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

Triaxial ellipsoid — where it appears

A body with three unequal axes, on which latitude depends on longitude and no auxiliary latitude exists. Its coordinate lines cannot be made orthogonal, its area needs quadrature, and no substitution turns a spherical equal-area formula into an equal-area one on it.

Named by 8 essays across 3 fields — each of them below, with the objects they name alongside it.

How much curvature varies from place to place, body by body. The Gaussian curvature along a meridian, each body against its own smallest value, so the curves are comparable. Written from the bodies actually drawn: Earth varies by 1.0135, Mars varies by 1.0239, Vesta varies by 2.71, Phobos varies by 4.16, from the flattest place on each to the sharpest. A surface of constant curvature is a sphere and nothing else is, so the impossibility argument this field is built on has a local version on every real body: how flat a patch is depends on where the patch is.

Curvature that varies from place to place

The impossibility this site is built on was argued on a sphere, where the curvature is one number. On the Earth it varies by 1.35 per cent, on Jupiter by 31, on Vesta by a factor of 2.7 — and on a body with three axes it varies along a parallel, which no formula in latitude can express.

impossibility · Curvature The equator, with its ellipticity exaggerated forty thousand times. The dashed circle is the equator every projection formula on this site assumes. The solid curve is the equator satellite geodesy reports, drawn with its departure multiplied by 40,000 so that seventy metres on a six-thousand-kilometre radius can be seen at all. The long axis is at 14.9° west and the short one ninety degrees from it, and the difference between them is 70.0 metres — a real quantity, about the height of a twenty-storey building, on a body every geodetic computation treats as a surface of revolution.

The equator is not a circle either

Eleven rungs price what pretending the Earth is a sphere costs, and every one of them replaces the sphere with a surface of revolution — a body whose equator is a circle. It is not. The two equatorial radii differ by seventy metres, the two surfaces part by thirty-five, and the auxiliary latitudes every ellipsoidal formula is written in stop existing.

wrong · Ellipsoid On a triaxial body, latitude depends on longitude. Walk round each body at a constant planetocentric latitude of 45° and watch the direction of the surface normal, which is what the planetographic latitude is. On Mars it does not move at all — that is what having an axis of revolution means. On Vesta it swings by 1.40° and on Phobos it swings by 6.40°. A body without an axis has no latitude that is a function of position alone, and every coordinate on it is a convention with a body-fixed frame attached.

A body that is not an ellipsoid

Vesta's three axes are 286.3, 278.6 and 223.2 kilometres, all different, so it has no axis of revolution — and on such a body the planetographic latitude of a point at 45° planetocentric swings by 1.4° as one walks round it in longitude, and by 6.4° on Phobos. Latitude stops being a function of position.

datums · Bodies 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 conformal map of Vesta, in the coordinates that make one possible. The body's own isothermal coordinate, used as the map — which is Mercator's construction carried out on a body that has no axis to be Mercator about. The lines are Jacobi's ellipsoidal coordinates, which are the surface's lines of curvature, at 30° and 20° spacing; the two quadratures that turn them into an isothermal pair are one-dimensional. The measured angular deformation away from the marked points is 1.1e-6°, which is this site's noise floor, and the areal factor spans a factor of 162 — conformal, and emphatically not equal-area. The four marks are the umbilics, where the coordinate collapses and the map has nothing to say.

A conformal map of a body with three axes

This site built an equal-area map of a triaxial body and wrote down what it could not do: the conformal one, which needs an isothermal coordinate that a surface with no axis of revolution was said not to have. It has one, Jacobi found it in 1839 for a different reason, and it takes two one-dimensional integrals.

datums · Bodies A conformal map solved rather than written down. The same patch of parameters on two bodies, mapped to the plane by solving the discrete Cauchy–Riemann equations — one complex equation per triangle, 1568 triangles, least squares, conjugate gradients, and no formula for either surface. Left: a sphere, where the answer is known in closed form and is not used. Right: a body with a bump on it, which has no isothermal coordinate and therefore no closed form at all. The parameter lines cross at right angles in both, to a median of 0.60° and 1.25° of angular deformation.

A conformal map of a body that is not a quadric

Jacobi's ellipsoidal coordinates give a triaxial body a conformal map by two quadratures, and this collection wrote down what that argument uses: the surface has to be a quadric. A real body is not. Solving the discrete Cauchy–Riemann equations instead — one complex equation per triangle, two thousand triangles, conjugate gradients — gives a conformal map of a bumped body to a median of 1.10°, converging at first order in the mesh, with the areal factor spreading by 3.07 and refusing to converge at all.

datums · Bodies Seven cuts of the same size, in different places on one body. The body's own colatitude and longitude, with the seven windows drawn on it. Each covers the same surface area to 0.24 per cent — the longitude extent is divided by sin θ and a scale is then solved per window — so their sizes are held and only their positions differ. Each is shaded by the areal spread of the conformal map solved on it, listed beside the grid and running from 1.03 to 1.36. The darker windows are the worse ones, and they are the ones over the body's lobes.

Cuts of the same size in different places

Where a body is cut decides how well it can be mapped, by a factor of ten — established with five windows of five different sizes, so *where* and *how much* were confounded and the factor could have been entirely about extent. Held to the same surface area to a quarter of a per cent, the answer survives at a factor of 1.32, and what predicts it is the curvature the window encloses.

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.

Closed formPlanetary datumShape modelFirst fundamental formConformalitySurface normalAuxiliary latitudeCauchy riemannConventionGauss–Bonnet theoremGaussian curvatureIsothermal coordinates

All concepts