Several years ago I wrote a tool to do quaternions using J expressions. I
got tired of the awkwardness of entering quaternions as lists of numbers and
wanted to express things more like J. I just tested and it still works, but
it requires permitting x. and y. names in explicit definitions. I wrote this
before the dropping of the dots in the names and before the quaternion
script was available.
You can enter expressions using J primitives. Numbers are represented as an
extension of J's complex numbers - 1i2j3k4. Looking at the output can get
hard to read as the letters can get lost. A couple of examples.
+:4 2j7+2 3k8^/1i2j3k4 2
4.67873i_0.82789j_1.24184k_1.65578 16
4.10623i0.048993j14.0248k0.0605291 _106j14k96
*/~1 0i1 0j1 0k1 NB. Quaternion multiplication table.
1 0i1 0j1 0k1
0i1 _1 0k1 0j_1
0j1 0k_1 _1 0i1
0k1 0j1 0i_1 _1
I took a shot at defining the circular functions in quaternions. I'm not
sure that they are correct. And I am stumped on matrix inverse.
If anyone wants the scripts I would be glad to send them. Any suggestions or
corrections would be appreciated.
On Sat, Apr 26, 2008 at 12:12 PM, J. Patrick Harrington <[EMAIL PROTECTED]>
wrote:
> On Sat, 19 Apr 2008, Chris Burke wrote:
>
> See also:
> >
> > open '~system/packages/math/quatern.ijs'
> >
> > I just got around to looking at the quaternion routines
> in j602/system/packages/math/quatern.ijs. By representing
> the quaternions as 4x4 matrices, they are quite compact. I wondered if I
> should use them. The verb for multiplying
> quaternions given there is just
>
> mp=. +/ . *
> j=. ,: =i.4
> j=. j, 0 1 0 0, _1 0 0 0, 0 0 0 _1,: 0 0 1 0
> j=. j, 0 0 1 0, 0 0 0 1, _1 0 0 0,: 0 _1 0 0
> j=. j, 0 0 0 1, 0 0 _1 0, 0 1 0 0 ,: _1 0 0 0
> qnmat=. mp & j
> qnmul=: (mp qnmat) f. "1
>
> An array of quaternions is stored as a (n 4) dimension
> element. I thought I should compare this with what I
> had used for this mode of storage, which was
>
> dot=: [: +/"1 * NB. dot products of vector arrays pm1=: 1&|."1 pm2=:
> 2&|."1 pm12=: 4 : '(pm1 x)*pm2 y'
> cross=: pm12 - pm12~ NB. cross products of vectors
> NB. Multiply quaternions: q1 QM q2 --> q3
> QM=: 4 : 0
> s1=. 0{"1 x
> v1=. }."1 x
> s2=. 0{"1 y
> v2=. }."1 y
> s=. (s1*s2) - v1 dot v2
> v=. (s1*v2)+(s2*v1)+(v1 cross v2)
> s ,&.|: v NB. catenate under transpose
> )
>
> Since I'm interested in the speed of the operations for large
> arrays, let's compare the verbs:
>
> rand=: [: ? ] # 0"_
> q1=. 200000 4 $ rand 800000
> q2=. 200000 4 $ rand 800000
> ts'q=. q1 qnmul q2'
> 0.313453 8.39142e6
> ts'qq=. a QM b'
> 0.195947 6.50142e7
>
> The output is the same:
> 12345{q
> _0.949615 _0.673075 0.278588 0.426178
> 12345{qq
> _0.949615 _0.673075 0.278588 0.426178
>
> but qnmul is about 50% slower than QM, though it uses much
> less space.
>
> But what I normally do is store the quaternions as (4 n)
> arrays, as I've found that faster. Thus I use
>
> dot2=: [: +/ * cross2=: ((1: |.[)*(_1: |. ]))-((_1: |.[)*(1:|.]))
> QM2=: 4 : 0
> s1=. 0{ x
> v1=. }. x
> s2=. 0{ y
> v2=. }. y
> s=. (s1*s2) - v1 dot2 v2
> s,(s1*"1 v2)+(s2*"1 v1)+(v1 cross2 v2)
> )
>
> Comparing,
> q1r=. |: q1
> q2r=. |: q2
> ` ts'qq2=. q1r QM2 q2r'
> 0.139955 7.34037e7
> 12345{"1 qq2
> _0.949615 _0.673075 0.278588 0.426178
>
> Which is more than twice as fast as qnmul (at the expense
> of 10 times the space). Could qnmul be tweaked to get better speed on
> large arrays?
>
> Patrick
>
>
> ----------------------------------------------------------------------
> For information about J forums see http://www.jsoftware.com/forums.htm
>
>
----------------------------------------------------------------------
For information about J forums see http://www.jsoftware.com/forums.htm
