Illustrated essays on map projections with the distortion measured rather than described: every indicatrix labelled with its computed scale factors, every claimed property verified before it is printed, and the impossibility derived from the curvature rather than asserted. https://www.cartographic-projection.com/ 2026-08-05T01:27:01.910Z https://www.cartographic-projection.com/essays/web-mercator-is-not-conformal/ 2026-08-05T01:27:01.910Z

It carries almost every map on the internet, it is named after the projection whose entire purpose is preserving angles, and it does not preserve angles. The machinery here found that without being told to look.

https://www.cartographic-projection.com/essays/no-map-is-faithful/ 2026-08-05T01:27:01.910Z

Not "no map yet", and not "no map at page size". Gaussian curvature can be computed from inside a surface, a sphere has some and a plane has none, and that closes the question permanently.

https://www.cartographic-projection.com/essays/tissots-indicatrix/ 2026-08-05T01:27:01.910Z

A tiny circle on the sphere becomes an ellipse on the map, and the ellipse's two axes are the whole story. Almost every published indicatrix is drawn without them, which discards the content and keeps the decoration.

https://www.cartographic-projection.com/essays/the-shortest-route-is-not-straight/ 2026-08-05T01:27:01.910Z

The shortest path between two points on a sphere is an arc of a great circle, and on almost every map it is a curve. The straight line on a Mercator chart is a different route entirely, and on some journeys it is twenty-eight per cent longer.

https://www.cartographic-projection.com/essays/cylinders-cones-and-planes/ 2026-08-05T01:27:01.910Z

The standard taxonomy sorts projections by the shape of paper they were notionally rolled from. It is memorable, it is how everyone is taught, and it says almost nothing about the properties anyone actually chooses on.

https://www.cartographic-projection.com/essays/every-projection-minimises-something/ 2026-08-05T01:27:01.910Z

A projection is the solution to an optimisation problem, and naming the objective explains more than naming the family. Some objectives are exact constraints, some are least-squares fits, and one is a table of numbers a man adjusted until it looked right.

https://www.cartographic-projection.com/essays/why-mercator-exists/ 2026-08-05T01:27:01.910Z

A ship can hold a compass bearing and cannot easily hold a great circle. Mercator is the answer to one question — what must a map do so that a constant bearing is a straight line — and it answers it exactly.

https://www.cartographic-projection.com/essays/mercator-against-peters/ 2026-08-05T01:27:01.910Z

The most-argued question in cartography, conducted almost entirely without anyone measuring anything. Both projections are exactly what they claim, each destroys what the other keeps, and the numbers are computable in either direction.

https://www.cartographic-projection.com/essays/which-projection-is-best/ 2026-08-05T01:27:01.910Z

An incomplete question, and the incompleteness is the answer. Every projection preserves something and destroys something else, so the comparison worth making is between a projection and a purpose, not between two projections.

https://www.cartographic-projection.com/essays/what-can-be-unrolled/ 2026-08-05T01:27:01.910Z

A cylinder is obviously curved and is intrinsically flat, so it lays out on a table with nothing stretched. A sphere is not. The distinction is exactly zero Gaussian curvature, and it is why paper tubes exist and paper globes do not.

https://www.cartographic-projection.com/essays/what-survives-a-change-of-coordinates/ 2026-08-05T01:27:01.910Z

The scale along the meridian is a property of the map and the grid together. The principal scale factors are a property of the map alone. Only the second kind describes the projection, and the two are routinely quoted as though they were the same thing.

https://www.cartographic-projection.com/essays/what-a-standard-parallel-buys/ 2026-08-05T01:27:01.910Z

A standard parallel is a line where the projection is exact. Choosing one does not reduce the distortion — it decides where the distortion is zero and lets everything grow away from it.

https://www.cartographic-projection.com/essays/the-trade-off-is-two-lines/ 2026-08-05T01:27:01.910Z

Conformal means the two principal scales are equal. Equal-area means their product is one. Both at once forces both to one, which is an isometry, which the curvature forbids. That is the entire argument.

https://www.cartographic-projection.com/essays/the-two-ways-a-map-is-wrong/ 2026-08-05T01:27:01.910Z

Angles and areas fail independently. A projection can be perfect about one and catastrophic about the other, and a single number summarising "how distorted" a map is has already thrown away the distinction that matters.

https://www.cartographic-projection.com/essays/the-gnomonic-companion/ 2026-08-05T01:27:01.910Z

One projection turns every great circle into a straight line, and it is the only one that does. It shows less than half the sphere, distorts enormously, and was indispensable for three centuries because of that single exact property.

https://www.cartographic-projection.com/essays/compromise-projections/ 2026-08-05T01:27:01.910Z

A projection that preserves nothing exactly can distort everything less than one that preserves something exactly. For a general-purpose world map that is the right trade, and it is why the two most widely used ones today have no exact property at all.

https://www.cartographic-projection.com/essays/the-aspect-is-a-free-choice/ 2026-08-05T01:27:01.910Z

A projection's distortion pattern is fixed relative to its own axis, and where that axis points is entirely up to the cartographer. Rotating it is the cheapest available improvement and it is the one most often left unmade.

https://www.cartographic-projection.com/essays/the-projection-that-shows-true-size/ 2026-08-05T01:27:01.910Z

There is no such thing, and the phrase hides a real question. Equal-area projections preserve area and destroy shape; nothing preserves size in the sense the phrase implies; and the tools that make the point best are not maps at all.

https://www.cartographic-projection.com/essays/measuring-instead-of-naming/ 2026-08-05T01:27:01.910Z

A projection is called conformal because that is its name. Running the definition as a computation over several hundred points takes about twenty lines, catches a projection the whole internet uses, and is almost never done.