Discovering Stereographic Projection

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When I studied math in graduate school, I was entranced by the beautiful geometric worlds that mathematicians discovered and explored. Once I got my PhD, I left mathematics to work in the tech industry. During my studies, I discovered that life in academia was not what I had imagined.

Despite leaving academia, I haven't lost my fondness for it. Now that I have some years of professional software experience under my belt, I thought it would be fun to use my coding skills to bring these fantastic geometric worlds to life, via interactive 3D visualizations, like the one you see above. You can interact with the visualization by dragging and zooming on the canvas and by sliding the parameter controls.

Stereographic projection

The visualization above depicts a mapping known as stereographic projection that maps all of the points of the sphere, except for its north pole onto the plane. The way that the mapping works is as follows: Given a point on the sphere, we draw a line through the point and the north pole. Stereographic projection maps this point to the point where that line intersects the plane.

From the visualization it is clear that points are mapped one-to-one and cover the plane. For people who aren't familiar with these kinds of constructions, it is surprising that something bounded like the sphere can be mapped in such a way as to cover an infinite plane!

Here's a brief explanation of the control parameters. Theta and phi are angular parameters commonly used to specify a point on the sphere. If you draw a line from the origin to a point on the sphere, theta is the angle between that line and the vertical z-axis. So as theta goes from 0 to π, the point of the sphere goes from the north pole to the south pole along a vertical plane. As for phi, if we project the radius line onto the blue plane, phi is the angle that the projected line makes with the x-axis. So as phi goes from 0 to 2π, the point goes around the sphere along a horizontal plane.

You can see that points near the north pole are mapped very far away from the origin. As a result, stereographic projection allows us to interpret the north pole as a point at infinity. So stereographic projection can be viewed as a way of extending the plane to have a point at infinity in a very simple and concrete way. This can be very useful when we want to analyze functions on the plane that have a natural way of being extended to the point at infinity, such as Möbius transformations.

By playing with the visualization, we can discover interesting things. I wondered what kind of shapes we would get when projecting circles on the sphere to the plane. As it turns out, circles get mapped to circles! When you try to imagine this in your mind, it isn't at all obvious. However, when viewing the visualization on the screen, it is totally clear. I think this is a great example of how visualizations can shine light on certain relationships, that would have been obscure otherwise

It might not be obvious how the parameters are used to specify a circle on the sphere. I used a point on the sphere (given by theta and phi) to specify the central axis of the circle and a parameter between 0 and 1 to set the distance of the center of the circle from the origin.

The power of visualization

I haven't been worried about writing too precisely or in too much detail, because I want to focus on what is most novel and interesting about this blog, which is the geometric visualizations. There are many resources out there where details are provided. With the visualizations in hand, it should be much easier to read through and understand the details if and when the time comes.

Traditionally these mapping and transformations are described with a text description, and very little visual aid, perhaps a single image. For this reason, many people think that these geometric ideas are very complicated and hard to understand. But in fact, when visualized, it becomes clear that everything is actually very simple.