Once upon a time, I was a math major. I shunned engineering as not pure enough — I wanted to be a *scientist*. And I didn’t see that most of the really good mathematics is developed in support of other sciences, particularly physics, and *engineering*. Fourier analysis. Eigenfunctions. Complex numbers. And linear algebra.

I’ve always loved computer graphics, and especially plotting of 3D clouds of points. In order to visualize a cloud of points in 3D, one has to generate a right-eye picture and a left-eye picture, and somehow present these so the correct eye gets the correct picture. The last bit is easy – put the pictures next to each other, the R picture on the L and vice versa, and cross your eyes to get it all working. There are a bunch of other techniques, but this is my own favorite. (Yeah, I haven’t played with 3D goggles yet, other than the venerable old Viewmasters, which were absolute magic when I was 7 years old.)

The other part is a little more involved, mathematically. To generate an image from the perspective of a given eye, map the cloud so that the x-y plane of the display is perpendicular to the eyepoint vector. Then, if you want perspective, shrink the cloud in the distance by moving each point closer to the eyepoint vector by a proportion that varies with the eye distance.

Then do it all again for the other eyepoint.

I’ve been aware of quaternions since my math days, but have never looked closely at them. They can be viewed from the perspective of complex analysis, but are better viewed as linear algebra.

What’s cool about them (from my limited view) is that they provide an elegant and simple way of rotating vectors in 3-space. Here’s the reference that I used to learn how. And here’s the inevitable Wikipedia link ( and another ) to provide more of an introduction.