Cylindrical projection, an Archimedes' result, and Duistermaat-Heckman Theorem

I am always amazed by how accurately some old world map made long time ago depicted the globe. Cartography, or simply map-making, has long been an established form of art and science since navigation to the New World became more and more active. There are several cartography techniques which serve to describe certain geometrical information of the world geography. Cylindrical projection is one of them.

The principle of cylindrical projection is simple indeed. Let us imagine that the globe is a perfect sphere which is inscribed in a cylinder with its height and radius of cross section equal to the radius of the globe. Position this geometric configuration in the three dimensional coordinate system in such a way that the -axis passes through the center of both the sphere and the cross section of the cylinder. Cylindrical projection is just the horizontal radial projection from the sphere onto the curved surface of the cylinder. Put in a more mathematical way, cylindrical projection maps a point with -coordinate to the intersection of the cylindrical surface and the horizontal ray(parallel to the -plane) emanating from and passing through that point.

World maps made by cylindrical projection is obtained by flattening out the image of cylindrical projection on the cylindrical surface. For the convenience of archiving, it has been more common to draw world maps on a flat sheet of paper than on a sphere. But paper maps have one major drawback--they greatly distort the actual geography of the world. More precisely, distances on any paper map are not proportional to actual distances. This is a simple consequence of Gauss's Theorema Egregium, which states that the Gaussian curvature of any surface is expressible in terms of its metric. Since a sphere has constant positive curvature, whereas a plane has vanishing curvature, there is no map between them preserving distance. You may observe that the closer to the Poles a place is, the more distortedly it is presented on a map made by cylindrical projection. For instance, the Antarctica appears to be much more elongated on such maps.

But cylindrical projection is not completely without any merit. In fact cylindrical projection is area-preserving. Legend has it that this result was discovered to Archimedes. It is rather mysterious to me how Archimedes derived this fact. Anyway I will show below a proof which amounts to a 'change of variables', but is rephrased in differential-form terminology.

It suffices to show that the cylindrical projection induces a pullback which maps the area form of the cylinder to that of the inscribed sphere. The spherical coordinates are

whereas the cylindrical coordinates are

So . Note that the area form of the sphere is , while that of the cylinder is . Its pullback by is
So we are done.

For those who enjoy understanding math from a vantage point of view, note that this result of Archimedes' turns out to be a particular case of a theorem in symplectic geometry known as Duistermaat-Heckman Theorem. It says that given a Hamiltonian -space where is a torus and the symplectic form of , then the Radon-Nikodym derivative of the pushforward of the canonical measure (given by , 2n=dimensional of ) through the moment map with respect to the Lebesgue measure of is piecewise polynomial, i.e. for any measurable ,
for some piecewise polynomial in . Here the sphere is a Hamiltonian -space, where its symplectic form is just the area form, acts on the sphere by rotating about the vertical axis at unit speed and the moment map is the height function. The Radon-Nikodym derivative in this case is the constant .

Ideal Map (Part 1)

How is a map which is a flat sheet of paper representing the actual curved surface of the Earth?

For example, how can we tell the distance between HK and Beijing from a world map? When we draw a line on the map joining HK and Beijing, what is the path that is traced on the globe? If 2 straight line segments have the same length on the map, must the 2 corresponding paths have the same length on the globe?

Actually, the above problems depend on how the map was created.

When we create a map of the globe (or some portion of the globe), we might want our map to possess some nice properties such as:

1. The map should be planar (or flat), so that it is more convenient for us to laid it down on a table and draw on it.
2. Any “straight line segment”* on the globe should correspond to a straight line segment on the planar map, so that it is easy for us to trace a shortest path between any 2 points (by using a ruler to draw a straight line segment on the map for instance).
3. The shape of any region on the globe is preserved on the map. For example, an equiangular triangle on the map should correspond to an equiangular triangle** on the globe.
4. The map has a fixed scale, which means the lengths of any 2 paths on the globe is decreased by the same factor on the corresponding paths on the map.

Here comes a problem which intrigued many Navigators and Mathematicians hundreds of years ago, and is central in Cartography.

Can we create a map which possesses one or more of the above attributes?

Before we reveal the answers, please think about it first!

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* For simplicity, suppose the globe is a perfect sphere. We know that a sphere is curved, so what is a “straight line” on a sphere? Imagine we’re walking “straight” on large sphere, our path actually traces an arc of a great circle (which is a circle which has the same center as the sphere it is lying on). It can be proved, by undergraduate calculus, that a shortest path between 2 points on a sphere is the minor arc of a great circle. Hence, a “straight line segment” on a sphere is actually an arc of a great circle.

**A triangle on a sphere is a region on the sphere which is bounded by 3 straight lines (i.e. 3 arcs of great circles) on the sphere. An angle at a point P on the sphere formed by 2 great arcs meeting at P is the angle between the tangents of the 2 arcs at P.