The Viewable Sphere: Mathematics Meets Photography

Dr. Bruce Torrence had a lecture about photography that incorporated math. He began by speaking about the "viewable sphere" . The viewable sphere is something very much like the picture above. The reason this came about is because a spherical photograph is difficult to print, so they have this concept which is similar to the concept of the panel photo app on the iPhone (sorry if that's not the right name for it). They call this concept the Equirectangular Projection. Using this concept is the easiest way to change images (it uses the same format as maps do--longitude and latitude).

There are four steps that need to be taken in order to do an equirectangular projection:
Step 1: Take an ordinary photo. (Later an application of mathematical transformation to the digital image will turn it into an equirectangular projection).
Step 2: Take another photo from exactly the same vantage point.
Step 3: Continue to shoot photos covering the entire viewable sphere.
Step 4: Stitch together to make a complete equirectangular panorama (The equirectangular panorama will be exactly twice as wide as tall).

A neat trick that Dr. Torrence suggested is if you are doing it handheld, use a string with a penny tied to the end and tie the other end to the camera to keep a mark on the ground in order to keep the distance the same. It is very important for all the photos to be the same height!

The horizon in the 3D scene corresponds to the equator of the viewable sphere while the sky (which is everything above the horizon on the sphere) lands outside the horizon circle.

It can be shown that any circle in the stereographic image plane corresponds to a circle on a viewable shpere. A line in the stereographic image plane corresponds to a circle through the North Pole in the viewable sphere. Any line in the original scene corresponds to a great circle on the viewable sphere corresponds to a line/circle in a stereographic image. Lines are helpful because it means you can tilt the sphere which allows us to emphasize certain features while keeping the horizon circular.

Other interesting tid-bits are if you tilt the sphere completely upside down, you get a tunnel wall. If you tilt the sphere 90 degrees, the horizon (equator) becomes a straight line.

I really enjoyed this talk, I thought the images were fascinating and I learned a lot! I had not seen pictures like these before and I learned that it takes a lot of sophistication. I would definitely encourage others to look some of these pictures up because they are really amazing!