Every essay
The impossibility
Gaussian curvature is intrinsic, a sphere has some and a plane has none, so no map can be faithful. A theorem rather than an engineering limit.
No map is faithful
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.
What can be unrolled
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.
The trade-off is two lines
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.
Measuring distortion
Scale along the meridian, scale along the parallel, and the two invariants that survive a change of coordinates. Tissot's indicatrix drawn from the derivatives and labelled with its numbers.
Tissot's indicatrix
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.
What survives a change of coordinates
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.
The two ways a map is wrong
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.
Measuring instead of naming
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.
The families
Cylindrical, conic, azimuthal — a taxonomy of construction that says surprisingly little about the properties anyone actually cares about.
Cylinders, cones and planes
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.
What a standard parallel buys
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.
The aspect is a free choice
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.
What each projection optimises
Every projection minimises something. Naming the objective is more honest than naming the shape of the paper it was notionally rolled from.
Every projection minimises something
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.
Which projection is best
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.
Compromise projections
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.
Paths and directions
Great circles, rhumb lines, and the question that produced Mercator: what has to be true of a map for a constant compass bearing to be a straight line?
The shortest route is not straight
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.
Why Mercator exists
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.
The gnomonic companion
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.
What is taught wrongly
The most-argued subject in cartography, and most of the argument is conducted without measuring anything — including the projection that fails the property named in its own title.
Web Mercator is not conformal
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.
Mercator against Peters
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.
The projection that shows true size
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.