MA 391 Composition and Communication

Stereographic projection is used in geometry as a mapping function. It allows for a sphere to be projected onto a plane which is a way of picturing an 3D object (such as a sphere) as a 2D picture (a plane). The projection is defined on the entire sphere except for at one point, the projection point. The projection point is located at the topmost point of the sphere and is the point where the projection will be sourced. Mathematics Behind the Projection Here is how stereographic projections work! Imagine a sphere above a plane we want to project on to and the top of that sphere (equivalent to the north pole on a globe) will be out projection point N. For any point P’ on the plane, there is a point P on the sphere that is found by drawing a straight line from N to P’. For this week’s project, I have created a sphere with a design cut out that will demonstrate an example of stereographic projection. The design I have chosen is three circles overlapping to create Disney Micky Mouse E...

Flat Earthers consist of a group of people who believe the Earth to be flat. Their argument being that when a human is standing on Earth, the land around them appears to be be flat. In their argument against a round Earth, the claim is that if the Earth was round and you were standing on it, you would be standing at a tilt and you do not. Stereographic projection, in my opinion, helps the flat earther's theory ever-so slightly. The idea that a spherical, round, globe can be projected onto a 2-dimensional, flat map comes from the idea of stereographical projection. And that, in turn, could be why the flat earther's continue to push their theory. What the flat earther's might not be smart enough to fully grasp the concept of, is the actual idea and specific details of stereographic projection. They see a 2-dimensional map of the Earth and they're sold on the idea of a flat Earth, but what they don't notice are the size diff...

Introduction What is stereographic projection? You are probably already familiar with this topic without realizing it. We know the earth is round which is why we have globes to show the earth. However, we are also able to see the earth on a piece of paper if we want. This would be considered a stereographic projection. Stereographic projection projects a sphere onto a plane. We move from a 3-dimensional object to seeing it on the 2nd dimension. We can see our initial item, the earth, is 3-dimensional and gets projected and gets turned into a map which is 2-dimensional. Stereographic Projection We will look at an example but before that lets get to know more about stereographic projection. The entire sphere is projected onto the plane except the one point on the top. If we are projecting the earth this point would be the north pole. The north pole would be known as the projection point. In the picture we can see the blue lines that are being projected...

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...

How do you portray an infinite plane using a finite object? Stereographic projection, that's how. Stereographic projection makes use of a one-to-one and onto function with an infinite plane as its domain and a finite sphere (minus the top point) as its codomain. The infinite plane gets mapped onto a sphere by taking the point on the sphere that is furthest away from the plane and connecting it to any point on the plane with a line. The point on the sphere that the line intersects is the corresponding point to the point on the plane. Because the plane gets infinitely shrinked as it reaches the top of the sphere, the entire plane gets mapped to the sphere. If there is something graphed on the plane and you wanted it to be mapped onto the sphere, all you would need to do is connect every point on the graph to the top of the sphere and display the points of intersection on the sphere. Because the function is one-to-one and onto, the inverse exists, too. Take the top poin...

Projection is already ubiquitous in life. Projection is used in life. For example, in class, teachers will use projection to show the content of this lesson, and leaders at work will also use it as an auxiliary tool to show the next goal. Projection is widely used in life. It is a great auxiliary tool. Why? Because of its special properties, it can be adjusted by people, and the image can be enlarged and reduced by changing the distance. For example, the picture we watched in the cinema is a long-distance projection. These are all very well realized. Through the way of projection, the pattern on the cardboard can be projected by letting light pass through some cardboard with special patterns. These are all projected from one plane to another, and stereographic projection is a projection that projects a spherical surface to a plane. For example, the map is the stereographic projection of the earth. There will be a large gap between the actual projected image and the sphere ...

This week, we made our last step away from Calculus topics, to something more in the realm of pure maths. Our topic for this week is "Stereo-graphic Projection", which if you do not know what that even means, it is okay! Despite not knowing exactly what this means, yet, you know many examples of this! The major one is what you would see on the globe. This globe is a projection of the world, represented on a sphere. As you may know, the proportions on globes are not always true to relative sizes of countries/continents/oceans, as they easily get distorted with this projection. A stereo-graphic projection, for a base definition, is a projection from a sphere onto a plane. This is done with some mapping, or function, that takes objects on this sphere onto a plane (and also vis-ver-sa since this map is bijective). Anyway, that this looks like is imagine you are standing at the north pole on the globe. Next, take a light that can shine through the earth, and this lig...

Introduction If you’ve compared a globe and a paper map, you’ve probably noticed that despite displaying the same information, the two have slight differences. For example, on most paper maps Greenland appears rather large, even though this is not the case on an actual globe. Such distortions are unavoidable when trying to make a spherical surface flat. Imagine trying to cut open an inflatable beach ball in such a way that it forms a perfect square or rectangle- it’s impossible! Therefore, in order to transform a globe into a map, we use a function called a stereographic projection. This post will explore how this is done and the distortions that result from it. Stereographic Projections The function to perform a stereographic projection is fairly simple. First, imagine we position our globe above the map we want to make. For any point \( a \) on our map, its resulting location on the globe, \( f(a) \), is found by drawing a line from \( a \)...

Introduction So far, we've spent a lot of time navigating the world of calculus. To begin moving away from calculus, let's discuss ruled surfaces! Ruled surfaces are generated by connecting two curves with straight lines in three-dimensional space. Below, you can see how the given surface is made up of straight lines connecting the points on the left and right curves. If the number of straight lines connecting these curves is infinite, we see a solid surface. We're going to dive into ruled surfaces with an example I find interesting not just because of its shape but also its applications to biology. (Source: http://www.grad.hr/geomteh3d/Plohe/pravcasta.png) The curves I chose to define my surface For an interesting example, I chose the ruled surface defined by the parameterized curve \[x(t)=rcos^5(t)\] \[y(t)=rsin^5(t)\] \[z(t)=0.4t\] and the vertical line \[z(t)=0.4t.\] Why I chose the surface and curves I did ...

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...

In mathematics, a surface is ruled if at every point on the surface, there is a distinct straight line on the surface. To better illustrate this concept, I give a picture of a ruled surface  provided by user Ag2gaeh  from the Wikipedia page regarding this topic.  Point P lies on the surface, which has been constructed using only straight lines between the two arbitrary black curves. It is also possible to have a surface which has two distinct lines that lie on it at every point. These surfaces are doubly ruled. A famous example of a doubly ruled surface is the hyperboloid of one sheet. Below is an image of the hyperboloid of one sheet defined by two sets of distinct lines, rendered in OpenScad using code written by Dr. Kate Ponto. Since a hyperboloid of one sheet can be constructed using only straight lines, this allows for their application in architecture and engineering since the doubly ruled structure provides exceptional support. Many people are able to rec...