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

Four ways to build the same equal-area projection

Drawn here at the parameters it defaults to, with every essay that calls it.
Four ways to build the same equal-area projection. The cylindrical equal-area projection's own areal scale factor, measured from its Jacobian against the metric of the body it is drawn for. On a sphere with the spherical formula it is one everywhere, which is the control. Feed the same formula the geodetic latitude a coordinate actually carries and measure against the ellipsoid it refers to, and it is 1.00674 on the equator and 0.99332 at 88° — a spread of 1.34 per cent, on a projection whose entire purpose is that there is no spread. Rescaling to make the totals agree does not repair it. The authalic northing q/2 does, exactly.

The cylindrical equal-area projection's own areal scale factor, measured from its Jacobian against the metric of the body it is drawn for. On a sphere with the spherical formula it is one everywhere, which is the control. Feed the same formula the geodetic latitude a coordinate actually carries and measure against the ellipsoid it refers to, and it is 1.00674 on the equator and 0.99332 at 88° — a spread of 1.34 per cent, on a projection whose entire purpose is that there is no spread. Rescaling to make the totals agree does not repair it. The authalic northing q/2 does, exactly.

It is one of 13 figures drawn by the same construction, and the drawing above is this one with nothing changed.

2 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.

Four ways to build the same equal-area projection. The cylindrical equal-area projection's own areal scale factor, measured from its Jacobian against the metric of the body it is drawn for. On a sphere with the spherical formula it is one everywhere, which is the control. Feed the same formula the geodetic latitude a coordinate actually carries and measure against the ellipsoid it refers to, and it is 1.00674 on the equator and 0.99332 at 88° — a spread of 1.34 per cent, on a projection whose entire purpose is that there is no spread. Rescaling to make the totals agree does not repair it. The authalic northing q/2 does, exactly. What is taught wrongly

Equal-area on the wrong body

The site's headline is that Web Mercator puts geodetic latitudes into a spherical conformal formula and stops being conformal. The same sentence is true with "equal-area" in it and nobody says it: the areal factor is 1.00674 at the equator, 0.99332 at 88°, and averages to almost exactly one — so every check that adds up areas passes while every cell is wrong.

 Four radii of the Earth, and one that is a range. The four constants called the mean radius of the Earth, on a scale of kilometres, with the range of the local Gaussian radius √(MN) drawn behind them. Three of the four agree to about a part per million; the rectifying radius is 3560 metres smaller, which is 559 parts per million. The Gaussian radius spans twelve times that range on its own, which is why there is no such thing as the conformal sphere. The families

Four radii of the Earth

The ellipsoid is usually handled with an auxiliary latitude. Every such construction also needs a radius, and the radius that makes each property exact is a different number: the published 6371 km is right for an area to half a part per million and wrong for a meridian distance by 559 — while the radius a conformal map needs is not a constant at all, and spans 6,739.

The whole library