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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.




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