Tetszoleges tengely koruli forgatas (elmelet es gyakorlat)
                 Irta Kaproncai Tamas [tomcat@szif.hu]


Tetszoleges tengely koruli forgatas (elmelet)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

  - Hogyan lehet altalanos helyzetu sikban forgatni egy pontot, tehat van
egy origon atmeno normalvektorral adott sikunk, aztan egy pontunk meg egy
szogunk. Hogyan tovabb???

A problema megoldasa a klasszikus Rodrigues-keplet:
(U vektor korul fi szoggel elforgatas tr. matrixa)

Rot(U,fi)= Cos(fi)*I + (1-Cos(fi))[UoU] + Sin(fi)[UX]

I az egysegmatrix     [ 1 0 0 ]
                      [ 0 1 0 ]
                      [ 0 0 1 ]

[UoU] = U-nak onmagaval vett diadikus szorzata

          [axbx axby axbz]
( [AoB] = [aybx ayby aybz] , A=B=U )
          [azbx azby azbz] 

[UX] = U-val valo vektorszorzas transzformacios matrixa

         [  0  -uz  uy ]
( [UX] = [  uz  0  -ux ] )
         [ -uy  ux  0  ]


Ezutan pl. egy P(px,py,pz) pont fi-vel valo elforgatottja U korul:

P'(px',py',pz') = Rot(U,fi) * P(px,py,pz)


Tetszoleges tengely koruli forgatas (gyakorlat)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Szamoljuk ki a transzformacios matrixot:

a = Cos(fi)
b = 1-Cos(fi)
c = Sin(fi)

                [ 1 0 0 ]       [  ux2 uxuy uxuz ]       [  0  -uz  uy ]
Rot(U,fi) = a * [ 0 1 0 ] + b * [ uyux  uy2 uyuz ] + c * [  uz  0  -ux ]
                [ 0 0 1 ]       [ uzux uzuy  uz2 ]       [ -uy  ux  0  ]

Az eredmeny:

[   a+b*ux2   b*uxuy-c*uz b*uxuz+c*uy ]
[ b*uyux+c*uz   a+b*uy2   b*uyuz-c*ux ]
[ b*uzux-c*uy b*uzuy+c*ux   a+b*uz2   ]

Ez azonos a FvDFH konyv 227. oldalan talalhato (5.97) keplettel:

[   ux2+Cosfi(1-ux2)    uxuy(1-Cosfi)-uzSinfi uzux(1-Cosfi)+uySinfi ]
[ uxuy(1-Cosfi)+uzSinfi   uy2+Cosfi(1-uy2)    uyuz(1-Cosfi)-yxSinfi ]
[ uzux(1-Cosfi)-uySinfi uyuz(1-Cosfi)+yxSinfi   uz2+Cosfi(1-uz2)    ]

Itt a megirt rutin is NASM szintaktikaval:

%macro  ARotM33 2
        FSINCOS                  ;;Cos(fi) Sin(fi)
        FLD1                     ;;1 Cos(fi) Sin(fi)
        FSUB    ST0,ST1          ;;1-Cos(fi) Cos(fi) Sin(fi)
        FLD     QWORD [%2+z]     ;;uz b a c
        FMUL    ST0,ST3          ;;c*uz b a c
        FLD     QWORD [%2+y]     ;;uy c*uz b a c
        FMUL    QWORD [%2+x]     ;;uxuy c*uz b a c
        FMUL    ST0,ST2          ;;b*uxuy c*uz b a c
        FSUBRP  ST1,ST0          ;;b*uxuy-c*uz b a c
        FLD     QWORD [%2+y]     ;;uy b*uxuy-c*uz b a c
        FMUL    ST0,ST4          ;;c*uy b*uxuy-c*uz b a c
        FLD     QWORD [%2+z]     ;;uz c*uy b*uxuy-c*uz b a c
        FMUL    QWORD [%2+x]     ;;uxuz c*uy b*uxuy-c*uz b a c
        FMUL    ST0,ST3          ;;b*uxuz c*uy b*uxuy-c*uz b a c
        FADDP   ST1,ST0          ;;b*uxuz+c*uy b*uxuy-c*uz b a c
        FLD     QWORD [%2+x]     ;;ux b*uxuz+c*uy b*uxuy-c*uz b a c
        FMUL    ST0,ST0          ;;ux2 b*uxuz+c*uy b*uxuy-c*uz b a c
        FMUL    ST0,ST3          ;;b*ux2 b*uxuz+c*uy b*uxuy-c*uz b a c
        FADD    ST0,ST4          ;;a+b*ux2 b*uxuz+c*uy b*uxuy-c*uz b a c
        FXCH    ST2              ;;b*uxuy-c*uz b*uxuz+c*uy a+b*ux2 b a c
        FSTP    QWORD [%1+M33.12];;b*uxuz+c*uy a+b*ux2 b a c
        FSTP    QWORD [%1+M33.13];;a+b*ux2 b a c
        FSTP    QWORD [%1+M33.11];;b a c
        FLD     QWORD [%2+z]     ;;uz b a c
        FMUL    ST0,ST3          ;;c*uz b a c
        FLD     QWORD [%2+x]     ;;ux c*uz b a c
        FMUL    QWORD [%2+y]     ;;uyux c*uz b a c
        FMUL    ST0,ST2          ;;b*uyux c*uz b a c
        FADDP   ST1,ST0          ;;b*uyux+c*uz b a c
        FLD     QWORD [%2+x]     ;;ux b*uyux+c*uz b a c
        FMUL    ST0,ST4          ;;c*ux b*uyux+c*uz b a c
        FLD     QWORD [%2+z]     ;;uz c*ux b*uyux+c*uz b a c
        FMUL    QWORD [%2+y]     ;;uyuz c*ux b*uyux+c*uz b a c
        FMUL    ST0,ST3          ;;b*uyuz c*ux b*uyux+c*uz b a c
        FSUBRP  ST1,ST0          ;;b*uyuz-c*ux b*uyux+c*uz b a c
        FLD     QWORD [%2+y]     ;;uy b*uyuz-c*ux b*uyux+c*uz b a c
        FMUL    ST0,ST0          ;;uy2 b*uyuz-c*ux b*uyux+c*uz b a c
        FMUL    ST0,ST3          ;;b*uy2 b*uyuz-c*ux b*uyux+c*uz b a c
        FADD    ST0,ST4          ;;a+b*uy2 b*uyuz-c*ux b*uyux+c*uz b a c
        FXCH    ST2              ;;b*uyux+c*uz b*uyuz-c*ux a+b*uy2 b a c
        FSTP    QWORD [%1+M33.21];;b*uyuz-c*ux a+b*uy2 b a c
        FSTP    QWORD [%1+M33.23];;a+b*uy2 b a c
        FSTP    QWORD [%1+M33.22];;b a c
        FLD     QWORD [%2+y]     ;;uy b a c
        FMUL    ST0,ST3          ;;c*uy b a c
        FLD     QWORD [%2+x]     ;;ux c*uy b a c
        FMUL    QWORD [%2+z]     ;;uzux c*uy b a c
        FMUL    ST0,ST2          ;;b*uzux c*uy b a c
        FSUBRP  ST1,ST0          ;;b*uzux-c*uy b a c
        FLD     QWORD [%2+x]     ;;ux b*uzux-c*uy b a c
        FMUL    ST0,ST4          ;;c*ux b*uzux-c*uy b a c
        FLD     QWORD [%2+y]     ;;uy c*ux b*uzux-c*uy b a c
        FMUL    QWORD [%2+z]     ;;uzuy c*ux b*uzux-c*uy b a c
        FMUL    ST0,ST3          ;;b*uzuy c*ux b*uzux-c*uy b a c
        FADDP   ST1,ST0          ;;b*uzuy+c*ux b*uzux-c*uy b a c
        FLD     QWORD [%2+z]     ;;uz b*uzuy+c*ux b*uzux-c*uy b a c
        FMUL    ST0,ST0          ;;uz2 b*uzuy+c*ux b*uzux-c*uy b a c
        FMUL    ST0,ST3          ;;b*uz2 b*uzuy+c*ux b*uzux-c*uy b a c
        FADD    ST0,ST4          ;;a+b*uz2 b*uzuy+c*ux b*uzux-c*uy b a c
        FXCH    ST2              ;;b*uzux-c*uy b*uzuy+c*ux a+b*uz2 b a c
        FSTP    QWORD [%1+M33.31];;b*uzuy+c*ux a+b*uz2 b a c
        FSTP    QWORD [%1+M33.32];;a+b*uz2 b a c
        FSTP    QWORD [%1+M33.33];;b a c
        FCOMP   ST0
        FCOMPP
%endmacro

%macro  ARotM33 3
        FLD     QWORD [%3]       ;;fi
        ARotM33 %1,%2
%endmacro