphere projected face by face onto an icosahedron and the solid cut open along 11 of its 30 edges. The face projection here is the gnomonic, which draws every great circle as a straight line within a face, and the worst angular deformation inside a face is 6.6°. Each cut is a place where two pieces of the world that touch are drawn apart; each corner of the solid is a place where the surface has curvature and the map has an angle deficit.

polyhedral-figure

3 figures · 16 essays 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-figure

11 essays Mollweide. The graticule of the Mollweide projection at 30° of longitude and 15° of latitude. equal-area in an ellipse, at the cost of the corners. It is equal-area.

projection-figure

17 figures · 46 essays 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-figure

12 essays The same six projections, ranked over Europe and Chile. Each column orders the projections by Kavrayskiy's regional criterion — the root-mean-square departure of the two principal scales from unity, integrated over the region with the area element. The lines cross, which is the point: Lambert conformal conic leads over Europe and comes fifth over Chile. A table of projections ordered by distortion is a table about somebody's region.

region-figure

7 figures · 38 essays New York to Madrid on Mercator. Two routes. The great circle is 5768 km and is the shortest path on the sphere. The rhumb line holds a single compass bearing the whole way and is 5939 km — 171 km further, or 3.0 per cent. On Mercator the rhumb line departs from straight by 9.6e-16 of its own length.

route-figure

12 figures · 32 essays 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-figure

6 essays 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-figure

12 figures · 17 essays Flexion over Mollweide. The root-mean-square flexion over all directions, sampled at 375 points and drawn as a disc whose size is the value. It reaches 10.19 per radian of arc near 78°S 173°W. No first-order quantity — h, k, a, b, the areal factor or ω — carries any of this information.

secondorder-figure

4 figures · 10 essays

Every figure, by name

A family is one object; each figure below is one question asked of it, and each has a page of its own. 253 in total.

All 384 essaysFields31 laddersThreads50 tested claims713 named objectsSearch