Unwrapping the Egg

Several years ago I was asked to create a web page that would allow young children to paint a Ukrainian Easter Egg(Pysanka). They would do this by selecting a color from a pallet of pre-selected colors. After selecting a colour, they could fill in any area by simply clicking on the image.

Not long after the initial version was released, I was asked to create some sort of preview of how this image would look like on an egg.

This was one of my first experiences using WebGL on a project. To show the egg, I generated a polygonal revolved sphere and the radius to “1 + 0.2 * theta”; where theta is the current latitude angle. This created the desired egg shape.

I then placed the coloured image onto the with respect to the eggs normal vectors and whether we were mirroring the image on two sides of the egg or four.

Finally I added three lights to make the egg pop a little bit. On the right side, there is a white light to view the true colors of the egg, and on the left side there is a yellow and blue light. These two lights are positioned to mimic the coloring of the Ukrainian flag and was done to tie it into the rest of the site.

For a few years, this is how the page was used; but, recently I was asked if we could make a physical 3D version of the egg available.

The first thought was, could we generate a file that can be used for 3D printing . The answer to this is of course yes, the egg itself would be a relatively safe object to print, but since this site was for young children(at about Grade 3 in elementary school) the chances that they would have access to a 3D printer were quite low.

We decided the best option would be to print out a two dimensional version of the egg which could then be folded into a 3D model. This was a simple enough solution, what could possibly be difficult about this? Turns out, a little something called Gaussian curvature would show that this is technically impossible.

In short a sphere(or egg) has a positive curvature while a piece of paper has zero curvature. When trying to transform shapes while keeping the same surface area, this total curvature has to remain the same; otherwise you would see rips along the shape where the tension was too great.

So, we knew this would never be a perfect solution but an approximation would work just fine. Luckily I was able to find a great paper, titled “Wrapping spheres with flat paper”, which addressed this issue (https://www.sciencedirect.com/science/article/pii/S0925772109000182). In this they described several methods of generating this approximation while limiting materials.

I extracted their n-petal method of rendering a sphere on a sheet of paper. To do this I used their formula to find the outer edges of the petals. I would then interpolate between that edge and the middle to find where each vertex of this shape would be placed. I then rotated each petal based on their angle on the sphere. Finally I increased the height of each petal by 10%. This gave the printout a look which resembled an egg(even if it wasn’t the dimension in the 3D visual) while still being easy to cut out and tape together.

Finally I had to create a border around the image to make it easier to cut out. This was done by creating a stencil with the current printout, then drawing black versions of the printout with an offset of two in each direction.