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

Vesta, with its own coordinate lines

Drawn here at the parameters it defaults to, with every essay that calls it.
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°.

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°.

It is drawn by planetary-figure with show: "triaxial-map" — one member of a family of 5 figures that share a generator, so the drawing above is what that generator returns when it is asked for this one and given nothing else.

3 essays call it. Every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about.

Where it is called

Changing this changes every one of these figures.

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. The impossibility

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.

 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°. What the numbers refer to

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.

 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. What the numbers refer to

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.

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