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Stereographic Projection

Introduction
Cartography is the study and procedure of making and understanding maps. Cartography is the underlying foundation that reality can be created or modeled in ways that communicate spatial information effectively. All this information relates to the topic of interest this week, stereographic projection. Stereographic projection is the idea of mapping some kind of function that projects a sphere onto a plane. The most well-known example of this is the globe and projecting that onto a map. A map of Earth is rather inaccurate due to distortion, for bodies of land that reside near the North or South pole are actually significantly smaller than they appear. Today, we will be further understanding just how stereographic projections work and looking at a specific example of one too.
Stereographic Projection?
As stated before, stereographic projection is the idea of mapping a function that projects a sphere onto a plane. When trying to visualize sterographic projection, imagine having a birds eye view and looking straight down on an area. To visualize this idea, you could create something like this:
This is the most simple way to visualize exactly how sterographic projections work. A function that is a sphere is mapped directly on to a plane. The image supports this idea, for the sphere is mapped down from a three-dimensional space to a two-dimensional space, and the sphere is almosted flattened out. Now, what if we used the same idea from above, but instead of having a solid sphere that is directly mapped down, we have a sphere with some "holes" in it. This is what we are going to be looking at to allow us to further comprehend stereographic projection.
Stereographic Projection Example
Now, we are going to look at a specific example that includes a sphere with some "holes" in it. For this example, I chose a raindrop like shape as my holes in my sphere. Below is an example what the result would be when you project the sphere onto a plane. I had a lot of difficulty actually creating this design, but I hope it comes out well when I print it.
That is the result of what the mapping of the function would look like when the sphere is projected onto a plane. Now, when we add in the sphere, the raindrop like shape will actually create respective holes in the sphere. Once we flash a flashlight into the top of the sphere, we will be able to see the stereographic projection that we just saw above! The image below is what the sphere would look like before it is projected into the plane.
Why These Functions?
I chose this example for multiple reasons. First, I really like the way that the stereographic projection played out with this shape. I feel like the raindrop like shape was an interesting one to choose and showed just how creative you can be with stereographic projections. Also, I felt like this example would allow someone who has grasped a foundation of stereographic projections to further understand the topic. Once printed, my shape will come out to approximately, 2in x 2in x 2in in size.

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