Lecture 2 15

Figure 1.11. Stereographic projection from the sphere to the plane.

as much of the surface as possible, without regard to distortions near

the edges, this approach represents publishing an atlas, with many

smaller maps, each zoomed in on a small neighbourhood of each point

in order to minimise distortions.

Orthogonal projections, whether to coordinate planes or tangent

planes, form only a subset of the class of local coordinates on sur-

faces; there are many other members of this class besides. In the case

of a sphere, one well-known example of local coordinates is stereo-

graphic projection (Figure 1.11), which gives a

diffeomorphism6

from

the sphere minus a point to the plane.

Another example is given by the use of the familiar system of

longitude and latitude to locate points on the surface of the earth;

these resemble polar coordinates, mapping the sphere minus a point

onto the open disc (Figure 1.12). The north pole is the centre of the

disc, while the (deleted) south pole is its boundary; lines of longitude

(meridians) become radii of the disc, while lines of latitude (parallels)

become concentric circles around the origin.

However, if we want to measure distances on the sphere using any

of these local coordinates, we cannot simply use the usual Euclidean

distance in the disc or the plane—for example, the polar coordinates

mentioned in the last example preserve distances along lines of longi-

tude (radii), but distort distances along lines of latitude (circles cen-

tred at the origin). This is especially true near the boundary of the

6That

is, a bijective differentiable map with differentiable inverse. See Lecture

17 for more details.