Why no flat map of the Earth can preserve all distances
PAPER II — MATHEMATICS · QUESTION 33 OF 35
In plain terms: The free official manual of map projections — what each one keeps and what it must give up, in full.
The question as set
Prove that no plane map of the sphere preserves all distances, and say what Mercator, the equal-area projections, and the geodesic-preserving projections each surrender instead.
The question in Paper II · Its section on the reference page
The question in context
This is question 33 of the 35 set in Paper II — Mathematics of The Truly Hardest Exam in the World, the examination of The College of All Minds. In the College’s subject index it is filed under Geodesy and navigation and Geometry and topology and Maps and representation. Like every question of the College, it carries three references of record, verified live at publication and described twice — technically, and in plain terms — so the ground can be judged before it is walked. The response required is an argument, not a survey, of at most three thousand words, as technical as the question demands and no more.
The references of record
Map Projections — A Working Manual
The catalogue of surrenders: every projection given with its distortions, formulae, and the property it elects to preserve.
In plain terms: The free official manual of map projections — what each one keeps and what it must give up, in full.
General Investigations of Curved Surfaces
The theorem itself: curvature intrinsic to the surface, whence no isometry of sphere to plane can exist.
In plain terms: The free original of the result that makes a perfect flat map impossible — not difficult, impossible.
The practical taxonomy: conformal, equal-area, equidistant and compromise, with the use each is fit for.
In plain terms: The free official primer distinguishing the projections by what they protect — a plain companion to the theorem.
Background: Theorema Egregium · Map projection