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 5 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 previous rung could only bound the conformal seam's corner 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 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.

Closed formConformalityVerificationConventionConvergence rateEqual-areaEstimatorNumerical integrationTopologyTrade-offAngular deformationAreal factor

All concepts