Here’s a hot wire foam cutter that I made from scraps, following a general design I saw in Make magazine a couple years ago. I picked up 20 feet of 30-gauge nichrome (nickel chromium) wire on ebay for $2.09, shipping included, and the rest of the stuff I happened to have on hand. Aside from the wire, I used a 12V power supply [see power details below] that I got from Radio Shack decades ago, a bit of peg board, some scraps of pine, a few nails, a foot or so of standard household electrical wire, and a steel rod.
Hot Wire Cutter
I used the ground wire to fasten the cable tightly against the rail.
Hot Wire Cutter, Side View
From the bottom, you can see how the ground wire loops around, and how the nichrome wire is attached.
Hot Wire Cutter, Bottom View
Here’s a name that I carved out of foam. I tried using carbon paper to transfer a printout to the styrofoam, but that wasn’t very effective. Instead I mostly followed the little dent made by the pencil as I tried to trace.
A Name Cut out of Styrofoam
I originally used a 12V power supply because that’s what I had on hand, and it didn’t look like it would pull enough current to melt the wire. At about 2 amps, though, it made the wire glow, and I knew that was more than enough heat for the styrofoam. So I decided to reuse an old camcorder power supply that was rated at 7.5V/1.6A. With this power supply, the contraption ended up drawing 1.1A and the wire was at a more “Goldilocks” temperature — not to hot and not too cold. It slows down the cutting a bit, but I think it also makes it more controllable.
Here are some great resources I found. I wish I’d found these before I started!