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

Where the middle of a 20° × 20° region is, in five planes

Drawn here at the parameters it defaults to, with every essay that calls it.
Where the middle of a 20° × 20° region is, in five planes. The region is drawn in longitude and latitude — which is itself a projection, and one of the ones being compared. Each filled mark is the shoelace centroid computed in one projected plane and inverted back to the ground; the hollow mark is the centre of area on the sphere, by integration. They spread over 273 kilometres. The equal-area member is 66 kilometres out, because a centroid is a first moment and preserving area says nothing about where the area sits.

The region is drawn in longitude and latitude — which is itself a projection, and one of the ones being compared. Each filled mark is the shoelace centroid computed in one projected plane and inverted back to the ground; the hollow mark is the centre of area on the sphere, by integration. They spread over 273 kilometres. The equal-area member is 66 kilometres out, because a centroid is a first moment and preserving area says nothing about where the area sits.

It is drawn by operation-figure with show: "centroid-planes" — one member of a family of 14 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.

1 essay calls 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.

Where the middle of a 20° × 20° region is, in five planes. The region is drawn in longitude and latitude — which is itself a projection, and one of the ones being compared. Each filled mark is the shoelace centroid computed in one projected plane and inverted back to the ground; the hollow mark is the centre of area on the sphere, by integration. They spread over 273 kilometres. The equal-area member is 66 kilometres out, because a centroid is a first moment and preserving area says nothing about where the area sits. What a machine does with it

A centroid belongs to a plane

Every renderer labels a region at its centroid, and every centroid is a shoelace over coordinates as stored — which is a statement about the plane they are in. Six planes put the middle of one 20° × 20° region up to 273 kilometres apart, the equal-area member is 66 kilometres out, and the disagreement falls as the square of the region's size.

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