I love pop-up cards. I love to make them, send them, and receive them. But as anyone who has tried to design their own cards knows, creating these things is really hard. You have to cut, glue, fold, test, modify, cut, glue, test, on and on forever in order to get the pieces to move correctly. If the card is even mildly complicated, designing the card can become an endless project.

I decided to create a program to help me design my cards, so that I could just move the pieces of paper around into the positions where I wanted them, and then the machine would figure out the correct shapes, fold lines, and glue points that are required.

It turns out that's not too hard to do. It also turns out that most of the many varieties of pop-up card mechanisms can be boiled down into just a few versions, and most of those share the same underlying geometry. That underlying geometry is quite elegant and nice, and you can write down the formulas on a 3-by-5 card and have room left over for a few doodles.

I've used my system to make a few different cards for fun, and to help me design one that I sent out recently when I moved. I'd love to get a more robust version of the program out into the hands of kids (and adults) so they can use it to make their own designs.

You can see some more examples of cards in my two-part CG&A column.

On the left is a typical pop-up card during the design phase. This was a moving card that I designed when I moved from the East coast to the West. A raised horizontal road wound through three vertical layers representing different slices of the United States countryside.

This was an ambitious project, because this was meant to be only the first page of a three-popup mini-book. The time required to get the design to work just right was prohibitive, and eventually I sent out only one page, shown on the right. This was originally meant to be the last page of the book. When you open the card, the envelope pops out of the mailbox (the envelope showed my new address). Because I'd eliminated the middle fold of the book (which gave my phone number on the inside of a phone that popped up), I put my new phone on the red flag that pops up above the card.

The point of the tool is that you don't really need to know anything about how pop-up cards work to make one. And you certainly don't need to know the underlying mathematics or geometry. But if you're going to program such a tool, you'd better have it worked out!

There are lots of fun pieces of geometry in this project. Perhaps the most interesting is that at one point we need to find the two points that are defined by three intersecting spheres

Here you can see three slightly-different spheres sitting above a plane. They all meet in one point (the other point of intersection is below the plane). The dotted line and the circle show the point in the plane we need to find. If you like these sorts of problems, you might want to think about how to find that point before reading on.

My solution uses a geometric idea called the radical axis. If you draw a set of triangles in the plane of the three circles above, you can create a set of three lines that join the regions where each pair of circles overlaps, as shown in the next figure.

The picture on the right suggests that the three lines joining the overlaps of the three circles all meet at a single point. Just looking at a picture isn't a proof, of course, but you can in fact prove that these three lines, which are the radical axes, indeed always do meet at a single point, and it's exactly the point that we want: the one that lies under the intersection of the three spheres.

With this and a few other geometrical results under our belts, we can write a program to design popup cards that will always work, regarless of complexity.

The tool includes a few additional helpful bits, such as packing the individual pieces tightly on the page so that you don't waste much paper when printing and cutting them out, printing very light folding lines so you can make precise and accurate folds, and color-coding the parts of the paper where the pieces glue together.

You can download a copy of the technical report from http://research.microsoft.com/scripts/pubs/view.asp?TR_ID=MSR-TR-98-03

Some of this work is covered under Patent 6,311,152.

You can also see some more examples in my CG&A column.