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

Parameterisation — where it appears

The choice of coordinates on a surface, which decides h and k and no invariant, and which on a triaxial body cannot be made orthogonal. Only the invariants describe the map, which is checked here by swapping the output axes and requiring the invariants to be unmoved and h and k to change.

Named by 6 essays across 4 fields — each of them below, with the objects they name alongside it.

The indicatrix at 30°, 40° on Gall–Peters. A circle on the sphere becomes this ellipse on the map. Its semi-axes are the two principal scale factors, a = 1.083 and b = 0.923; their product is the areal factor 1.000; and the maximum angular deformation is 9.16°. h and k are shown too, and depend on the coordinates rather than on the map.

What survives a change of coordinates

The scale along the meridian is a property of the map and the grid together. The principal scale factors are a property of the map alone. Only the second kind describes the projection, and the two are routinely quoted as though they were the same thing.

distortion · Tissot The same map on four differently prepared pages. Mollweide at 30°E 20°N, then the same projection with a rotation and a magnification applied to the page, then with one axis stretched by 1.6, then with a shear of 0.5. The similarity changes nothing: flexion, skewness, ω and the anisotropy are identical to every printed figure, and only the last column — the same turning measured per unit of page arc rather than per unit of ground arc — moves, by exactly the magnification. The stretch and the shear change all of them, and flexion by 35 per cent.

The second derivative is not an invariant

The first-order ladder established which quantities survive a change of coordinates and which are artefacts of the parameterisation. Asked of the second order, the answer is that flexion survives a rotation and a magnification of the page exactly, and survives nothing else: a stretch of 1.6 in one axis moves it by eleven per cent, a shear of 0.5 by thirty-five, and a shear of 3 leaves a ranking of eight world maps with a rank correlation of −0.07 to the one it started with.

distortion · Flexion 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 parameters are not reproducible and the map is. The three-parameter aspect search run at three grid resolutions and compared with the finest, twice over. Compared on the numbers it returns, the answers are 65° of pole apart. Compared on what they do to the region — the root-mean-square difference in angular deformation at every sample — they are 0.29° apart, against a map whose own deformation over that region averages about a degree. The disagreement recorded as a shortfall is a disagreement about coordinates for one map.

Report the map, not the parameters

The previous rung found the aspect search returning the same score to 2.3 per cent from poles sixty degrees of latitude apart, and recorded that as a shortfall: the answer was not reproducible. The shortfall assumed the disagreeing triples make disagreeing maps. They do not — the three answers agree on the distortion field to a quarter of the deformation the map already has.

choosing · Choosing Vesta, with its own coordinate lines. Vesta drawn in an orthographic projection — every figure here is a projection, including this one — with the parametric coordinate lines its published coordinates are written in. Its three semi-axes are 286.3, 278.6 and 223.2 km, so the equator is an ellipse rather than a circle and the body has no axis of revolution to define a latitude against. The surface's own radius runs from 223.2 to 286.2 km, a ratio of 1.282, and the outward normal departs from the direction out of the centre by up to 14.10°.

A map of a body with three axes

Every projection on this site is written in a latitude, and a body with three unequal axes has none: the coordinate lines are not perpendicular, Lambert's equal-area formula spreads areas by 28 per cent on Vesta, and the equal-area map that does work has a different one on every meridian.

datums · Bodies A section through a body with a neck, and the rays that leave it twice. An equatorial section of a stated contact binary — two lobes of radius 1 centred at ±1.5, joined by a neck of radius 0.35, with the origin in the neck. The lines are rays from the origin and the marks are where each one crosses the surface. two of the 25 drawn cross more than once, so along those directions there is no such thing as "the" radius, and a longitude and a latitude do not name a place.

A ray from the centre hits the surface twice

Eleven rungs map bodies that are lumpy, triaxial and turning at a drifting rate, and every one assumes the surface is star-shaped about the centre — which is what makes a longitude and a latitude a coordinate at all. A contact binary is not: on a stated body with a neck a third of a lobe wide, 10.9 per cent of the sky has no single radius, and the shape model everybody publishes fills the neck in and adds 1.67 per cent of the volume.

datums · Bodies

Named alongside it

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

Angular deformationAspectClosed formCoordinate independenceDegeneracyInvariantPlanetary datumPrincipal scale factorsTriaxial ellipsoidAffine transformationAnisotropyAuthalic latitude

All concepts