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

Vector field — where it appears

A quantity with a direction as well as a size, given at every point of a surface — a wind, a current, a plate velocity. Projecting one is a different problem from projecting a scalar field, because the operators that matter for it are the divergence and the curl rather than the gradient.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

Ground north, on a map that has the pole on it. Every arrow points along its own meridian, towards the pole, and the pole is at the centre. Walking once anticlockwise round any loop enclosing it turns the arrow once anticlockwise as well: the index is 1, counted as a winding number with no distance anywhere in the calculation. A field like this cannot be combed flat. There is no way to choose a page direction for north at every point of the neighbourhood without the choice tearing somewhere, and the somewhere is the point in the middle.

North cannot be up everywhere

Ground north is a field of arrows on the sphere, and a field of arrows on a sphere must vanish somewhere. The failure is not measured, it is counted: the indices of the zeros sum to two, obtained here as a winding number in seven different charts with no distance anywhere in the calculation, and it is the same two that Gauss–Bonnet gets by integrating curvature.

impossibility · Topology One current, two maps, and only one of them conserves it. A flow that has no sources or sinks anywhere on the sphere — it is the perpendicular of a stream function's gradient, so its divergence is zero by construction — drawn on two projections. The arrows on the right are the images of the same ground velocities as the ones on the left. On the equal-area map the drawn field is still divergence-free; on the other it is not, and the arrows drawn in the warning colour are where a reader measuring the picture would find a source or a sink that is not there.

A current drawn on a page has sources

Seven rungs project a scalar field and ask what the page does to its gradient. A wind or a current is the other half of what gets mapped, and the operator that matters for it is the divergence — which is preserved by an equal-area map exactly, by no other map at all, and by a conformal map least of anybody's expectation.

distortion · Gradient

Named alongside it

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

GradientInvariantAnisotropyAreal factorChartClosed formConformalityConservationContinuityConvergence angleCritical pointDivergence

All concepts