Essential Math WeblogThoughts on math for games

12/29/2004

Errata: Tensor Product

Filed under: Erratical,Mathematical — Jim @ 10:31 pm

William Brennan points out that on page 84, the equation

$(\mathbf{u}\bullet\mathbf{v})\mathbf{w} = \mathbf{u}(\mathbf{v}\otimes\mathbf{w})$

is not correct. There are two ways to rewrite this. In one the intended order is correct, but is missing the transpose operator to indicate that it’s a row vector:

$(\mathbf{u}\bullet\mathbf{v})\mathbf{w} = \mathbf{u}^T(\mathbf{v}\otimes\mathbf{w})$

Alternatively, you can generate the same result with a column vector by doing:

$(\mathbf{u}\bullet\mathbf{v})\mathbf{w} = (\mathbf{w}\otimes\mathbf{v})\mathbf{u}$

In Chapter 3, we simplify the Rodrigues formula (page 123) and general planar reflections (page 128) by using the tensor product. Since the arguments of the tensor product are the same in these cases, the ordering doesn’t matter. However, the ordering of the arguments needs to be reversed in the shear matrices on page 132, so

$\mathbf{H}_{\hat{\mathbf{n}},\mathbf{s}} = \left[\begin{array}{cc}\mathbf{I}+ \mathbf{s}\otimes\hat{\mathbf{n}} & \mathbf{0} \ \mathbf{0}^T & 1 \end{array}\right]$

and

$\mathbf{H}^{-1}_{\hat{\mathbf{n}},\mathbf{s}} = \left[\begin{array}{cc}\mathbf{I}- \mathbf{s}\otimes\hat{\mathbf{n}} & \mathbf{0} \ \mathbf{0}^T & 1 \end{array}\right]$

Other tensor product arguments may need to be reversed elsewhere in the text, though I can’t find any at this time.

Errata: Figure 1.9

Filed under: Erratical — Jim @ 9:23 am

Reported by Richard Ruth: In Figure 1.9 on Page 26, the label on the hypotenuse should be $\sqrt{x^2 + y^2}$.

12/28/2004

A New Beginning

Filed under: General — Jim @ 4:04 pm

Been a little busy lately trying to get a game out, but I did manage to squeeze in some time to create this new site using WordPress. It has some advantages in that I can display math and code a little easier, as you can see. I’m using MimeTeX for the math display — it’s got some issues with bold fonts, but otherwise I can enter the formulas I need in a LaTeX-like way without anyone needing to install MathML fonts. For the code, I’m using iG: Syntax Hiliter, with some tweaks to the colors.

Hopefully I’ll have some time in the next few weeks to type up some more errata, and what I and my co-lecturers are up to as we prep for GDC 2005.