Solar Halos and Sun Dogs talks about how to create dot patterns that capture the beautiful solar phenomena that occur when light passes through hexagonal ice crystals suspended in the air.

In the implementation section I mention that I use a wall-based ray-crystal intersection algorithm. This works fine, but Eric Haines has presented a simpler and faster algorithm in his article, "Fast Ray-Convex Polyhedron Intersection" in Graphics Gems II. According to Eric, the code in the book has a bug, so make sure you use the online version of the implementation.

Ronen Barzel observed that I don't describe how I rotate the crystals into place, and that different rotation methods will yield different statistical distributions of ice crystals, and thus statistically different types of images. He's right on all counts. My approach to orienting the crystals was simple: my interface let me pick independent minimum and maximum values for rotation about each of the X, Y, and Z angles. To orient a crystal, I picked uniformly-distributed random numbers in each of these ranges, and rotated the crystal sequentially around the X, Y, and Z axes. This will not generate a really uniform distribution of orientations, but I haven't seen any artifacts. I doubt that a more sophisticated method would yield images that were visually distinguishable from the ones I made, but if someone implements such a method I'd like to hear about how the results compare.

Errata: On page 85, towards the end of the second paragraph, the reference to Figure 2 should have been Figure 3. On page 85, I say that the viewing angle in the figures is 180 degrees, and then on page 86 I say it's 90 degrees. That latter should have been a half-angle of 90 degrees - the images are fisheye views of an entire hemisphere, so it's 180 degrees across a diameter, and 90 degrees across a radius. Thanks to Eric Haines and John Dill for pointing out these errors.

Computer-Generated Solar Halos and Sun Dogs picks up where the first column left off, and extends the simulation to smooth color images.

You can download an AVI film (1.06 Mb) containing 90 frames of a smooth, color sunrise, showing the upper and lower tangent arcs. Four frames from the movie are shown above. The frame number indicates the number of degrees made by the center of the sun with respect to the horizon. Thus, frame 0 is right on the horizon, so we shouldn't be able to see the lower half of the picture (though thanks to the magic of computer graphics, we can see what the sky might look like if the earth were invisible!). The camera tilts up with the sun, keeping it centered in the image, until the sun is overhead.

Frieze Groups talked about the applications of these linear symmetry patterns to computer graphics. I gave a descriptive "proof" of why there are only seven frieze groups, and discussed how to recognize which of the seven any given pattern was built upon.

Polyhedral Origami is a popularization of the work of Tomoko Fuse, and Rona Gurkewitz and Bennett Arnstein. The column shows how to build the five platonic polyhedra by simply folding up pieces of paper. I also have a lame example of a teapotahedron; I encourage anyone with a better origami teapot to send me a folding diagram and a picture of the result for a future column.

More Origami Solids takes us to three of the Archimedean solids: the truncated tetrahedron, the cuboctahedron, and the icosadodecahedron. I talk about how to fold these, and how they lie halfway between the duals formed by pairs of Platonic solids.

Unfortunately, due to space constraints about half of the column had to be cut; I plan to post the missing pieces (with pictures) here.

Hey, Buddy, How Do I Get Into the Siggraph Electronic Theatre? was a joint column written with Jim Blinn, providing some tips for getting your animation accepted for the Siggraph "film show".