## 2009年3月20日 星期五

### 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 $z$-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 $z$-coordinate $z_0$ to the intersection of the cylindrical surface and the horizontal ray(parallel to the $xy$-plane) emanating from $(0, 0, z_0)$ 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 $p:\mathbb{S}^2\to C$ induces a pullback $p^*$ which maps the area form of the cylinder to that of the inscribed sphere. The spherical coordinates are
$x=\sin\varphi\cos\theta, y=\sin\varphi\sin\theta, z=\cos\varphi$
whereas the cylindrical coordinates are
$x=\cos\theta, y=\sin\theta, z=z$
So $p(\varphi, \theta)=(\theta, \cos\varphi)$. Note that the area form of the sphere is $\sin\varphi d\varphi\wedge d\theta$, while that of the cylinder is $d\theta\wedge dz$. Its pullback by $p^*$ is
$p^*(d\theta\wedge dz)=d\theta\wedge d(\cos\varphi)=d\theta\wedge (-\sin\varphi d\varphi)=\sin\varphi d\varphi\wedge d\theta$ 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 $T$-space $(M, \omega)$ where $T$ is a torus and $\omega$ the symplectic form of $M$, then the Radon-Nikodym derivative of the pushforward of the canonical measure (given by $\frac{\omega^n}{n!}$, 2n=dimensional of $M$) through the moment map $\mu: M\to\mathfrak{t}^*$ with respect to the Lebesgue measure of $\mathfrak{t}^*$ is piecewise polynomial, i.e. for any measurable $U\subset\mathfrak{t}^*$,
$\int_{\mu^{-1}(U)}\frac{\omega^n}{n!}=\int_U p(t)dt$for some piecewise polynomial $p(t)$ in $t\in\mathfrak{t}^*$. Here the sphere is a Hamiltonian $\mathbb{S}^1$-space, where its symplectic form is just the area form, $\mathbb{S}^1$ 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 $2\pi$.