Geographers have devised all sorts of projections from the curved surface of a sphere in three-space to flat two-dimensional maps. We have already been studying one of the most useful mapping techniques, central projection from a viewing point to a horizontal plane. When the viewing point is at the top of a sphere that rests on the horizontal plane, central projection sends each point of the sphere to a unique point of the plane. This gives a mapping from the sphere to the plane that cartographers call stereographic projection. To describe this mapping in terms of light rays, we begin with a transparent globe resting on a plane and imagine a bright light situated at the north pole. For each point on the sphere, some ray of light will pass through the point and create an image on the horizontal plane. Thus every point of the globe, other than the north pole itself, will have an image point on the plane, and every point of the plane corresponds to exactly one point on the sphere. The rays from the north pole through the points of the equator form a right circular cone, which cuts the plane in a circle. Similarly, the image of any parallel of latitude on the sphere is a circle in the plane.
This projection provides a very accurate map of the Antarctic region, but the map becomes more distorted near the equator. Land masses in the northern hemisphere are further distorted, and the image of Greenland for example is huge. To get a reasonable image of Greenland, we can rotate the globe, keeping the plane and the light source fixed, so that Greenland is moved near the point where the sphere touches the plane. Of course once this is done, Antarctica has moved very close to the source of the light rays, and its image is greatly distorted.
One of the significant properties of stereographic projection is that it not only sends parallels of latitude to circles--it sends all circles on the sphere to circles in the plane. Only circles passing through the north pole are exceptions; these have straight lines as their images. This property makes it much easier to keep track of what happens to the southern hemisphere as we rotate the globe around a horizontal axis through the center of the sphere.
Before the rotation begins, the image of the southern hemisphere is the interior of a disc centered at the point where the sphere touches the plane. As we begin to rotate, the image of the equator is still a circle, moving off-center, and the image of the southern hemisphere is the interior of that off-center disc. Ultimately the equator rotates so far that it passes through the light source at the top point. The image of the equator is then a straight line, and the image of the southern hemisphere becomes an infinitely large half-plane. If we continue the rotation, the original southern hemisphere moves over the light source. Once the light source lies within the rotated southern hemisphere, the image of this region is the region outside the circular image of the equator. By the time the rotation has brought the equator back to a horizontal position, the southern and northern hemispheres have been interchanged and the images in the plane have been "turned inside out."
|Stereographic Projection from Four-Space|
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|The Polyhedral Torus in the Hypercube|