Skip to main content

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.

Comments

Popular posts from this blog

Do Over: Integration Over a Region in a Plane

Throughout the semester we have covered a variety of topics and how their mathematical orientation applies to real world scenarios. One topic we discussed, and I would like to revisit, is integration over a region in a plane which involves calculating a double integral. Integrating functions of two variables allows us to calculate the volume under the function in a 3D space. You can see a more in depth description and my previous example in my blog post, https://ukyma391.blogspot.com/2021/09/integration-for-over-regions-in-plane_27.html . I want to revisit this topic because in my previous attempt my volume calculations were incorrect, and my print lacked structural stability. I believed this print and calculation was the topic I could most improve on and wanted to give it another chance. What needed Improvement? The function used previously was f(x) = cos(xy) bounded on [-3,3] x [-1,3]. After solving for the estimated and actual volume, it was difficult to represent in a print...

Minimal Surfaces

Minimum surfaces can be described in many equivalent ways. Today, we are going to focus on minimum surfaces by defining it using curvature. A surface is a minimum surface if and only if the mean curvature at every point is zero. This means that every point on the surface is a saddle point with equal and opposite curvature allowing the smallest surface area possible to form. Curvature helps define a minimal surface by looking at the normal vector. For a surface in R 3 , there is a tangent plane at each point. At each point in the surface, there is a normal vector perpendicular to the tangent plane. Then, we can intersect any plane that contains the normal vector with the surface to get a curve. Therefore, the mean curvature of a surface is defined by the following equation. Where theta is an angle from a starting plane that contains the normal vector. For this week’s project, we will be demonstrating minimum surfaces with a frame and soap bubbles! How It Works Minimum surfac...

Ruled Surfaces : Trefoil

Ruled Surfaces : Trefoil A ruled surface is a surface that consists straight lines, called rulings, which lie upon the surface. These surfaces are formed of a set of points that are "swept" by a straight line. This is relatively intuitive once you see a good visual, but can be a bit abstract without that concrete example. A very basic example of a ruled surface is a cylinder; if we have a straight line and move it in a circle we create a cylinder made entirely of straight line. Note that the surface will only be a cylinder if all the lines are parallel. If the lines are not parallel we can create hyperboloids and cones depending on how much we have rotated. The rotation we are describing here is not a simple turning action, but more of a twisting motion—less like rotating a can by turning it and more like wringing out a washcloth by twisting it. Specifically, a cylinder is essentially two circles connected by rulings, if we keep one of the circles...