I was told that it would be a good idea to show an example of me doing this. That is a good idea, so here it is. I'm working on a triceratops puppet now, and I needed to make a body. Ergo, I made an ellipsoid.
Beware--I was making a pattern on a piece of black posterboard (it's all I had) with pencil to make it show up. Hopefully it's not too painful to see.
I determined that the height would be 6 inches (making a=3) and that the width would be 4 inches (making b=2). Following the formulas in this post, I found the height and width of the pattern.
I drew an extended line to the left. Somewhere on that line is the center of the circle that will have a radius making both the top and bottom of the height line and the far right point of the width line all points in the same arc.
I copied the angle on the left and mirrored it on the right. You can see where the two arcs intersect. I drew a line from the top of the height (the longer one; this picture has been rotated 90 degrees), through the intersection of the arcs, all the way to the dotted line. This formed the isosceles triangle that was needed.
Using the line that I just drew, I took a string and a pencil and created an arc, connecting the three points that I needed. Because of my 'simple' compass, the line wasn't nearly as smooth as I would have liked, but it'll work.
I recreated the same arc on the other side, leaving me with a pre-cut pattern.
The pattern after cutting.
I copied it eight times onto the foam.
I then cut out the pieces. The foam had been in my closet for a little while, so it was slightly curved, which actually worked to my advantage later.
I coated the sides of the pieces with contact cement and then began sticking them together. Here, you can see it halfway done.
The ellipsoid, halfway done.
The inside, halfway done.
I finished it off, resulting in this: the final product.
It worked perfectly. I cut off the top and part of the bottom to make the holes for my hand (since this will be the body of my puppet).
So that's how it works in reality, not just in theory.