🚀 go-pugleaf

In article <1992Apr13.161433.25003@ircam.fr>, francis@ircam.fr (Joseph
Francis) writes:
|> I believe quaternions are rational extensions into threespace of
|> imaginary numbers, you get from "x + iy" to "x + iy + jz" (or I guess
|> for EE's, it would be "x + jy" and "x + jy + kz") where i,j have
|> traditional square-root-of-negative-one meanings. I can't remember
|> exactly, it was only interesting to me for playing with 3d extensions
|> of the ordinary Mandelbrot set graphics generator functions.

As wrote berrier@sweetpea.jsc.nasa.gov (K. Berrier 283-4125) in a previous
article, a quaternion can be seen as a transformation matrix consisting of
a scalar and a vector, where the scalar is the angle of rotation about
the axis represented by the vector.

Actually, the quaternions are the first non commutative field which is an
extension of R; a quaternion can be written: a + ib + jc + kd , where
i^2=j^2=k^2=-1, ij=k, jk=i, ki=j, ji=-k, kj=-i, ik=-j
In other words, the field of quaternions can be seen as the affine space R^3
where (i,j,k) is an orthonormal base for the underlying vector structure,
and the product of any two combinations of i,j,k can be seen as the vector
product of the corresponding vectors in R^3.
One of the interesting things about quaternions is that they can be seen
as rotations, and the product of two quaternions yields the product in the
affine space R^3 of the two corresponding rotations.

Quaternions are only an example of hypercomplex fields; you can define the
octavions, which form the first non associative field which is an extension 
of R (you have to introduce 7 objects, like the previous i,j,k, with 
appropriate relations).

	Dominique Luzeaux