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Re: Rotation3D, MatrixRotation3D ?
*To*: mathgroup at smc.vnet.net
*Subject*: [mg30410] Re: Rotation3D, MatrixRotation3D ?
*From*: "ojg" <ole.jonny.gjoen at hitecvision.com>
*Date*: Fri, 17 Aug 2001 03:09:50 -0400 (EDT)
*References*: <9ld1m6$2gi$1@smc.vnet.net>
*Sender*: owner-wri-mathgroup at wolfram.com
Sounds interesting:)
Are your routines able to regenerate the auler angles from the rotation? My
problem is of a similar kind and might be good to look at:)
Thanks,
Johnny
"David Park" <djmp at earthlink.net> wrote in message
news:9ld1m6$2gi$1 at smc.vnet.net...
> Johnny,
>
> If you have Mathematica Version 4, I may be able to help you.
>
> My Mathematica friend, Rip Pelletier, and I have been studying rotations
> using Jack B. Kuipers` book "Quaternions and Rotation Sequences". I have
> been preparing a set of tutorials which basically help us to understand
> rotations by explaining it as clearly as possible. They are still somewhat
> of a work in progress, but I am far enough along to be able to attack the
> problems that you have posed. I am also working on a set graphics routines
> which illustrate rotation sequences and help to visualize the various
> sequences.
>
> We would be glad to share these with you if you wish. They do require
> Mathematica Version 4 and they also use my DrawingPaper and DrawingCube
> packages from my web site. They also use two packages
> Geometry`ExtendRotations and Geometry`RotationGraphics. You would have to
> install these packages in your AddOns/ExtraPackages folders.
>
> Here are some of the usage messages for the routines that are in the
> packages.
>
> Geometry`ExtendRotations`
>
> SetRotationMatrices[alibi|alias] will set the 3D axis rotation matrices \
> according to the alibi or alias form. The current setting is saved in the
\
> variable RotationMatrixForm. The alibi form matrices will rotate a vector
by
> \
> the right hand rule. The alias form matrices will rotate the coordinate
> frame \
> by a right hand rule.
>
> Rx[\[Theta]] is a 3D rotation matrix for rotation about the x axis.
> Ry[\[Theta]] is a 3D rotation matrix for rotation about the y axis.
> Rz[\[Theta]] is a 3D rotation matrix for rotation about the z axis.
> Rot[\[Theta]] is a 2D rotation matrix for rotation about the supposed z
> axis.
>
> RotationAxisAndAngle[rotationmatrix] will generate the axis of rotation
and
> \
> the associated rotation angle in radians for a 3 x 3 rotation matrix. An \
> equally valid answer is obtained by reversing the signs of both the angle
> and \
> the axis vector.
>
> EulerAngles[matrix, seqstring, form:alibi] will return the Euler angles \
> corresponding to a sequence of axes rotations specified by seqstring. An
> axes \
> sequence of "XYZ" means rotation around the X axes by \[Psi], followed by
\
> rotation about the Y axis by \[Theta], followed by rotation about the Z
axis
> \
> by \[Phi]. The rotations angles {\[Psi],\[Theta],\[Phi]} are returned. For
\
> sequences of the form ABA, 0 \[LessEqual] \[Theta] \[LessEqual] \[Pi]. For
\
> sequences of the form ABC, -\[Pi]/2 \[LessEqual] \[Theta] \[LessEqual] \
> \[Pi]/2. The answers are in terms of the standard alibi matrices, unless
\
> alias is specified in the optional argument.
>
> Geometry`RotationGraphics`
>
> AlibiAliasAnimation2D[vector, {\[Theta]min, \[Theta]max}, numframes:24] \
> produces a side by side animation of the alibi and alias interpretation of
a
> \
> 2D rotation.
>
> DrawVector3D[vectortip, name, l, options] will draw a vector from the
origin
> \
> going to vectortip. The vector will be labeled with name at l vectortip. \
> options can be used to pass options to the Arrow3D routine.
> DrawVector3D[vectortip, name, options] is the same as
> DrawVector3D[vectortip, \
> name, 1.1, options]
>
> DrawFrame3D[rotmat, {name1, name2, name3}, l, options] will draw frame
axes
> \
> rotated according to the rotation matrix, rotmat, rotating the axes
vectors
> \
> in the alibi mode. The axes will be labeled with the names at a fractional
\
> distance l along the axes vector. options can be used to pass options to
the
> \
> Arrow3D routine.
> DrawFrame3D[rotmat, {name1, name2, name3}, options] is the same as \
> DrawFrame3D[rotmat, {name1, name2, name3}, 1.1, options]
>
> DrawBook[rotmat, scale] will draw a book centered at the origin and
rotated
> \
> using the rotation matrix rotmat. scale can be used to change the overall
\
> size of the book. scale = 1 gives a book of dimensions 0.9 x 0.6 x 0.2.
The
> \
> book looks best with the following lighting option:
> Lighting \[Rule] True,
> AmbientLight \[Rule] GrayLevel[0.3],
> LightSources\[Rule]{{{1, 0, 1}, GrayLevel[0.4]},{{1, 1, 1}, \
> GrayLevel[0.4]},{{0, 1, 1},GrayLevel[0.4]}}
> Another possibility is to use the BookLighting option command.
>
> ThreeStepBookAnimation[{rot1, \[Theta]1max}, {rot2, \[Theta]2max}, {rot3,
\
> \[Theta]3max}] will produce an animation of the book, rotating
successively
> \
> according to the three rotation matrices by the specified angles.
>
> TwoBookAnimation[seq1, seq2] animates two side by side book rotation \
> sequences where each sequence is given by a sequence of the form {{rot1, \
> \[Theta]1max}, {rot2, \[Theta]2max}, {rot3, \[Theta]3max}}.
>
> AliasAnimation3D[{rot1, \[Theta]1max}, {rot2, \[Theta]2max}, {rot3, \
> \[Theta]3max}] will animate the a three step rotation of the frame using \
> alias form matrices. The routine sets the alias form for matrices and then
\
> restores the original form.
>
> So far, I have two completed tutorials.
> 1) 2D rotations, alibi and alias interptetations. Matrix form. Mathematica
> expressions for transforming row vectors, column vectors etc.
> 2) 3D rotations about axes. Illustration of the matrix forms and the
> mnemonic XYZXY for determining the directions of rotations and where to
put
> the +Sin and -Sin terms in the matrix.
> 3) 3D rotation sequences. Calculating Euler angles or axis of rotation.
> Still in progress, although the underlying routines are finished.
>
> We would be glad to share this work with anyone studying 3D rotations,
> especially if they are interested in contributing ideas to improve them.
>
> Rip has been jumping ahead, learning how to use quaternions, but so far I
> have been staying with rotation matrices.
>
> David Park
> djmp at earthlink.net
> http://home.earthlink.net/~djmp/
>
>
>
>
>
>
>
> > From: ojg [mailto:ole.jonny.gjoen at hitecvision.com]
To: mathgroup at smc.vnet.net
> >
> > Question regarding rotations.
> >
> > Some of the documentation found regarding this is not as far as I can
see
> > complete in the documentation, at least the subject is difficult enough
to
> > make me unsure once not 100% clear:)
> >
> > Fist, what are the defined "euler angles" in mathematica, and in
> > what order
> > are they applied?
> >
> > Second, of which side of the vector is the rotational matrix multiplied
?
> >
> > Third, is there a mathematica way to rotate around an abitrary
rotational
> > axis? If not, what would the mathematica matix be for this?
> >
> > My problem to solve is as follows: Given three rotational angles (a,b,c)
> > applied in order to the following three rotational axes: Y axis, X axis,
Z
> > axis. (usual right hand system). This rotation applied to any
> > vector v will
> > give you a vector V (first Y rotation applied on v, etc).
> >
> >
> > Now, given a rotational matrix with pure numerical values in, I
> > need to find
> > the three angles, and I need a general formulae for this solution taking
> > care of the special cases.
> >
> > Thanks,
> > Johnny
> >
> >
> >
>
>
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