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

Planetographic latitude — where it appears

The angle between a body's equatorial plane and the normal to its reference surface, which is what an instrument levelled on that surface measures. It differs from the planetocentric one by an angle that depends on the flattening, and the two are routinely confused in published coordinates.

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

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.

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.

datums · Bodies 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.

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.

datums · Bodies 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

Named alongside it

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

Planetary datumClosed formConventionEllipsoidFlatteningPlanetocentric latitudeReference frameAreal scaleAxis orderConformalityDimensional analysisGeodetic latitude

All concepts