### How to make a paper spherical panorama

Photos usually only show a rectangular fragment of the scene when the image was taken. Typical panoramic images display more of a landscape, but they still don’t capture the whole picture. Even a 360˚ panorama only captures a horizontal circle, leaving out the ground and sky.

What if you have a nice scene that you’d like to convey in full? I’ll show you how to make a spherical panorama, making use of all 4π steradians with software, a printer, and some paper.

My example is an unseasonal Christmas scene because I played with this a few months ago. Maybe we can imagine this is for a Christmas-themed celebration held in winter in the southern hemisphere.

#### Tools for creating panoramic images

The first free tool I’d recommend is Microsoft Photosynth, which generates panoramas and other immersive images and videos. You can use Photosynth on a computer or as an app on a smartphone or tablet. The iPhone app automatically snaps pictures as you aim the camera in different directions, and it stitches them into a panorama. Here is an interactive panorama I made of the inside of my house:

The second is Hugin, a free tool for the computer that gives you more freedom to construct your panorama, although it has a higher learning curve. You can import photos from any camera, and the software will stitch them together. You can also help it find matches between the photos to help it stitch better, and change settings to make the panorama as seamless as possible. You can also choose different projections, which allows you to create some interesting artistic effects. For example, one of my favorite projections is called the stereographic projection, which is employed here as in this “small world” effect reminiscent of *Le Petit Prince,* taken in my backyard.

#### Figuring out coordinates and projections

These tools can make a 2D representation of a scene, but we want a 3D sphere. You’re probably familiar with the problem of turning a globe into a map of the world: you have to make a projection to translate the 3D globe into the 2D map. Here we want to go in the other direction.

For this project, I prefer the equirectangular projection because it has the nice property that the horizontal coordinate is the longitude and the vertical coordinate is the latitude. This makes the mathematics of the mapping much simpler. Now the mapping to the sphere is

$$\begin{bmatrix}x\\y\\z\end{bmatrix}=\begin{bmatrix}\sin{\theta}\; \cos{\phi}\\ \sin{\theta}\;\sin{\phi}\\ \cos{\theta}\end{bmatrix},$$

where \(\theta\) and \(\phi\) are the standard spherical coordinates. But having these coordinates isn’t enough; I want to make a physical model of this sphere.

The solution is to mathematically find the correct shape for 10 (or some large number) of petal-shaped images, such that the spherical image can be printed on several flat pieces of paper, and then rolled together to form an approximation of a sphere. The sphere is partitioned into some number of sectors (regions with \(\phi\) constrained between \(\frac{2\pi i}{n}\) and \(\frac{2\pi (i+1)}{n}\)). The paper petals will then join, or form a cobordism, between these endpoints.

After plotting and subsequently printing the correct boundary function in Mathematica, here is what the petals look like when they are joined at the point that will become the zenith:

#### Building the photo sphere

Now that we have the shape planned, we can take a panoramic image and project it down to the petaled surface. You can make a full sphere panoramic image (as an equirectangular map) using the software of your choice. The height should be an even number of pixels.

Then what I did was import the photo into Mathematica and write a program to process it. If you’d like to try my Mathematica code for this, you can import this text file into Mathematica.

I decided to alternate the directions of the petals to minimize paper waste (as well as cost).

I cut out the petals, and I rolled each of them to make it easier to join them together.

I then put tape on the white surface (the inside) to hold the petals into the spherical shape. The result was two hemispheres:

Here’s the other side:

It’s beginning to look like something! Now it’s just a matter of gluing the two hemispheres together:

And it’s done!

#### Things to try

You can use fishing line to hang the sphere. When it rotates, occasionally the sphere appears to turn inside out, and it looks like you are viewing the room through a circular portal through a fisheye lens.

It would be fun to improve on this idea using less reflective paper to reduce glare, and to try to make the seam at the equator less conspicuous.