[Jmol-developers] Re: Rx,Ry,Rz from T (was show Center)

[Bob H. <ha...@st...>]

[This is not really something for the Jmol-users list, but since it
started that way, I feel I should finish it there as well.]

Final analysis:

Jmol use of transformation matrix and production
of sequential "rotate x <Rx>; rotate y <Ry>; rotate z <Rz>"
from a transformation matrix.

For the record...

Define pure rotation 3x3 tansform:

  T= R_z*R_y*R_x

such that

  T * InputCoord = ViewCoord

  (rotate cw first x, then y, then z)

  where R_x =

  [

   1   0         0
   0  cos(rx) -sin(rx)
   0  sin(rx)  cos(rx)

  ]

  where R_y =

  [

   cos(ry)   0  sin(ry)
     0       1     0
  -sin(ry)   0  cos(ry)

  ]

  where R_z =

  [

   cos(rz) -sin(rz)  0
   sin(rz)  cos(rz)  0
     0        0      1

  ]

then with T:

  [

   m00 m01 m02
   m10 m11 m12
   m20 m21 m22

  ]

or in terms of sines (s) and cosines (c):

T=

[

  cy*cz  sx*sy*cz-cx*sz  cx*sy*cz+sx*sz
  cy*sz  sx*sy*sz+cx*cz  cx*sy*sz-sx*cz
  -sy    sx*cy	        cx*cy

]

Then we have:

Ry=-asin(m20)

if(m20==-1||m20==1){
   Rx=-atan2(m12,m11)
   Rz=0
}else{
   Rx=atan2(m21,m22)
   Rz=atan2(m10,m00)
}

This logic pushes all the rotation into Rx when Ry=+/-90.
(In that case, a rotation about x and a rotation about z amount to
a rotation about the same axis.)



  Spreadsheet random input validation is at

  http://www.stolaf.edu/people/hansonr/jmol/test/transform.xls

  (requires add-in matrix.xla)

-- 
Robert M. Hanson, ha...@st..., 507-646-3107
Professor of Chemistry, St. Olaf College 1520 St. Olaf Ave., Northfield, MN 55057
mailto:ha...@st... http://www.stolaf.edu/people/hansonr