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Stereographic Projection of a Diamond

Something I have always found fascinating is representing representing three dimensional images in two dimensions. The most common example of this can be found in photography and video recording. The events recorded clearly happen in 3-dimenisons, but it can only be replayed on a 2-dimensional screen (barring any awesome technology that lies on the horizon). Stereographic projection is especially useful for wide-angle views captured by a camera with a fish-eye lens like the picture seen here:
Credit to NASA astronaut Andrew Morgan for the amazing picture of the southeastern Mediterranean coastline. The advantage of stereographic projection in cases like this is the ability to capture large areas within a photo at the cost of distortion. The distorition is a result of mapping an originally flat image to a spherical image; in the example above, this allows for more of the coastline to fit in the single picture when compared to a flat image.

Ok, But What is Sterogrpahic Projection?

The core mathematical idea that drives the process of photos like the one above is mapping a spherical image to a plane. One way to think about this concept is wrapping the square coordinate plane around a sphere. Each point on the plane now maps to a point on the sphere, represented by its position on the sphere. Another way to form this sphere from the plane is to construct a sphere that lies on the z-axis. Any point on the sphere can be mapped to the xy plane by drawing a line that starts at the north pole of the sphere to a point on the plane. The point at which a line intersects the sphere maps to the endpoint of that line on the plane. This idea provides a function that can map a 3-D object to a flat plane. It is important to note that the entire surface of the sphere is defined except for the north pole, the starting point of the line used to define the planar image.

Properties of the Projection

An interesting visual application of stereographic projection is the size of shapes on a sphere. The earlier definition of a function that uses stereographic projection is the intersection on the sphere with a line from the pole of the sphere to the xy plane. Another property of this projection is that uniformally spaced points on the sphere are not uniform when mapped to the plane. This phenomenon actually explains the distortion of fisheye lenses and why flat maps of the Earth are unable to be completely accurate. This means that uniform spacing can only exist on the spherical pre-image or the planar image.

Diamonds on the Sphere

Let's consider a case in which uniform spacing on the plane is desired. Let's create a sphere that, on its surface, has diamonds of varying sizes. If a light were shone through the north pole, the projection of light through all of the diamond shaped holes onto a surface would show an array of diamonds that were all the same size. The following image shows the CAD design for such a sphere:
The red dots in the image show the destination of the diamond shaped light for each hole in the sphere.

The diamond shape consists of a trapezoid and an iscosceles triange that is translated such that the short side of the triangle and the long side of the trapezoid share a line. This was not terribly difficult to implement, but required elements of geometry to get close to what I wanted followed by some guess and check. I chose a diamond shape because it consisted of simple geometric shapes and I found it aesthetically pleasing. Additionally, whenever I buy a diamond ring for my partner I can use this concept to justify that since the tiny rock and the giant rock have the same image they are actually the same size. Still working on I'm gonna explain my way throught that one.

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