| Probably the simplest reference would be Kevin Shoemake's article
in one of the Siggraph proceedings on "Animating Rotations with Quaternion
Curves". Here, he gives references to a few other papers on them.
I think it was 1982 or around there... you'll have to look it up.
Glassner's book "Graphics Gems" also discusses them briefly. Also,
in Courant and Hilbert's book on mathematical physics, vol I they
show how quaternions and orthogonal matrices are connected by the
Cayley transform. They even appeared in Hackmem.
There is always Hamilton's massive book on quaternions...
- Jim
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| If you have NCSA Mosaic you might want to check out:
http://robocop.modmath.cs.cmu.edu:8001/htbin/mjwgenformII
For a "genetic art" display. Nine computer-generated images are shown
and you can "vote" on which ones you like best. Every 10 votes the
winners are paired genetically and new images are created.
What does this have to do with quaternions? Well, that's how the images
are created, the (i,j,k) coefficients map to the (R,G,B) color scheme.
It's all explained in subsequent pages. Quoting from the explanation:
"The previous art exhibit used functions over the reals, this one uses
functions over the quaternions (4 dimensional vectors) and gets color
R/G/B values directly from three entries of the vectors. Quaternions
were chosen because:
* They form a vector space (over the reals) large enough to
conveniently hold the three vales: R,G and B.
* They form an associative division algebra over the reals (thus
formulas over the quaternions should have some of the same properties
as the formulas used in the first exhibit).
* Any properly orthogonal mapping of euclidian 3-space can be viewed
as the action of a quaternion automorphism on the purely imaginary
quaternions (the functions orth1 and orth2 in the current
implementation).
* The quaternions are somewhat unique (e.g. any normed associative
real division algebra is isomorphic to the reals, complex numbers or
the quaternions)."
I find the last point most interesting. Does this mean the
8-dimensional counterpart ("Octernions"?) fails to satisfy some
conditions, or was the term "quaternions" used to refer generically
to the class of 2^Nth roots of unity.?
If you poke around in the subsequent pages you'll find the code that
generates the art, including a rather complete quaternion arithmetic
package in C. All GPL'ed (GNU Public License, free to copy and use)
John
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| RE: -.1
� I find the last point most interesting. Does this mean the
� 8-dimensional counterpart ("Octernions"?) fails to satisfy some
� conditions, or was the term "quaternions" used to refer generically
� to the class of 2^Nth roots of unity.?
As I recall, the 8-dimensional beasts are called Caley (?) numbers and
are not associative. And there are no higher-order division algebras.
-- Jeff
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