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

Composite method — where it appears

A projection assembled from more than one construction, joined along stated parallels. Goode's homolosine is the standard example, using the sinusoidal near the equator and Mollweide beyond it, and the join is continuous in position and not in derivative.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

London to Tokyo, seen four ways. The 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.

The gnomonic companion

One projection turns every great circle into a straight line, and it is the only one that does. It shows less than half the sphere, distorts enormously, and was indispensable for three centuries because of that single exact property.

paths · Paths London to Tokyo in three straight legs. The great circle, and the route a plan of three constant-heading legs actually follows between waypoints on it. The two touch at the waypoints and part between them by up to 456 km, and the flown route is 253 km longer than the direct one. The headings are 50°, 109°, 149°. 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 Orthographic the rhumb line departs from straight by 2.4e-1 of its own length.

Flying a curve in straight legs

Nobody steers a great circle, because a great circle requires the heading to change continuously. What is actually flown is a handful of constant-heading legs between waypoints on it, and the gap between plan and curve falls as the square of the number of legs.

paths · Paths The same composite with a strip spliced in, and the meridians no longer turn a corner. Goode's composite redrawn with an equal-area strip of 8° of latitude spliced between the sinusoidal and the Mollweide cap, shaded. The strip's parallel half-length is the unique cubic matching the sinusoidal's value and slope at 36.74° and the Mollweide's at 44.74°, so every meridian arrives and leaves along one direction and the 11.2° corner the plain join turns at 60° of longitude is gone. Nothing else about the map is changed: the cap is the unmodified Mollweide, and the composite is equal-area throughout.

An equal-area strip removes the corner and charges nothing for it

Goode's seam matches three quantities and breaks the fourth: every meridian but the central one turns a corner there. A third map spliced between the two removes it — four conditions on one function, satisfied uniquely by a cubic — and the expected price does not arrive. The spliced composite carries less angular deformation across the seam than the plain join, not more: 34.7° against 37.7° at sixty degrees of longitude. What it does leave is the same defect one derivative up, a jump in curvature falling as the reciprocal of the strip's width.

families · Families Goode's composite with a strip of a generalised Eckert II, whose meridians are straight. The sinusoidal to 32.28°, a strip of the generalised Eckert II projection with r = 0.8311 from there to 49.27°, and the unscaled Mollweide beyond, both hemispheres; the strip's edges are dashed. Inside the strip every meridian is a straight line, as Eckert II's are, and it leaves the sinusoidal's meridian and joins the Mollweide's along their own tangents, so the 11.9° corner Goode's join turns at 180° of longitude is gone. The composite is equal-area throughout, and its strip is 17.0° wide and centred at 40.78°, four hundredths of a degree from Goode's own join.

Only Eckert II's family can be the strip, and its meridians are straight

The cubic that smooths Goode's seam is four conditions on the length of a parallel, and equal area turns them into four conditions on how the parallels are spaced. The sinusoidal's spacing is flat, so a strip can leave it only where its own spacing is momentarily flat — and of eleven named equal-area maps, only Eckert II's spacing ever turns over. One member of its family meets all four conditions exactly, over seventeen degrees centred four hundredths of a degree from Goode's join, with every meridian straight. The cubic over the same span is the same map to four parts in a hundred thousand.

families · Families

Named alongside it

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

ConstraintContinuityDegrees of freedomEqual-areaGoode homolosineGreat circleMercatorNavigationPseudocylindricalSinusoidalVerificationAngular deformation

All concepts