Re: [Algorithms] How to derive transformation matrices

[<ro...@do...>]

I wrote

>What I am trying to stress here is an idea that is foreign to most
>graphics people, but which is extremely important to understanding
>what is going on geometrically, an idea that has been mentioned
>recently in another thread in this list. It is the idea that, however
>much you need them in the end for computing, coordinate systems are an
>enormous hindrance to understanding intrinsic geometry.
>

Well, maybe the light is beginning to dawn on some graphics people in
some quarters.  I don't have time to go to Siggraph this year, but I
see in the program a description of a course called "Geometric
algebra".  This is from the description:

"Geometric algebra is a new fundamental language for the mathematics
of computer graphics, modeling, and interactive techniques. It is
especially useful for handling geometric problems, since it allows for
intrinsic (coordinate-free) and dimensionally seamless descriptions of
geometry..."

Note the use of the term "intrinsic....geometry".  Further, one of the
best ways to make a discussion manifestly coordinate invariant is to
make it  "coordinate-free".

Further, in IEEE Computer Graphics and Applications, May/June 1999,
there is a good tutorial by James R. Miller, entitled "Vector Geometry
for Computer Graphics".  It broaches some of the same ideas of using
coordinate-free vector representations as an aid to understanding.

These ideas are just beginning to dawn on the vanguard of theorists of
computer graphics. They were at the essential core of the training I
received in math grad school a couple of decades ago.