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

A body that is not the Earth

Drawing the planet shows nothing — a picture of Mars is a picture of Mars whatever convention its coordinates use — so every mode draws the difference instead. body-latitude is the two latitudes of one point, body-projection the error a spherical formula makes on another flattening, body-figure the figures of the bodies themselves, and triaxial-map and triaxial-conformal the case where two equatorial radii differ and the normal swings as one walks round.

Drawing the planet shows nothing — a picture of Mars is a picture of Mars whatever convention its coordinates use — so every mode draws the difference instead. body-latitude is the two latitudes of one point, body-projection the error a spherical formula makes on another flattening, body-figure the figures of the bodies themselves, and triaxial-map and triaxial-conformal the case where two equatorial radii differ and the normal swings as one walks round.

Every one of these is the same generator answering a different question, which is why they share a file, a set of colour roles and a set of assertions. Changing it changes all 5.

15 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.

What it draws

One picture per question the family answers. Each has a page of its own.

How far apart the two conventions are. The difference between the two latitudes against latitude, on four bodies. Every curve is zero at the equator and at the pole and peaks near 45°, where it is the body's flattening in radians — 0.19° for the Earth, 0.34° for Mars and 3.84° for Jupiter. The shape is the same on all of them because the formula is the same; only the scale differs, and the scale is one number.

body-latitude

How far apart the two conventions are The error of a spherical formula is twice the flattening. The worst angular deformation of the spherical Mercator formulae applied to each body's own latitudes, against that body's flattening, on logarithmic axes. The points are measured by differencing the projection; the line is 2f radians, which is not fitted. The two agree to a third of a per cent for Mercury, the Earth and Mars, and depart by 3.3 and 5.1 per cent for Jupiter and Saturn, whose flattenings are too large for the first term to be the whole story. The site's headline number — 0.3848° on Earth — is an instance of this law rather than a fact about Web Mercator.

body-projection

The error of a spherical formula is twice the flattening 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.

body-figure

On a triaxial body, latitude depends on longitude 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°.

triaxial-map

Vesta, with its own coordinate lines 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.

triaxial-conformal

A conformal map of Vesta, in the coordinates that make one possible

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.

 Two latitudes for one point on Mars. A cross-section of Mars with its flattening exaggerated 6× so the two angles can be told apart. The line to the centre defines the planetocentric latitude and the outward normal defines the planetographic one; at 45° they differ by 0.338° on the real body, which is 20 kilometres along the surface. Both numbers are published for Mars, and a coordinate that does not say which it is is ambiguous by that much. What the numbers refer to

A coordinate on another body

Mars publishes two latitudes for every point and they differ by up to 0.338°, which is twenty kilometres of ground — almost exactly the same distance as the Earth's own 0.192°, because Mars is smaller by nearly the same factor. The libraries that compute either have known both numbers since this collection's early essays and had never been pointed anywhere but here.

 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. What is taught wrongly

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.

 The error of a spherical formula is twice the flattening. The worst angular deformation of the spherical Mercator formulae applied to each body's own latitudes, against that body's flattening, on logarithmic axes. The points are measured by differencing the projection; the line is 2f radians, which is not fitted. The two agree to a third of a per cent for Mercury, the Earth and Mars, and depart by 3.3 and 5.1 per cent for Jupiter and Saturn, whose flattenings are too large for the first term to be the whole story. The site's headline number — 0.3848° on Earth — is an instance of this law rather than a fact about Web Mercator. What the numbers refer to

The same projection on a different body

A projection's distortion does not depend on the body's size at all — only on its flattening — so Web Mercator's 0.3848° of angular deformation is not a fact about Web Mercator. It is 2f radians, and the same mistake made on Mars measures 0.6765°, on Jupiter 7.68°, and on a body scaled ten times larger than Mars the identical number to twelve figures.

 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. What the numbers refer to

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.

 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.

 Shape predicted from field, against shape as published. Clairaut's theorem gives a body's flattening from two numbers of its gravity field: J₂, which is how its mass is arranged, and m = ω²a³/GM, which is how fast it spins. For a body in hydrostatic equilibrium the prediction is the shape, and the diagonal is where such a body sits. Earth is on it to 0.05 per cent — 12 metres at the pole, out of twenty-one kilometres of flattening. Mars is 12.5 per cent off it, which is 2.23 kilometres, and the excess is Tharsis: a body carrying a continent-sized volcanic load is not a fluid figure, so its ellipsoid is not one of its own level surfaces, and its zero of height has to be chosen rather than found. What the numbers refer to

A body with no sea level

On Earth the zero of height is found rather than chosen — water settles onto the equipotential surface by itself. Nowhere else has one, and the difference is measurable: Clairaut's theorem predicts the Earth's flattening from its own gravity field to twelve metres at the pole and misses Mars's by 2.2 kilometres.

 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. What the numbers refer to

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.

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