Skip to main content

Stereographic Projection

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 Ears. I chose this shape because I thought Mickey Mouse would be a fun shape to make and I believe this might be a technique Disney Parks use. For example, they might use it when projecting their light show on the castle at Magic Kingdom, a way to create park maps, or for special effects in their rides.

For the model, on the top half of the sphere are four cutouts with a circle at the very top. The circle is located at the point [0, 0, 20] and represents the projection point, so our light source will be shining through this hole to project the design onto a wall.  

How To Create the Stereographic Projection

To start this model, I first created what the projection on the plane should look like without the sphere. Using the projection point as the starting point, I created a loop of Mickey Mouse ears that project onto the xy plane. Each mouse represents a projection from a different angle with the projection point. Here is what the xy projection plane looks at from the top and bottom view.




Each mouse shape was created by projecting three cylinders overlapping each other onto the plane. Initially, it was challenging finding the right balance between overlapping coordinates without the shape looking like a bubble, but eventually found a good distance and decreased the radius of the spheres that form the ears. Each point on the plane will be like the point P’ described above.

Now that we have the projection plane mapped out, it is time to add the sphere that will map out this projection when a light is shined through the projection point N. When creating the sphere, the intersection between the initial projection (all the points P’) and the sphere will be removed creating the holes seen in the picture, which is like the point P described at the beginning. We now have a sphere that will project an array of Mickey Mouse ears.




Comments

Post a Comment

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...
Post a Comment
Read more

Knot 9-31

Knot 9-31 A knot is mathematics is defined as a closed, non-self-intersecting curve that is placed in three dimensions and cannot be the "unknot". The main difference between a knot in the real world and a known in mathematics is that a knot in mathematics does not contain any extra strands. The example following will help visualize this. Today, I have specifically chosen knot 9-31 from the Knot Atlas. This knot is very unique and contains some very interesting properties that we are going to look into. The Crossing Number For my knot today, I chose knot 9-31 from the Knot Atlas. This knot contains 9 crossings! The Unknotting Number The unknotting number is exactly what is sounds like. This is the minimum number of times the knot must be passed through itself to untie it. Luckily, the Knot Atlas is super useful and provides the unknotting number for us, but I still...
Post a Comment
Read more

Do Over: Integration for Over Regions in the Plane

Introduction Earlier in the semester, we visited the topic of integration for over region regions in the plane. This was our first experience with taking integration in the third dimension. To do this, we had to use double integrals to calculate the volume of a region between two surfaces. This topic seems relatively straightforward and anyone who has taken a higher calculus class knows of double integrals. Today, we are going to revisit this topic for a couple of reasons. Why Double Integrals Again? As mentioned before, I am revisiting this topic for a couple of reasons. One of the main reasons is due to how the 3D print turned out. Due to the function, I chose, the final rectangular prism was stand alone. When printing, this resulted in a sloppy prism that lacked structural integrity. The image below is a reference to my original design that includes the prism that had the issue. The main cause of ...
Post a Comment
Read more