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Friday, July 23, 2010

M208 - What's missing ?

Suppose you want to do some 3D graphics programming. Think of anything between writing low level code in the game-industry and developing educational flash movies, JavaFX or whatever is buzzing at the moment. Math wise you need to be very good in Linear Algebra. You must be able to implement the elements of the group O(3). In MS221 we learned everything about O(2) and SO(2). MS221 means you can do anything in R2. I expected that M208 covered everything in R3. This is not true, unfortunately. M208 does not cover the rotations around a line for example.

Rotating around a line takes a translation a rotation around one of the x,y or z-axis, another rotation around one of the x,y or z-axis and another translation. The two angles of rotation must be calculated first btw. - The exact strategy has to be learned from some mathematics book about the subject.
Rotations in R3 become very simple if you perceive R3 as part of R4 which can be mapped to the space of quaternions. A rotation then becomes simply a calculation of three quaternions.

Remark 1. Quaternions are buzzing. They did so between 1850-1900 or so when they became rather obscure. Things changed: nowadays programming languages with a 3D graphics API have standard classes for quaternions. Quaternions are things we ( as mathematicians ) must know about. That's why they are painfully absent in M208, the course where they belong most imho. Quaternions should be standard knowledge especially since they are very easy to learn. They fit in nicely in the sections on complex numbers.

Remark 2. Mathematics books require a different reading and study protocol than Open University course packs. I like the course packs. They are very efficient. Are fun to read, etc. They do their job. Transferring knowledge from the pack to the brain of the student. If you are only used to OU course packs picking up knowledge from books ( not even mentioning articles ) is very difficult. Imho at least one topic should be learned from a book in standard mathematics format, i.e. in ( densed ) Definition - Theorem - Proof - Example - Exercise format.

You must have guessed that I have been studying the quaternions. They are very interesting indeed. The quaternions aren't final though. Expect the octonions too. After that sedonions come in the picture, I believe that these numbers can be divided by zero. ( Wouldn't that be interesting?!

Example: Hopf Fibration article.

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