Fourier Polygons discusses a fascinating relationship between the Fourier analysis of signals and polygons, which Alvy Ray Smith told me about. It turns out that if you think of the vertices of points (x,y) as complex numbers a+bi, then you can decompose polygons into "basis" polygons just as you can decompose signals into basis signals like sines and cosines. It's a really cool and beautiful thing.

String Crossing looks at those string-art figures you may have made at camp. You hammer a bunch of nails into a board, and then tie metallic string from every nail to every other nail. Graph theorists call this a complete graph. How many crossings are there in such a pattern? The path to the answer involves noticing and making use of all sorts of unexpected patterns that keep showing up in the formulas and geometry.

An Open and Shut Case tackles the issue of camera shutters. In computer graphics we usually assume that our shutters are perfect: they open up completely in no time at all, and then shut down in an equally instantaneous moment. But real shutters aren't like that. Iris shutters open up to reveal an enlarging hole, which shuts down as they close. Guillotine shutters have a plate that drops down (or moves sideways) to let light pass to the film. Motion-picture shutters use a couple of pie-slices at a fixed angle to each other. It turns out that these different shutter geometries can really influence the images we make. If we want our computer-generated effects to match up with live-action images, we need to pay as much attention to film shutters as we do now to interlace for video projects.

O Say, Can You See? looks at the efficiencies of different kinds of windshield wipers. Some go left and right in a short arc, others sweep out a half-circle, and some move left and right with almost no vertical motion. Some rough approximations and simulations shows that the types of wipers we have on most of our cars are the best compromise between keeping the rain off and not blocking the driver's view of the road.

Celtic Knots Part I starts off a 3-part series on this topic, which has always fascinated me both technically and artistically. I talk about the construction technique pioneered by George Bain and his sons, and show how it can be efficiently implemented in a computerized assistant. The goal isn't to turn the computer loose making art; it's to design a tool that helps us make art that satisfies our vision. I show how to draw pleasing curves for a variety of situations, and how to determine the proper overlapping and underlapping of a knot.

Celtic Knots Part II gets further into the theory of knotwork. I introduce my idea of "snakes" derived from knotwork graphs, and show how to make a set of tiles that, when placed together properly, will create a Celtic knot. I also talk about knotwork on grids that aren't square, which opens up all sorts of new design possibilities.