RodriguesRotationFormular

hi all

I rewrite yogen.mac ,named  quaternion.mac.
maybe it will increase usability,i think.
please put it share/contrib/quaternion.mac.
RodriguesRotationFormular is OK.
(%i1) load("quaternion.mac");
(%o1)                           quaternion.mac
(%i2) declare([_e,_f,_g,t],scalar)$
(%i3) declare([_a,_b,_c],scalar)$
(%i4)  x:_e*i+_f*j+_g*k$
(%i5)  y:_a*i+_b*j+_c*k$
(%i6) expand4((cos(t)+x*sin(t)).conj4(cos(t)+x*sin(t)))$
(%i7) load("coeflist.lisp")$
(%i8) load("format.lisp")$
(%i9) format(%o6,%poly('sin(t)^2,'cos(t)^2),factor);
                         2       2     2     2       2
(%o9)                 sin (t) (_g  + _f  + _e ) + cos (t)
(%i10) expand4((cos(t/2)+x*sin(t/2)).y.(cos(t/2)-x*sin(t/2)))$
(%i11) format(%,%poly(i,j,k),factor)$
(%i12)  coefmatrix([coeff(%,i),coeff(%,j),coeff(%,k)],[_a,_b,_c])$
(%i13) trigsimp(subst([_g^2=1-_f^2-_e^2],%))$
(%i14) trigreduce(%)$

(%i15) format(%,%poly(_g,_f,_e),factor);
                [                            2     ]
                [    cos(t) - (cos(t) - 1) _e      ]
(%o15)  Col 1 = [                                  ]
                [  sin(t) _g - (cos(t) - 1) _e _f  ]
                [                                  ]
                [ - (cos(t) - 1) _e _g - sin(t) _f ]
         [ - sin(t) _g - (cos(t) - 1) _e _f ]
         [                                  ]
 Col 2 = [                            2     ]
         [    cos(t) - (cos(t) - 1) _f      ]
         [                                  ]
         [  sin(t) _e - (cos(t) - 1) _f _g  ]
         [     sin(t) _f - (cos(t) - 1) _e _g      ]
         [                                         ]
 Col 3 = [    - (cos(t) - 1) _f _g - sin(t) _e     ]
         [                                         ]
         [                2                  2     ]
         [ (cos(t) - 1) _f  + (cos(t) - 1) _e  + 1 ]


basematrix1 and basematrix2 are defined in this file.
basematrix1:[matrix([0,-%i],[-%i,0]),matrix([0,-1],[1,0]),matrix([-%i,0],[0,%i])];
this means ,basematrix1 is representing [i,j,k] with matrix.

(%i16) matcoeff4(matrix([1,-2*%i-3],[3-2*%i,1]),basematrix1);
                    [ 1  0 ]  [ 2  0 ]  [ 3  0 ]  [ 0  0 ]
(%o16)             [[      ], [      ], [      ], [      ]]
                    [ 0  1 ]  [ 0  2 ]  [ 0  3 ]  [ 0  0 ]
this means ,matrix([1,-2*%i-3],[3-2*%i,1])is representing [1+2*i+3*j] with
basematrix

(%i17) matcoeff4(matrix([1,0],[0,1]),basematrix1);
                    [ 1  0 ]  [ 0  0 ]  [ 0  0 ]  [ 0  0 ]
(%o17)             [[      ], [      ], [      ], [      ]]
                    [ 0  1 ]  [ 0  0 ]  [ 0  0 ]  [ 0  0 ]
so matrix([1,0],[0,1])is 1
(%i18) matcoeff4(matrix([0,%i],[%i,0]),basematrix1);
                  [ 0  0 ]  [ - 1   0  ]  [ 0  0 ]  [ 0  0 ]
(%o18)           [[      ], [          ], [      ], [      ]]
                  [ 0  0 ]  [  0   - 1 ]  [ 0  0 ]  [ 0  0 ]
so matrix([0,%i],[%i,0]) is -i
but next example matrix([1,-2],[3,1]) can not be represent by quaternion
(%i19) matcoeff4(matrix([1,-2],[3,1]),basematrix1);
                             [ %i     ]  [ 5    ]
                             [ --  0  ]  [ -  0 ]
                   [ 1  0 ]  [ 2      ]  [ 2    ]  [ 0  0 ]
(%o19)            [[      ], [        ], [      ], [      ]]
                   [ 0  1 ]  [     %i ]  [    5 ]  [ 0  0 ]
                             [ 0   -- ]  [ 0  - ]
                             [     2  ]  [    2 ]
if to represent this is possible,it 1+(%i/2)*i+5/2*j,do you think?
last you may set matrix_element_mult to "*"
(%i20) matrix_element_mult:"*";
(%o20)                                 *
thanks
Gosei Furuya
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