Chronology Current Month Current Thread Current Date
[Year List] [Month List (current year)] [Date Index] [Thread Index] [Thread Prev] [Thread Next] [Date Prev] [Date Next]

[Phys-L] N-Dimensional Rotations, Including Boosts



Hi --

I have just put up a greatly-revised version of my paper on how to represent
N-dimensional rotations, including boosts. Take your choice:
http://www.av8n.com/physics/rotations.htm
http://www.av8n.com/physics/rotations.pdf

Improvements since the previous version include:

*) I added some additional introductory examples of rotors and rotations.

*) I clarified the relationship between the product-of-vectors representation
and the matrix representation:
-- If you are trying to keep track of rotations _per se_, such as for
a rapidly maneuvering aircraft, you really ought to represent rotations
in terms of Clifford algebra (or, equivalently, quaternions).
-- In contrast, if you have a single, fixed rotation operator, and wish
to use it to rotate a huge number of vectors, the matrix representation
is more efficient.

*) The perl program is more flexible in its input, and more informative in
its output. It will print out the matrix representation as well as the
VRML representation, if you ask it.

*) I restructed the review, discussing the pros and cons of four different
ways to represent rotations: Clifford algebra, matrices, Rodrigues vectors,
and Euler angles.

======================

I now have quite a few documents that use Clifford algebra. I got tired having
multiple half-baked introductions to the topic scattered all over the place,
so I collected my thoughts on Clifford algebra all in one place, and greatly
expanded it and cleaned it up. I'm not saying it's great, but it's a huuuge
improvement over what I had before. Take your choice:
http://www.av8n.com/physics/clifford-intro.htm
http://www.av8n.com/physics/clifford-intro.pdf

Notable features of this new document include:

*) I emphasize treating vectors as real geometric objects, objects in their
own right, independent of any basis. (This will come as a shock to those
who think vectors are always /defined/ as a big list of components, but
usually it is easier *and* more powerful than the basis+components approach.

*) I intended this document to be understandable even to students who have
little more than a modest familiarity with vector algebra. From there,
it is only a few small steps to generalize it to Clifford algebra. In
particular, students do not need to have ever heard of matrices.

*) I introduce the concept of the /gorm/ of an object. The gorm of a
vector V is V·V. Other authors have incautiously called this the
norm squared, which is problematic in the case of timelike vectors,
where the gorm is negative and not easily understood as the square
of anything. This is the answer to a question that has been bugging
me for years.


I've rewritten these papers eleventeen times over the last few weeks,
trying to get them to the point were I wasn't embarrassed by them. If
anybody has suggestions for further improvements, I'd be glad to hear
them.


=====================================================================

"I don't think Shakespeare is such a great writer. I mean look
at all this; it's just loaded with clichés."
_______________________________________________
Phys-L mailing list
Phys-L@electron.physics.buffalo.edu
https://www.physics.buffalo.edu/mailman/listinfo/phys-l