       Re: Re: Rotation3D, MatrixRotation3D ?

• To: mathgroup at smc.vnet.net
• Subject: [mg30432] Re: [mg30410] Re: Rotation3D, MatrixRotation3D ?
• From: Andrzej Kozlowski <andrzej at tuins.ac.jp>
• Date: Sat, 18 Aug 2001 04:04:57 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

```My previous message in this thread actually contains (implicitly) two
ways to do that. The first way is very straight forward and consists
simply of the combination of Euler's theorem and the Mathematica
function Solve. For example, let's construct a rotation matrix:

In:=
Needs["Geometry`Rotations`"]

In:=
B=RotationMatrix3D[Pi/3,Pi/4,Pi/6];

In:=
Solve[ RotationMatrix3D[\[Psi], 0, 0].
RotationMatrix3D[0, \[Theta], 0].RotationMatrix3D[\[Phi], 0,
0]\[Equal]
B,{\[Phi],\[Theta],\[Psi]}]

From In:=
Solve::ifun: Inverse functions are being used by Solve, so some
solutions may \
not be found.

Out=
{{\[Psi] -> Pi/6, \[Theta] -> Pi/4, \[Phi] -> Pi/3}}

One has to be careful here to get the names of the angles right (or
equivalently their order), I had it wrong in my message since
Mathematica convention is not the same as I am used to.

This method is not 100% reliable because Mathematica is not 100%
reliable when solving trigonometric equations. There is also a second
method, basically described at the end of my previous message. If you
really need it I send you the details  or you can look it up in a good
linear algebra book, e.g. Kostrykin and Manin, "Linear Algebra and
Geometry".

Andrzej Kozlowski
Toyama International University
JAPAN
http://platon.c.u-tokyo.ac.jp/andrzej/

On Friday, August 17, 2001, at 04:09  PM, ojg wrote:

> 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
>>
>>
>>
>>
>>
>>
>>
>>> From: ojg [mailto:ole.jonny.gjoen at hitecvision.com]
To: mathgroup at smc.vnet.net
> 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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