sandbox · exhibit 0007

The size of the lie

A sphere has curvature and a sheet of paper does not, so no map of the world is right about everything: Gauss proved in 1827 that flattening one must stretch it, and every projection since is a decision about where to put the damage. Twelve of them are drawn below. The page does not take any of their claims on trust — it differentiates each map where it stands, and reads the distortion back out of the arithmetic.

Projection

Shading

Drawn on top

— —

—

drag to turn the globe

Area, worst

—

Angle, worst

—

Area, typical

—

Angle, typical

—

Zürich → Tokyo: the straight line on this map

—

What this page is claiming

Every number above comes out of the same twelve functions, so the honest thing is to check them against arithmetic that was written separately and does not share a line with them: analytic scale factors from the textbook, areas measured by counting a polygon instead of differentiating a map, distances measured on the sphere, and one theorem that no projection here is allowed to escape.

Why a flat map cannot be right

Take any point on a globe and a small circle around it. A map sends that circle to some closed curve; if the circle is small enough, that curve is an ellipse — Tissot's — and it is the whole story of what the map does there. If the ellipse is a circle, the map preserves angles at that point. If its area is right, the map preserves area. The theorem is that no map of a sphere can do both anywhere except at isolated points, because a map that did would be an isometry, and Gauss' Theorema Egregium says an isometry cannot change curvature: the sphere has 1/R², the sheet has 0.

So the twelve maps here divide into the ones that keep the ellipse round and let it grow — Mercator, stereographic, the conformal conic — and the ones that keep its area and let it lean — Lambert, Mollweide, Hammer, Albers — and the compromises, which do neither and are shaped by taste. The page measures both quantities for all of them. Nothing in the measurement reads the label.

What the numbers mean

Area is the ellipse's area divided by the true area of that patch of sphere: 1.00× is honest, 4× means the map has quadrupled that piece of the world. Angle is the maximum angular deformation ω — the largest error the map can make in any bearing taken at that point. Zero is conformal. The typical values are area-weighted medians over the sphere, so a projection cannot look good by being terrible only where it is small.

The route is the same argument in kilometres. Zürich to Tokyo is 9,577 km on the sphere; the straight line between the two dots on the map is a different path, and the page inverts the projection along it to find out how long that path really is. On the gnomonic it is the same route — that is what a gnomonic is for, and the only reason anyone tolerates the rest of its behaviour. On the others it is longer, and the excess is printed. No straight line on any map can beat the great circle, which is a pleasant thing to be able to check rather than assert.

How the ellipses are drawn

Twice, from two different arithmetics. The filled shape is the actual image of a small circle on the sphere, sampled and projected point by point. The dashed outline is the ellipse predicted by the derivatives of the map at the centre — semi-axes a and b, from the four partial derivatives and nothing else. They lie on top of each other because Tissot was right, and if a projection here were mistyped they would come apart at the seams.

Exhibit 0007 · every projection, the sphere, and the checks are in this file; it fetches nothing. Formulas after Snyder, Map Projections — A Working Manual (USGS Professional Paper 1395). The written half is entry 0007.