To Scale!

In our last visit to 3-d math land, we moved from the 2x2 and 3x3 matrices we used to learn how matrices function to the full 4x4 matrix that we all know and love to hate from 3d applications. This time I’d like to add support for scaling to our matrices so we can round out the ways matrices work.

Adventures in the 4th dimension

In our last discussion of 3d math, we started to plumb the mysteries of the matrix. Along the way we discovered two important facts: First, that it’s possible to write an article about matrices with only the merest smidge of a Keanu Reeves mention and second (almost as important), that matrices are just a convention for applying dot products in series. We walked through the derivation of matrices for a series of dot products and shows how hat simple operation allows you to do rotations in two and three dimensions.

Naturally, any TA reading this will be knows there's more. We all know that the matrices we’re most familiar with — the transform matrices that drive animation and modeling — do more than rotate. So this this time out we’re going talk about how translation — spatial offsets — can be packed into matrices.  And we're going to do it in a truly brain bending way.  Sort of.

If none of this sounds familiar, you may want to return to the previous post in the series before continuing.

Dot Matrix

We started our math review with a look at the dot product, and started out by showing how dots work in a minimalist way. This time out we’ll do the same thing the most basic component of 3d math - the matrix.

Dot's all, folks

Last time out I went on (probably a bit too long) on the virtues of the dot product - the operation which takes two lists of numbers and multiplies them to create a single product. The highlight of the whole thing was the cosine dot product - the handy fact that the dot product of two normalized vectors is the cosine of the angle between them.

Now that the theory is out of the way, it’s time to highlight some of the zillions of applications for this handy little operation.

Bagels and Coffee, or, the vector dot product and you

I’ve been boning up on my math lately.
Like most TA’s I’ve cobbled together a bag of tricks from different situations I’ve dealt with over the years, but I’ve never really gone back to shore up my shaky high school trigonometry and pre-calculus. It’s certainly possible (at least, I hope it is!) to be a good TA with only seat-of-the-pants math skills — after all, we have parenting and scaling and all the other cool tricks in our apps to do the heavy lifting for us. Still, I’ve been finding that paying more attention to the math fundamentals is helping me solve problems more efficiently and elegantly than my patented hack-and-slash techniques did.
So, I’m starting an occasional series on some basic math concepts that I hope will be useful to other TA’s. I know it’s been helpful to me - there’s nothing that concentrates the mind like putting something out there on the internet for public commentary - it’s really forces you to think things through… At least, as long as you’re not on Twitter.