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 generators, each rendered at its defaults.

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 p4 plate and the p6m plate 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 19 of them so far.

cell-shapes

One 20° × 10° cell at 60°, under four projectionsThe same patch of the sphere, drawn by four projections and scaled to fit. The shapes differ and so do the areas: the figure under each panel is the areal inflation relative to the same cell at the equator, so 1.00 means the projection treats the two fairly.Mercator5.62× areaGall–Peters1.00× areaMollweide1.00× areaEquirectangular2.36× areathe cell at 60°scaled to fit, areas as stated

conformality-audit

Every projection in the library, measured against both propertiesMaximum 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.10⁻⁶10⁻⁴10⁻²110010⁻¹¹10⁻⁸10⁻⁵10⁻²1010000maximum angular deformation / degreesmaximum areal errornothing here, everMercatorStereographicWeb MercatorGall–PetersMollweideEquirectangulartolerances: 0.0001° and 0.00000121 projections measured

curvature-field

Gaussian curvature across a torusCurvature along a cross-section of a torus. curvature of both signs — positive outside, negative inside, zero on two circles. Where the curve crosses zero the surface is momentarily flat in the intrinsic sense, and a strip along that circle could be unrolled without stretching.0+4.2position around the cross-sectionpositive: dome · negative: saddleon a torus

developable

A cylinder unrolls exactly; a sphere does notBoth pictures show the same grid. On the left it is wrapped round a cylinder, on the right it is laid flat, and every distance in the grid is the same in both — the circumference is 2π and so is the width of the rectangle, checked to 10⁻⁹. This is possible because a cylinder has zero Gaussian curvature. No corresponding picture exists for a sphere.wrappedunrolled — every distance unchanged→K = 0 on bothcircumference 2π = width 2π

distortion-compare

Angular deformation against latitude, four projectionsThe same quantity for mercator, gallPeters, mollweide, winkelTripel, 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.0°20°40°60°80°MercatorGall–PetersMollweideWinkellatitudeangular deformationalong a meridian

distortion-profile

How Mercator distorts, by latitudeAngular deformation and areal scale plotted against latitude along the meridian at 0°. On Mercator the angular deformation reaches 0.0° and the areal factor reaches 91.5.-60°-30°0°30°60°angular deformation, to 2°areal factor, to 91.5×latitudetwo independent distortionsalong the 0° meridian

excess-chart

What a constant compass bearing costsThe extra distance of the rhumb line over the great circle, for five journeys. It runs from almost nothing on a nearly north–south route to 28 per cent on a high-latitude east–west one. Every number is computed from the two distance formulae rather than quoted.Cape Town – London+0.0%3 kmNew York – Madrid+3.0%171 kmSydney – Santiago+12.7%1437 kmLondon – Tokyo+18.2%1737 kmAnchorage – London+28.2%2031 kmextra distance over the shortest routegreat circle vs rhumb linecomputed, not quoted

family-table

Construction against propertyEvery 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.conformalequal-areacompromiseclaim failscylindricalMercatorMercatorLambertGall–PetersBehrmannEquirectangularMillerWebpseudocylindricalSinusoidalMollweideEckertRobinsonpseudoazimuthalHammerWinkelazimuthalStereographicLambertOrthographicGnomonicAzimuthalconicLambertAlbersrows: how it is builtcolumns: what it preserves

indicatrix-compare

The same point at 30°, 55° under four projectionsOne circle on the sphere, four projections, four ellipses. A conformal projection keeps it circular and lets the area run; an equal-area projection keeps the area and lets the shape go. Nothing keeps both, and the dashed circle shows what keeping both would look like.Mercatorω 0°area 3.04×Gall–Petersω 24°area 1.00×Mollweideω 21°area 1.00×Winkelω 11°area 1.14×dashed: undistortedscaled to fit

indicatrix-detail

The indicatrix at 45°, 55° on MercatorA 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.a = 1.74b = 1.74an infinitesimal circle, projectedmeasured at this pointh1.7434scale along the meridiank1.7434scale along the parallela1.7434larger principal scaleb1.7434smaller principal scalea·b3.0396areal scale factorω0.00°maximum angular deformationdashed: undistorteddrawn in Mercator

indicatrix-map

Tissot's indicatrix across MercatorA 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.dashed: an undistorted circledrawn in Mercator

inflation-chart

How much each projection inflates a cell, by latitudeFive patches of the sphere, each 20° by 10°, and the factor by which each projection enlarges them relative to the equatorial one. An equal-area projection sits flat on 1. Mercator reaches 15.4× at 70°, which is the mechanism behind every complaint about the size of Greenland.equator23°45°60°70°1×MercatorEquirectangularMillerGall–Peterslatitude of the cell1× means treated fairlyrelative to the equator

projection-grid

6 projections of the same sphereThe same graticule under equirectangular, mercator, mollweide, sinusoidal, robinson, winkelTripel. Every one of them is a faithful drawing of the same object and no two agree, because each has chosen a different thing to preserve.EquirectangularMercatorMollweideSinusoidalRobinsonWinkel tripelsame sphere, same graticuleno two agree

projection-map

MollweideThe 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.equal-areadrawn in Mollweide

route-map

New York to Madrid on MercatorTwo 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.New YorkMadridsolid: shortest · dashed: constant bearing+3.0% for the rhumbdrawn in Mercator

route-projections

London to Tokyo, seen four waysThe same two routes on four projections. The gnomonic projection renders every great circle as an exactly straight line, which is what it is for; Mercator renders every rhumb line straight instead. Neither path changed — only the map did.Mercatorgreat circle bows 3.7e-1Gnomonicgreat circle bows not at allEquirectangulargreat circle bows 2.0e-1Orthographicgreat circle bows 9.0e-2solid: shortest · dashed: constant bearingone pair of routes

standard-parallel

What a standard parallel buysThree equal-area cylindrical projections differing only in where they are exact. Each has zero angular deformation at its own standard parallel and grows away from it in both directions. Choosing a standard parallel is choosing which latitudes to treat well, and there is no choice that treats them all well.0°20°40°60°80°exact at equatorexact at 30°exact at 45°latitudeangular deformationall three are equal-area

surface-curvature

Six surfaces and their Gaussian curvatureCurvature 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; the cylinder and the cone can, and the sphere cannot.planeK = 0 — unrolls flatcylinderK = 0 — unrolls flatconeK = 0 — unrolls flatsphereK = 1.00torusK = 1.79pseudosphereK = -1.00K computed at each centretwo routes, one answer

world-cells

Five identical cells on MercatorFive patches, each 20° of longitude by 10° of latitude. On the sphere the higher ones are genuinely smaller, because the meridians converge. On Mercator the cell at 70° comes out 15.4 times larger than the equatorial one relative to its true size.1.0×1.3×2.4×5.6×15.4×each cell is 20° × 10°drawn in Mercator