Interactive Pop-Up Card Design, Part 1 talks about how to build a system to you create your own pop-up cards. These things are a lot of fun to make, give, and receive, but they're notoriously difficult to design so that hey they just right. This column gives the basics of the essential geometry. The idea is not to create cards to be viewed on the computer, but rather to use the computer to help us create real paper cards to give to each other in real life. For more details, click here.

Interactive Pop-Up Card Design, Part 2 continues the discussion, and talks about how to actually build a Pop-Up Card Design Assistant. I also discuss important features like placing guide lines and conserving on paper when printing out the final cards. For more details, click here.

Putting the PiecesTogether was inspired by news coverage of the Enron scandal. A famous event in the early stages of the breaking story focused on how Enron shredded key documents, and then gave the shredded paper to its laid-off employees to use as packing materials when taking their personal belongings home from the office. One woman showed up in a Senate hearing with several big boxes of strips of paper that had been shredded using conventional shredding machines. It seemed to me that it shouldn't be that hard to put the strips back together again and reassemble the evidence. It turns out that I was right. This column discusses how to build a system that is fast and accurate enough to reassemble a big box of shredded documents in just a few minutes on your PC.

Towards the end of the article I mention how it would be fun to reassemble broken pottery. It turns out that the SHAPE lab at Brown University has been doing research into just this problem, under the direction of David Cooper and Frederic Fol Leymarie. You can read about their work here. Some of their recent papers are here, and look at a bibliography of recent papers in the field here.

Duck! talks about the V-shaped waves that form behind a fast-moving body on deep water. I show how to compute these waves for objects moving both along straight and curves paths, and also show how one can create a five-pointed star with them.

Robert Stepanek found a typo in the equation for shallow Kelvin waves. The expression for Ky(u) should be:
Ky(u) = -Psi/4 (sin(u) + sin(3u))

Getting the Picture looks at different ways to create an interesting "approximate" version of a nice starting image. You might want to do this to add visual interest or complexity to a scene, or to use as source material for an interesting transition. I looked at how we might create a version of a starting image using circles, half-circles, triangles, boxes, meshes, and even Voronoi diagrams. The whole thing was driven by a simple adaptive refinement algorithm. You can see movies of the different figures being created by going here. Aaron Hertzmann at NYU has carried on some interesting work in this area; see his CGA 2001 paper here.

Digital Weaving, Part I starts the discussion of weaving. I talk about the basic idea behind how looms work, and how to read a weaver's draft. I also discuss The Weaving Equation, which is a nice little way I found to wrap up what's happening in a draft into a compact mathematical experssion. By making a few additional definitions, we can greatly simplify that equation into just a couple of quick table lookups and a single logical AND operation.