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

Stereographic — where it appears

The azimuthal projection whose radial function is 2 tan(ρ/2), which makes it conformal everywhere and sends circles to circles. It is the chart in which a sphere's coordinate becomes a complex number, which is why analytic functions of it are conformal maps of the sphere.

Named by 10 essays across 3 fields — each of them below, with the objects they name alongside it.

The world after a map of degree 2. The graticule of the sphere, sent through the square of the stereographic coordinate and then drawn in Mollweide. Every parallel is still a closed curve and every meridian still runs pole to pole, and the whole world has been wrapped round the sphere 2 times: the 2 meridians that used to be 180° apart now lie on top of one another. Nothing has been torn. The degree measured by integrating the area the map sweeps is 2.0004.

How many times, not whether

Three rungs of this ladder answer yes or no and have no other kind of answer. The degree is the first quantity here that counts: a continuous map of the sphere to itself covers it a whole number of times, injectivity forces that number to ±1, and the number is recoverable three ways — by counting preimages, by integrating swept area, and from the rate at which the preimages coalesce.

impossibility · Topology One construction, one number. A ray from a point on the equatorial plane, through the sphere, onto a tangent cylinder. Where the centre sits is the whole of the family: on the axis it gives tan φ, at the far side of the sphere 2 tan(φ/2), and at infinity sin φ. The same expression with angular distance in place of latitude, and a tangent plane in place of a cylinder, gives the gnomonic, the stereographic and the orthographic. The surface decides whether the answer is read as a height or as a radius and nothing else.

The developable surface was never necessary

Eight essays on this ladder start from cylinders, cones and planes, and none of them has asked which named projection is actually a projection onto one. Fitting a one-parameter perspective construction to eleven of them: six come back exact, and the three landmarks are the gnomonic at h = 0, the stereographic at h = 1 and the orthographic at h = ∞.

families · Families The whole sphere, in two sheets. Two stereographic charts, one centred on each pole, each carried to 100° from its own centre. Two caps cover a sphere exactly when each reaches 90°, so this is the minimal atlas with a little overlap, and the heavy circle in each is the equator — the one curve both sheets contain. Nothing smaller works: one chart cannot cover the sphere at all, which is what this field's first rung proves. What the counting also fixes is a price: the worst point of any two-chart atlas is at 90° from a centre, where a conformal chart's areal factor is exactly 6 and an equal-area one's angular deformation is 49.07°.

Two charts are enough, and one is not

The topological minimum for an atlas of the sphere is two sheets, and the counting fixes a price nobody chose: the worst point of any two-chart atlas is 90° from a chart's centre, where a conformal chart's areal factor is exactly 4 and an equal-area one's angular deformation is 38.94°. A national series has a hundred and twenty thousand sheets, and a hundred and twenty thousand minus two of them are bought by accuracy.

impossibility · Topology One of these is a corner. The corner reported at the exact conformal seam, and the corner reported at a cube's gnomonic seam fifteen degrees along its edge, each measured over a shrinking arc. The gnomonic's is 21.4572° at every span from twenty-four degrees down to three — the same number to four decimals — because it is a corner. The conformal one halves whenever the span does, fitted exponent 0.990, because a tangent read from a finite chord of a curved image departs from the true tangent in proportion to the chord. It is not a corner; it is the instrument.

The exact map says the seam is smooth

The conformal seam's corner could only be bounded at about two degrees, because the series it was fitted with holds its boundary condition to three parts in a thousand. An exact map exists — the stereographic projection composed with ∫dt/√(1 − t⁴) — and it settles it: the corner falls in exact proportion to the arc the tangent is read over, fitted exponent 0.99, while the gnomonic's 21.4572° is the same to four decimals at every span.

families · Polyhedral 50° S to 40° S, drawn conformally without a cut and with one. The band of latitude from 50° S to 40° S, drawn two ways that both keep every angle. On the left it is not cut: the only such map with least scale variation is the stereographic from the south pole, the ring closes up, and its scale varies by a factor of 1.0750 across the band. On the right it is cut along one meridian and opened into a conformal conic with cone constant 0.7080: the ring spans only 254.9° and leaves a gap of 105.1°, and its scale varies by 1.0038. Not cutting costs a factor of 1.071 in scale variation.

A ring can be drawn whole, and only one way

A band of latitude has a hole in it, and a region with a hole has a number no angle-keeping map can change. Draw the band without cutting it and that number forces the map: the stereographic from one pole is the least-varying picture there is. One cut along a meridian frees the cone constant, and the scale variation the cut buys back runs from nothing on a polar cap to a factor of 2.13 between the tropics.

impossibility · Topology A sea with two islands in it, and the three circles it already is. A stated sea 60° across with two islands 12° across, their centres 12° from its own, drawn on the globe and drawn again under the stereographic projection centred on the sea. Stereographic projection carries every circle on the sphere to a circle in the plane, so the flat picture is bounded by three exact circles with no fitting and no iteration: the sea's edge at radius 0.5359, and the two islands at radius 0.1060 with their centres 0.2108 from the origin. Normalised, the region's three conformal invariants are 0.2361, 0.6848, 0.1221 — three numbers, which is what a region with two holes carries and no angle-keeping map can change.

Two islands are three numbers, and the proof does not survive them

A sea with one island at its centre has one conformal invariant and a forced best map. Put two islands in it and there are three invariants, readable in closed form because stereographic projection carries circles to circles — and the forcing is gone. The theorem that picked the map needed the region to be unchanged by rotation, not to have one hole, and a single harmonic term two hundredths in size beats the stereographic by 0.19 per cent the moment an island moves off centre.

impossibility · Topology The same two islands, and the cut that buys most is the central one's. A sea 60° across with a large island A, 12° across, and a small one B, 6° across, on opposite sides of its centre, drawn in the plane the stereographic makes of it. Each dashed line is one way of spending a single cut — from A to the open sea, from B to the open sea, or from A to B — and the solid line is the one that buys most, as a share of what two cuts buy. Left, A 18° from the centre and B 6°: A 9% · B 78% · between 21%. Right, A 6° and B 18°: A 81% · B 12% · between 11%. The better cut is the central island's in both, whichever island is large, and a cut joining the islands buys 21% and 11%.

One cut belongs to the island that would cost most alone

A sea with two islands pays a few per cent for being drawn whole by an angle-keeping map, and one cut buys back part of it. Which island to cut is a design choice that leaves the ground exactly where it was, and it is not decided by size: across twenty arrangements of a large island and a small one, the cut worth most goes to the island that would cost the sea most if it were alone in it — in eighteen of twenty, the other two near-ties — which is usually the more central one. A cut joining the two islands buys almost nothing, and the better cut is nearly always the longer.

impossibility · Topology The two islands dearest alone are the wrong pair to cut when they sit on the same side. One of forty three-island seas, 60° across, drawn in the plane the stereographic makes of it; beside each island, what it would cost the sea if it were alone in it — P 0.97%, Q 0.72%, R 1.68%. Left: the two dearest islands, P and R, cut to the open sea, which buys 75% of what three cuts buy. Right: the best pair, Q and R, which buys 97%. The dearest islands are 69° apart as seen from the centre, and their single cuts overlap by 40 points.

The second cut goes across the sea

With two islands and one cut, the island that would cost the sea most alone names the cut. With three islands and two cuts, the obvious rule — cut the two dearest — names the best pair in only 23 of 40 seas, and loses up to 23 points of what three cuts would buy. It fails when the two dearest sit on the same side of the sea, where their cuts do each other's work. The best pair keeps the dearest island in 37 of 40; its partner is the island across the centre from it, which is right in 32 of 40 and never costs more than 12 points. A cut joining two islands is never worth a second cut to the sea.

impossibility · Topology With five islands the two dearest are still the wrong pair when they share a bearing. One of 20 five-island seas, 60° across, drawn in the plane the stereographic makes of it; below, each island's price to the sea alone — P 0.73%, Q 1.30%, R 0.36%, S 1.96%, T 2.74%. Left: the two dearest, S, T, cut to the open sea, which buys 77% of what five cuts buy. Right: the best pair, Q, T, which buys 100%; the dearest island and then the one farthest from it in bearing — Q, T — buy 100%. One cut buys 77%, three 100%, four 100%.

A sea with five islands is seldom worth five cuts

With three islands in a sea, the second cut went across the sea from the first, and the number of cuts worth spending looked like the number of directions the islands point. With four and five islands that survives in part and breaks in part. Choosing the dearest island and then, each time, the island farthest in bearing from every island already cut names the best pair of cuts in 28 seas of 40, where the two dearest name it in 16 — and it fails at three cuts of five, because the best three cuts often do not contain the best two. The dearest island is no longer reliably among the best pair. And 39 seas of 40 are worth fewer cuts than they have islands: the count follows the widest gap in the islands' bearings more closely than it follows how many islands there are.

impossibility · Topology Three ways to cover a sphere with 900 points. The three samplers this collection's numbers are computed with, each with about 900 points, drawn on the Mollweide projection so that equal areas on the sphere are equal areas on the page and the crowding is the samplers' rather than the map's. The graticule piles points at the poles; the equal-area rings space them evenly by area and unevenly by distance; the Fibonacci lattice trades a little of each. None of them is equal-area, and no finite set is.

There is no equal-area lattice on a sphere

Every number in this collection is an average over a point set, and the three point sets available all fail to be equal-area in different ways. The equal-area ring sampler this site has used since its early essays is the one whose outermost ring sits half a step inside the rim — which is how a measured scale spread once came in 3.42 parts in a thousand below a proved bound.

distortion · Sampling

Named alongside it

The objects these essays reach for when they reach for this one.

ConformalityVerificationTopologyInvariantSeamChebyshev's criterionScale factorDegrees of freedomMinimaxPurposeClosed formCone constant

All concepts