360 degree photos and Math.

  If you follow any crowd funding places, especially the ones with cutting edge technology, you'll have seen several offering fish eye lenses for various phones. I was recently offered a fish eye lens to pop on my iPhone to make 360 degree pictures and videos.  I don't own an iPhone so it makes no sense to buy it.

I've been trying to figure out how to incorporate 360 degree material into my class but its taken me a bit.  However, I  have found a few things. 360 degree videos are often referred to as spherical videos which use Mobius transformations for purposes of editing.

which are used to map a one to one mapping from one domain to another domain. 

Mobius transformations are used to map a one to one from on domain to another.  In addition, these are transformations of the sphere such as making regular rotations of the sphere, zoom like transformations, and other similar effects.  The first transformation turns the pixel coordinates into angles, the second transformation takes it from the equirectangular projection while the third deals with complex numbers.

On the other hand if you look at a 360 degree photo also known as a fish eye projection, it appears distorted but its not. Its actually a three dimensional projection onto a two dimensional plane.  This is what gives it the peculiar look.  There are programs which convert 360 degree photos into landscape shots so they look more "normal."

In simpler terms, the math involved takes an image which is circular in shape and creates a more rectangular shape through the use of "uncurling" the lines. Think of it this way, your source image is  2l by 2l and you want to make it so its destination image is 4l by 1l. Only the pixels inside the inscribed circle of the source make it to the destination image.

The pixels along the top of the destination image come from the circumference of the source image. The formula for the math conversion is (4l-x)/4l * 2pi where x is the Cartesian X axis.  In addition, when iterating from left to right on the destination image is the same as going clockwise on the source image. 

The radius (remember a circle has a radius) is calculated as l - y where y is the Cartesian Y axis.  So iterating from top to bottom on the destination image is the same as going from the edge to the center of the source image.

There are software programs out there with the math already present so you can convert your photos without having to do the math but its nice to know what these programs do when the conversion is carried out.

Let me know what you think.  I'd love to hear.