OK, so I'm a complete amateur, working on my first puppets right now. I am, however, a mathematician (or I pretend to be). I recently realized that I needed to make a body for my puppet. I wanted to make a football shape, but I didn't want to fork up the cash for a Project Puppet design. I decided to do some math to find out how to turn a 2-d surface (namely, foam) into a 3-d shape (namely, an ellipsoid). What follows is how I went about it. Be forewarned that there will be math and formulas, which may be boring to all one of you that will read this post. I'll start with a sphere and then move on to the prolate and oblate spheroids (stretched sphere, squashed sphere).
If you don't want to read the process, just go to the end of the post, and you will find simple directions on how to make a pattern.
Take a look at my handy picture. You will see that in a sphere the distance from the center to the surface on all three axes (X, Y, Z or length, width, and height) is the same. I've labelled it R (for 'radius', since I'm clever). R will make our calculations very convenient, as you will see if you make it to the spheroids.
When we make a sphere out of foam, we will need to focus on the surface. (since it will be hollow). We will cut it into kayak-shaped pieces, just like the longitude lines on a globe (see picture to the left). I will use eight pieces (though you can easily to more, and less easily do less; I wouldn't recommend less than five or six).
How will we figure out the measurements for these pieces?
First, consider how tall you want to make your sphere. How big will the puppet's head or body be? That length is the length from the top to the bottom of the sphere, which, as you can easily see, is 2R. This allows you to figure out R (by dividing the length you want by 2).
Now, have you ever noticed that if you cut a sphere in half along any of the axes (X,Y, or Z), the circle that you get is the exact same size? That's important here. We're going to cut the sphere along the X axis (with an X/Z plane). We get a circle (which I've highlighted in blue). You can see that the height of our sphere piece will be half the circumference of that circle.
That's handy for us. The circle has a radius of R, which means that it has a circumference of 2πR (see here for that formula). We want half of that, which is πR. This will be the height of our piece. I have drawn it out in the picture to the left.
What will the width be, you ask? Great question! I am making this out of eight pieces, which means that the width of each piece should be one-eighth (1/8) of the entire length of the sphere's equator. Note that if you are making it out of more or fewer pieces, the fraction will reflect the number of pieces you are making (so, ten pieces, 1/10; four pieces, 1/4). Because this is a sphere, once again, the length of the sphere's equator is equal to 2πR, which means that the length of our piece will be 2πR/8 = πR/4. This is important. It means that the width of our pattern will be one-fourth (1/4) the height. Since a sphere is symmetric in all directions when cut through the center, the width and the height will bisect each other. I've drawn it out to the left. Oh, and don't worry about π right now. We're just going to estimate it as 3 at the end.
Well, this is great. We have the height and the width. we have just one more problem. The sides must be curved, since the sphere itself is curved on its surface. We need to figure out how to connect the endpoints of the width and the height. See figure to the left.
To find this curve is surprisingly simple. We need a circle that has a center on the same axis as our width (the short line). The distance from that center to the end of our width line (the radius of that circle) must be the same as distance from the center of that circle to the top and the bottom of the height line. How can we do this? We need an isoceles triangle, with the two equal sides going from the far end of our width line and from the top of the height line to the center of that circle. Using a compass and straightedge, it's not too hard. Just copy angle M (see picture) to become angle N.
Now, using the compass, make a curve from the bottom of the height line to the top of the height line. Copy that to the other side, and you have your pattern. Make eight of these out of sheet foam, and, presto! you have a sphere.
Check in soon to see how to make a spheroid.