Previously, we have been discussing surfaces that are relatively easy to think about, and now we are going to consider a way to view a relatively simple surface, namely a sphere. We will be doing this via stereographic projections. But what is a stereographic projection? If you hear the word projection and immediately think of a (cartographic) map you are on the right track—as some maps are, in fact, stereographic projections. More formally, a stereographic projection is a mapping that projects a sphere onto a plane; so, by projecting a point(s) \(P\) on the surface of the sphere from the sphere's north pole \(N\) to point \(P'\) on a plane tangent to the sphere's south pole \(S\) as shown in the image below— note \(O\) denotes the origin.
This allows use to view what is technically a three-dimensional image in two dimensions; which is very useful in cartography, geology, etc. However, there are certain constraints to stereographic projections that can limit their usefulness in these fields and must be acknowledged when using a stereographic projection. These constraints are namely that while it preserves the accuracy of the angles at which curves meet it does not preserve distances or the areas of figures. A stereographic projection of the world can be found below—note the relative size of the various land masses.
Observe how the areas and distances on the map above are most accurate near the north and south poles—this is a feature that we will also see in our model, in that, nearest to the pole or center the areas of our shapes will be relatively accurate, but as we get further from where the pole intersects the tangent plane the areas get more inaccurate and, specifically, they get larger. This phenomenon can also be seen in the map above. The British Isles are so small as to almost be invisible while the Japanese home islands appear much larger; when they are only about 25,000 mi\(^2\) larger than the British Isles—the British Isles are 121,684 mi\(^2\) and the Japanese home islands are 145,937 mi\(^2\).
Now that we have defined what a stereographic projection is, some of their uses, and some of the constraints that must be considered when using a stereographic projection, we can begin to discuss our model of one such projection. When considering what to model, I immediately knew that I wanted to model the night sky but also knew that creating an actual reproduction of the sky—like you might find in a planetarium projector—was far above my 3D printing skill level. So, I went to the next best thing, stars. I originally wanted to use eight-pointed stars but realized that would be difficult to construct—both from my perspective in a coding sense and the 3D printer in a physical sense—thus I went with four-pointed stars. When light is projected through our model we will get a projection of uniformly spaced four-pointed stars which will look something like the image below.
This will give us some semblance of that planetarium projector that I mentioned earlier, as if we have a dark enough room and a bright enough light we will see this projection of stars on the ceiling—or other flat surface—we make tangent to the south pole of our sphere. Additionally, you may have noticed that our projection is of uniformly spaced and identically sized four-pointed stars, which would seem to contradict what we said previously in regard to stereographic projections not maintaining distances or areas; however, we compensated for that limitation of our stereographic projection in our 3D model of our sphere. That sphere can be seen below
As you can see in the first image the stars on the sphere are small and closer together nearer the north pole and as they approach and reach the south pole, shown in the second image, that they increase in size and get closer together and approach the true size and spacing of the stars in our projection.
The actual physical model of this sphere in approximately 1.5 inches square.
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